# embeddings_queue

created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd342-934e-11ee-bdb6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "P(T2)which are based on csumandTas follows:\nP(T1) =T\u2212clamp (csum)\n2T, P(T2) =T+clamp (csum)\n2T,\nwhere the clamp function restricts its argument to [\u2212T, T]such\nthat the probabilities lie in the range [0,1]. As the number\nof activated clauses (those producing c= 1) approaches T,\nthe probability of the TAsin that clause receiving feedback\ndecreases. In order to generate a randomized choice based on\nthese probabilities, we could assign booleans as follows:\nq1=rand< P(T1),q2=rand< P(T2)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd343-934e-11ee-87da-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "q1=rand< P(T1),q2=rand< P(T2)\nwhere rand is a real number in the range [0,1]. Since P(T1)\nandP(T2)are complementary in the range [0,1], q1and q2\nare also complementary in the binary domain and therefore\nq2=q1. We therefore introduce a single new signal for FB2,\npT, which takes the place of the preceding probabilities. The\ngeneration of this signal will be discussed in Section V.\nC.FB3:TA-level Feedback\nInputs: FB2,inc,c,x,ps\nOutput: TAaction (inaction, penalty or reward)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd344-934e-11ee-8e5d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "Output: TAaction (inaction, penalty or reward)\nThe third and \ufb01nal feedback stage translates the feedback\ntype from FB2into a TAaction for the speci\ufb01ed TAbased on:\nthe TA\u2019s current action, encoded as include in signal inc; the\ncurrent clause output, c; the feature of complemented feature\n(forf) associated with the TA,x; and a probabilistic choice\ncomponent, ps.\nThe FB3 output is chosen according to Table I, where\nps= 1 denotes that the higher probability (s\u22121)/soption is"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd345-934e-11ee-846a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "chosen from the payoff matrix. ps= 0 denotes that the lower\nprobability 1/sis chosen. The generation of signal pswill be\ndiscussed in Section V.\nD. Synthesis of Feedback Circuits\nUsing one-hot encoding in the feedback circuits main-\ntains direct compatibility with the inference datapath and\nensures speed independence, which is important for hazard-\nfreeness. Signals inc,c,xare already one-hot encoded (dual-\nrail) for the inference circuits in Section III, so these en-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd346-934e-11ee-bb79-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "codings are reused here in the learning circuits. Further-\nmore, we encode FB1as follows, and FB2similarly: FB1=\n{FB1T2,FB1T1,FB1none}.FB3describes TAactions and is encoded\nas: FB3={FB3r,FB3p,FB3i}, for reward, penalty, inactionrespectively. Note that rail orders do not matter as we refer to\nthe rails by name. In this way, to indicate T2feedback on FB1,\nwe would set FB1T2= 1,FB1T1= 0,FB1none= 0.\nUsing the previously de\ufb01ned logic, we can generate circuits"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd347-934e-11ee-8b35-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "for the feedback. The implementation of stage two feedback\ndepends on the clause polarity\u2014it being either negated (c neg=\n1) or non-negated (c neg= 0). Note that the stage one circuit is\ninstantiated once per TM, the stage two circuit once per clause ,\nand the stage three circuit once per TA.\nIn stage three, the FB2signal is shared throughout the TAs\nwithin the same clause. For this reason, we carefully design the\nlogic so that FB2signals are injected as close to the outputs as"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd348-934e-11ee-a068-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "possible, making the propagation path the shortest. Therefore\nwhen FB2= 0, the computation on the corresponding FB3rails\nwill conclude rapidly, and for all TAsin the clause.\nV. G ENERATION OF RANDOM BITSpT,ps\nWe introduced two distinct probabilistic choice mechanisms\nin Section IV. In Section IV-B we introduced the signal pT:\na probabilistic boolean chosen at the clause level which may\nforce FB2=none. This boolean is dependent on Tand csum."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd349-934e-11ee-a62c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "The probability of pT= 1varies at runtime according to csum.\nIn Section IV-C we introduced the signal ps, which is a random\nboolean generated for each TA. It is required under certain\ncircumstances to choose between two TAactions. This boolean\nis dependent on sand is \ufb01xed during runtime.\nFrom these requirements, we must be able to generate\nbiased random bits: bits where P(1)\u0338= 0.5(unbiased random\nbits have P(1) = 0 .5, such as those generated by a linear"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd34a-934e-11ee-84b6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "feedback shift register ( LFSR )). In addition, we must be able\nto vary P(1)at runtime for pT. Probabilistic choices are used\nin the TMto ensure diversity of learning. A pseudorandom\ngenerator is suf\ufb01cient to satisfy these needs, and bene\ufb01ts from\nhigher energy ef\ufb01ciency than a true random generator.\nDue to the requirement of PRBGs for each TA, and addition-\nally for each clause, their area and energy consumption are of\nutmost importance. We chose to implement a PRBG based on"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd34b-934e-11ee-b1a3-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "the principles of irregular sampling of a regular waveform, as\nthis allows us to minimize the overheads and take advantage\nof the asynchronous nature of the rest of the system.\nA. Asynchronous Sampling of Clock with Variable Duty Cycle\nWe take advantage of the asynchronous inference logic by\nusing asynchronous handshakes to sample a regular clock\nwaveform. A clock with 50 % duty cycle will generate un-\nbiased random bits. A clock can be generated using a ring"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd34c-934e-11ee-bbe9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 4, "chroma:document": "oscillator ( RO). To ensure random sampling, the clock and\nhandshake must be uncorrelated .ROscan be gated by adding a\nNAND orNOR gate into the ring. However this technique will\ndisadvantage us in this case as the ROwill always start up in\nthe same phase, therefore the clock and sampling signals may\nbecome correlated.\nWe take advantage of the properties of a purely inverter-\nbased RO. The ROcan be power gated using header/footer\ntransistor when entropy generation is not required. Such an RO"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd34d-934e-11ee-9e72-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "will start up in a non-deterministic phase according to thermal\nnoise and other effects in the inverters.\nTo generate biased bits for pTand ps, we need to vary the\nduty cycle of the clock. Agustin and Lopez-Vallejo [14] show\nhow to construct an ROsuch that each tap has a unique duty\ncycle. This is achieved by using inverters with asymmetric rise/\nfall times. For example, odd inverters have fast rise and slow\nfall times, and even inverters vice versa. This can be done via"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd34e-934e-11ee-9b58-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "transistor sizing (ie. in the silicon library), or altering supply\nvoltage to the inverters (ie. at implementation stage). Using\nthis ROsetup, we can multiplex between taps to obtain different\nclock duty cycles, and therefore alter our PRBG probability at\nruntime.\nWe have introduced a clock into our asynchronous circuit,\nand this may seem counterintuitive, however the load on this\nclock (and therefore its energy consumption) will be low\ncompared with fully synchronous designs where the clock"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd34f-934e-11ee-8984-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "drives large numbers of \ufb02ip-\ufb02ops. In Section V-B we will\ninvestigate the optimal number of PRBGs required. By taking\nthis distributed PRBG approach, we can vastly reduce the circuit\narea compared with a na\u00efve approach using one LFSR per TA.\nFigure 8 shows the circuit used to select the correct duty\ncycle clock and sample it with the asynchronous reqinput. The\nmutex ensures ackpand acknoutputs are mutually exclusive.\nThese outputs represent the random output bit with a dual-rail"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd350-934e-11ee-89c6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "encoding. That is for {ackp,ackn}:{0,1}represents logic-0,\nand{1,0}logic-1.{0,0}is the spacer ornull state used to\nseparate valid output values temporally.\nWhen the clkand reqsignals are both low, the output will\nbe in the spacer state. If a request is made (by asserting req)\nduring the low period of the clock, the output of the set-\ndominant latch is reset, and reqwins the mutex, resulting in\nacknasserting. If clkrises now, the output of the latch rises,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd351-934e-11ee-956e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "however the mutex is still held by req. A following deassertion\nofreqwill result in clkgaining the mutex, however ackpwill\nbe masked by the AND gate.\nIf a request is made during the high period of clk, the output\nof the latch will already be high and clkwill have won the\nmutex. On the rise of reqthe AND gate will unmask ackp. If\nclkfalls now, the output of the latch remains high since the R\ninput is inactive, and therefore ackpalso remains high."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd352-934e-11ee-a5cb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "The circuit is synthesized for the UMC 65LL cell library\nand the results are summarized in Table II. The asynchronous\nhandshake shows almost 10\u00d7saving in area compared to an 8-\nbitLFSR . Power and energy are also drastically decreased. The\nasynchronous handshake is well suited to energy-conscious,\npervasive applications.\nB. Optimal Number of PRBGs\nThe original TMalgorithm takes a probabilistic choice for\nevery TAupdate involving s. However Abeyrathna et al. [15]"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd353-934e-11ee-ad86-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "show that randomizing every dthTAupdate can save energy\nwith a minimal drop in test accuracy. With d= 1000 , an\naccuracy within 2 %ofd= 1 was maintained for three of the\nfour datasets tested.\nMutexS\u2022\nRQclk\ndc\u2192\nfrom\nRO\nreqackpacknFigure 8. Handshake circuit for the asynchronous pseudorandom bit generator\nwith dual-rail output.\nTable II\nPRBG SSYNTHESIZED FOR UMC 65 nm LOW-LEAKAGE SILICON LIBRARY .\nImplementation Area Cycle Time [ ns] Power [\u00b5 W] Energy [ fJ]\nLFSR8 92.2 0 .38 6 .41 2 .44"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd354-934e-11ee-95c0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "LFSR8 92.2 0 .38 6 .41 2 .44\nHandshake 9.72 0 .23 0 .621 0 .143\nFollowing on, we could hypothesize an optimal number\nofPRBGs to ful\ufb01ll the needs of a given TM. If every 100th\nTAupdate is randomized, we will need to produce at most\nNTAs/100probabilistic choices per learning phase.\nThis could result in a 100\u00d7reduction in PRBGs for an\nLFSR -based design, in exchange for arbitration overhead to\nshare one PRBG between multiple TAs. Similar could be said"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd355-934e-11ee-a64e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "for the asynchronous handshake design, although the already\nsmall size of the design means that arbitration overhead would\nalmost certainly result in a larger area than instantiating an\nasynchronous handshake PRBG for every TA.\nVI. A SYNCHRONOUS TSETLIN AUTOMATON\nIn a TM, the TAs consume most of the hardware re-\nsources [6]. The number of TAs is given by NTAs =\n2NClassesNClausesNInputs. Therefore the goal is to create a\nlightweight TAimplementation with a focus on area and"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd356-934e-11ee-a799-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "energy. The TAhardware must implement the \ufb01nite state\nautomaton described by the state diagram in Figure 2.\nThis section explores two methods for the design of a TA.\nFirstly a QDIapproach starting from an STGspeci\ufb01cation; and\nsecondly a BDapproach starting from a traditional synchronous\nspeci\ufb01cation, using matched delays to time the design. The\ndesigns will be compared with a synchronous implementation.\nA. Quasi Delay Insensitive Approach"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd357-934e-11ee-84c2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "A. Quasi Delay Insensitive Approach\nThe STGin Figure 9 represents a one-hot \ufb01nite state machine\n(FSM) for a two action, six-state TA(for the de\ufb01nition of STGs,\nsee [16]). The inputs pand rcome from the FB3rails FB3p\nand FB3rwhich were introduced in Section IV-D. The FB3irail\ndenotes inaction and is only used for completion detection.\nState bits are encoded in the x\u2217\u2217_0and x\u2217\u2217_1places at the\ntop of the graph. States x1\u2217(x2\u2217) are the action 1 (action 2)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd358-934e-11ee-ba68-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 5, "chroma:document": "states\u2014in these states the automaton is indicating the action\nto exclude (include) the feature or feature complement from\nthe clause composition. The x\u22171(x\u22173) states are the closest to\n(furthest from) the action decision boundary. As an example,\nwhen x13_1holds a token, the automaton is indicating action\n1 (exclude) strongly.\nA token at p0denotes the idle state for the automaton. It\noffers a choice between penalty and reward represented by"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd359-934e-11ee-88da-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "signal transitions p+and r+respectively. Each subsequent\nbranch from p+,r+, enables a transition in one of the action\noutput signals a1,a2. Depending on the current action and the\nprevious state, a state transition may also occur. The correct\nbranch is chosen based on read arcs from the x\u2217\u2217_1places. For\nexample in the 2nd left-most branch of r+, we transition from\nstate x12tox13. This happens by \ufb01rstly giving the token from\nplace x13_0tox13_1, and secondly giving token x12_1tox12_0."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd35a-934e-11ee-a043-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "After state transitions and action output have occurred, the\nacknowledge output, ack, makes a positive transition. Finally,\nthe previously given input and action signals return low,\nfollowed by ack\u2212and the token returns to p0. The automaton\nis now ready for the next penalty or reward input.\nThe internal transitions x\u2217\u2217encode some information about\nthe states in the STG. However, this is not enough to satisfy\ncomplete state coding ( CSC) required for synthesis of the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd35b-934e-11ee-8842-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "STG to logic gates [16]. In the previous example where we\ntransition from x12tox13, there is a point where tokens are\nheld by both x13_1and x12_1. The STGreaches this same state\nwhen transitioning in the opposite direction, from x12tox13.\nIn order to achieve CSC, we introduce internal signals to\nencode the direction of travel of the state: x\u2217L\u2217,x\u2217R\u2217; for left\nand right. We insert these signals into every branch to maintain\nuniformity in the STG. Although not all these internal signals"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd35c-934e-11ee-b2c7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "are strictly required, they ease scalability of the STGand help\nthe synthesis tool to minimize and share logic ef\ufb01ciently.\nThe STG in Figure 9 passes all veri\ufb01cation tasks required\nfor synthesis of a correct and hazard-free circuit in the\nWORKCRAFT framework: consistency, deadlock freeness, in-\nput properness, and output persistency. The circuit was syn-\nthesized and mapped to the target technology by the MPS AT\nbackend (the P ETRIFY backend did not produce a result in a"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd35d-934e-11ee-8599-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "reasonable amount of time).\nIn order to read the state of the TAwithout giving penalty or\nreward (during inference for example), we add a latch to the\noutput to store one of the porrsignals. The latch is controlled\nby the ackhandshake.\nThe STG state space expands exponentially as more signal\ntransitions are added. This makes it dif\ufb01cult for the synthesis\ntool to handle TAswith more than three action states. Addition-\nally there is a high possibility for human error when copying"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd35e-934e-11ee-a064-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "and editing branches of the STG. And as the number of TA\naction states increases, the graph becomes large and unwieldy\nto navigate. For these reasons we explore a BDapproach to\nasynchronous TAdesign.\nB. Bundled-Data Approach\nA synchronous one-hot FSM can be constructed using \ufb02ip-\n\ufb02ops as storage elements. The next state logic can be derived\nusing the adjacent states and p,rinputs as follows:\nx13=x13\u00b7r+x12\u00b7r\nx12=x11\u00b7r+x13\u00b7p\nx11=x12\u00b7p+x21\u00b7px21=x22\u00b7p+x11\u00b7p\nx22=x21\u00b7r+x23\u00b7p\nx23=x23\u00b7r+x22\u00b7r"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd35f-934e-11ee-b2c4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "x22=x21\u00b7r+x23\u00b7p\nx23=x23\u00b7r+x22\u00b7r\nThis synchronous circuit is converted into a self-timed one\nthrough the process of desynchronization [12], by decom-posing the \ufb02ip-\ufb02ops into master/slave latches and replacing\nthe clock with handshake controllers and delays matched to\nthe combinational logic paths. The timing of the resultant BD\ndesign is not as robust as the QDIapproach in Section VI-A\nsince the matched delays are \ufb01xed at design time and are"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd360-934e-11ee-b728-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "subject to process, variation and temperature variations.\nTo reduce the overhead of the handshake controllers, we\ngroup the TAsinto an array and use one handshake controller\nfor many latches. Depending on the overall size of TMfor\nthe target application, the TAscould be grouped in different\nways. A monolithic array will minimize controller overhead,\nhowever there comes a limit where buffers must be introduced\ninto the latch enable signals to maintain integrity. At this point,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd361-934e-11ee-95f8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "it may become desirable to group the TAsby clause.\nThe speci\ufb01cation of the BDdesign is easily scaled due to\nthe use of parameterizable Verilog code. The work\ufb02ow takes\nadvantage of mature synchronous tools for synthesis.\nC. Synthesis Results\nTiming and power analyses of asynchronous circuits need\nspecial consideration. In conventional synchronous static tim-\ning analysis, delay through logic gates is calculated using the\ninput slew and output load capacitance. Power estimations also"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd362-934e-11ee-8042-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "rely on these parameters. Synchronous tools cut combinational\nloops arbitrarily, therefore may be optimistic about slews,\ngate delays and internal power in asynchronous circuits which\ncontain such loops.\nTo obtain results for the QDIdesign we use a specialized\nasynchronous static timing analysis tool [17] which takes into\naccount the combinational loops in the timing calculation.\nThe tool operates on the gate-level netlist and propagates"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd363-934e-11ee-b63c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "slews iteratively through circuits with loops, therefore avoiding\noptimism in the slews and gate delays. In the case there are\nmultiple paths in the circuit which begin and end at the same\ninput/output combination, the tool uses the worst-case path to\ndetermine the cycle time.\nSince the UMC 65LL library does not contain C-elements,\nthe QDIcircuit is combinational with feedback loops. Table III\nshows a results summary of the TAimplementations. All results"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd364-934e-11ee-8378-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "are synthesis estimations and no layout is performed. In this\ncase, the QDIcircuit trades increased cycle time and energy\nfor increased robustness to process, variation and temperature\nvariations. The BDdesign strikes a good trade-off between\nperformance, area and elastic timing to integrate with the rest\nof the system. The layout process will introduce more power\nTable III\nRESULTS FOR THE SIX -STATE TSETLIN AUTOMATON SYNTHESIZED FOR\nTHE UMC 65 nm LOW-LEAKAGE SILICON LIBRARY ."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd365-934e-11ee-a4fa-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 6, "chroma:document": "THE UMC 65 nm LOW-LEAKAGE SILICON LIBRARY .\nImplementation AreaCycle Time\n[ns]Leakage\n[nW]Power\n[\u00b5W]Energy\n[fJ]\nSynchronous 87.8 0 .35 3 .08 4 .58 1 .60\nBundled-data386.0 0 .41 2 .89 5 .19 2 .13\nQDI 123 3 .35 3 .42 1 .58 5 .29\n4Averaged over the array."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd366-934e-11ee-ba86-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "Decision Boundary Transitions Rewards in End-states\nOne-hot Encoded States\nAction 1\n(exclude)\nStatesAction 2\n(include)\nStates\nDecision BoundaryFigure 9. Signal transition graph for a Tsetlin automaton with two actions and six states. Inputs: p,r. Outputs: a1,a2,ack. Other signals are internal.\nand latency into the synchronous and BDdesigns due to their\nclock trees, which the QDIdesign will not suffer from.\nVII. C ONCLUSION\nOur Petri net visualization of the TMarchitecture enables"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd367-934e-11ee-9f23-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "\ufb02exible hardware implementations by encapsulating both con-\ncurrency and serialization. By decoupling the TAstorage\nelements from the clauses, e.g. in a standalone array, the TM\ncould be serialized further by reusing clause and sum hardware\nacross multiple cycles, trading off throughput for reduced logic\narea (and leakage power).\nEarly propagation enabled by the RCD scheme used in the DR\ncircuits can be leveraged to trigger \ufb01ne-grained power gating"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd368-934e-11ee-9a0d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "in the system to further reduce energy consumption.\nDistributed generation of probabilistic choices maximizes\nconcurrency and decreases energy overheads compared to\na centralized approach. A simple PRBG implementation is\nsuf\ufb01cient and most suitable for our applications. Features\nof a clock source undesirable in synchronous circuits (jitter,\netc.) can be leveraged for stochasticity. Further study of PRBG\nproperties and their effects on TMlearning are a subject for\nfuture work.\nACKNOWLEDGMENT"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd369-934e-11ee-8130-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "future work.\nACKNOWLEDGMENT\nThe authors thank Christos Sotiriou and colleagues for help\nin asynchronous STAand for access and support for their tool.\nWe are also grateful to colleague Thomas Bunnam for fruitful\ndiscussions on ROsand RNG techniques. The authors would\nlike to acknowledge the funding support from EPSRC IAA\ngrant: Whisperable.\nREFERENCES\n[1] A. Wheeldon et al. , \u201cLow-Latency Asynchronous Logic Design for\nInference at the Edge,\u201d in Proc. Conf. Des. Autom. Test Eur. , Feb. 2021,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd36a-934e-11ee-9760-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "pp. 370\u2013373.[2] O.-C. Granmo, \u201cThe Tsetlin Machine \u2013 A Game Theoretic Bandit Driven\nApproach to Optimal Pattern Recognition with Propositional Logic,\u201d\narXiv:1804.01508v15 [cs.AI] , Apr. 2018.\n[3] G. T. Berge et al. , \u201cUsing the Tsetlin Machine to Learn Human-\nInterpretable Rules for High-Accuracy Text Categorization with Medical\nApplications,\u201d arXiv:1809.04547 [cs, stat] , Sep. 2018.\n[4] C. D. Blakely and O.-C. Granmo, \u201cClosed-Form Expressions for Global"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd36b-934e-11ee-aa95-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "and Local Interpretation of Tsetlin Machines with Applications to\nExplaining High-Dimensional Data,\u201d arXiv:2007.13885 [cs, stat] , Jul.\n2020.\n[5] B. Bhattarai, O.-C. Granmo, and L. Jiao, \u201cMeasuring the Novelty of\nNatural Language Text Using the Conjunctive Clauses of a Tsetlin\nMachine Text Classi\ufb01er,\u201d arXiv:2011.08755 [cs] , Nov. 2020.\n[6] A. Wheeldon et al. , \u201cLearning automata based energy-ef\ufb01cient AI\nhardware design for IoT applications,\u201d Philos Trans R Soc Math Phys"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd36c-934e-11ee-bc66-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "Eng Sci , vol. 378, no. 2182, Oct. 2020.\n[7] \u2014\u2014, \u201cSelf-timed, minimum latency circuits for the internet of things,\u201d\nIntegration , vol. 69, pp. 138\u2013146, Nov. 2019.\n[8] D. Sokolov et al. , \u201cAutomating the Design of Asynchronous Logic\nControl for AMS Electronics,\u201d IEEE Trans. Comput.-Aided Design\nIntegr. Circuits Syst. , vol. 39, no. 5, pp. 952\u2013965, 2020.\n[9] X. Zhang et al. , \u201cA Novel Power Delivery Method for Asynchronous\nLoads in Energy Harvesting Systems,\u201d J. Emerg. Technol. Comput. Syst. ,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd36d-934e-11ee-b492-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "vol. 7, no. 4, pp. 1\u201322, Dec. 2011.\n[10] D. Balsamo et al. , \u201cWearable and autonomous computing for future\nsmart cities: Open challenges,\u201d in 2017 25th Int. Conf. on Software\nTelecommun. Comput. Netw. (SoftCOM) , 2017, pp. 1\u20135.\n[11] \u201cThe workcraft framework.\u201d [Online]. Available: https://workcraft.org/\n[12] J. Cortadella et al. , \u201cDesynchronization: Synthesis of asynchronous\ncircuits from synchronous speci\ufb01cations,\u201d IEEE Trans Comput Integr"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd36e-934e-11ee-b46e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "Circuits Syst , vol. 25, no. 10, pp. 1904\u20131921, Oct. 2006.\n[13] K. S. Narendra and M. A. L. Thathachar, Learning Automata: An\nIntroduction . Englewood Cliffs, N.J: Prentice Hall, 1989.\n[14] J. Agustin and M. Lopez-Vallejo, \u201cAn in-depth analysis of ring oscilla-\ntors: Exploiting their con\ufb01gurable duty-cycle,\u201d IEEE Trans Circuits Syst\nRegul Pap , vol. 62, no. 10, pp. 2485\u20132494, Oct. 2015.\n[15] K. D. Abeyrathna et al. , \u201cA Novel Multi-Step Finite-State Automaton"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cd36f-934e-11ee-a714-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2109.00846.pdf", "page": 7, "chroma:document": "for Arbitrarily Deterministic Tsetlin Machine Learning,\u201d in Proc. 40th\nSGAI Int. Conf. Artif. Intell. , vol. 7, Jul. 2020.\n[16] J. Cortadella et al. ,Logic synthesis of asynchronous controllers and\ninterfaces . Springer-Verlag Berlin Heidelberg, 2002.\n[17] S. Simoglou et al. , \u201cGraph-based STA for asynchronous controllers,\u201d\nIntegration , vol. 75, pp. 91\u2013101, Nov. 2020."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf95a-934e-11ee-8aee-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "Tsetlin Machine Embedding: Representing Words Using Logical Expressions\nBimal Bhattarai ,Ole-Christoffer Granmo ,Lei Jiao ,Rohan Yadav andJivitesh Sharma\nCentre for AI Research (CAIR), University of Agder, Norway\n{bimal.bhattarai, ole.granmo, lei.jiao, rohan.yadav, jivitesh.sharma }@uia.no\nAbstract\nEmbedding words in vector space is a fundamen-\ntal \ufb01rst step in state-of-the-art natural language pro-\ncessing (NLP). Typical NLP solutions employ pre-\nde\ufb01ned vector representations to improve gener-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf95b-934e-11ee-9f23-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "de\ufb01ned vector representations to improve gener-\nalization by co-locating similar words in vector\nspace. For instance, Word2Vec is a self-supervised\npredictive model that captures the context of words\nusing a neural network. Similarly, GLoVe is a\npopular unsupervised model incorporating corpus-\nwide word co-occurrence statistics. Such word em-\nbedding has signi\ufb01cantly boosted important NLP\ntasks, including sentiment analysis, document clas-\nsi\ufb01cation, and machine translation. However, the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf95c-934e-11ee-b7b5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "si\ufb01cation, and machine translation. However, the\nembeddings are dense \ufb02oating-point vectors, mak-\ning them expensive to compute and dif\ufb01cult to in-\nterpret. In this paper, we instead propose to rep-\nresent the semantics of words with a few de\ufb01n-\ning words that are related using propositional logic.\nTo produce such logical embeddings, we introduce\na Tsetlin Machine-based autoencoder that learns\nlogical clauses self-supervised. The clauses con-\nsist of contextual words like \u201cblack,\u201d \u201ccup,\u201d and"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf95d-934e-11ee-8bb6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "sist of contextual words like \u201cblack,\u201d \u201ccup,\u201d and\n\u201chot\u201d to de\ufb01ne other words like \u201ccoffee,\u201d thus be-\ning human-understandable. We evaluate our em-\nbedding approach on several intrinsic and extrinsic\nbenchmarks, outperforming GLoVe on six classi\ufb01-\ncation tasks. Furthermore, we investigate the inter-\npretability of our embedding using the logical rep-\nresentations acquired during training. We also vi-\nsualize word clusters in vector space, demonstrat-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf95e-934e-11ee-988d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "ing how our logical embedding co-locate similar\nwords.1\n1 Introduction\nThe success of natural language processing (NLP) relies on\nadvances in word, sentence, and document representation. By\ncapturing word semantics and similarities, such representa-\ntions boost the performance of downstream tasks [Borgeaud\n1The Tsetlin Machine Autoencoder and logical word embedding\nimplementation is available here: https://github.com/cair/tmu.et al. , 2022 ], including clustering, topic modelling [Angelov,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf95f-934e-11ee-a795-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "2020 ], searching, and text mining [Huang et al. , 2020 ].\nWhile straightforward, the traditional bag-of-words encod-\ning does not consider the words\u2019 position, semantics, and\ncontext within a document. Distributed word representa-\ntion[Bengio et al. , 2000; Bojanowski et al. , 2017 ]addresses\nthis lack by encoding words as low-dimensional vectors, re-\nferred to as embeddings . The purpose is to co-locate simi-\nlar or contextually relevant words in vector space. There are"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf960-934e-11ee-b901-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "many algorithms for learning word embeddings. Contem-\nporary self-supervised techniques like Word2Vec [Mikolov\net al. , 2013 ], FastText [Bojanowski et al. , 2017 ], and\nGloVe [Pennington et al. , 2014 ]have demonstrated how to\nbuild embeddings from word co-occurrence, utilizing mas-\nsive training data. Introducing context-dependent embed-\ndings, the more sophisticated language models BERT [De-\nvlin et al. , 2019 ]and ELMO [Peters et al. , 2018 ]now per-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf961-934e-11ee-bc03-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "form remarkably well in downstream tasks [Reimers and\nGurevych, 2019 ]. However, they require signi\ufb01cant compu-\ntation power [Schwartz et al. , 2020 ].\nThe above approaches represent words as dense \ufb02oating\npoint vectors. Word2Vec, for instance, typically builds a 300-\ndimensional vector per word. The size and density of these\nvectors make them expensive to compute and dif\ufb01cult to in-\nterpret. Consider, for example, the word \u201cqueen.\u201d Represent-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf962-934e-11ee-a471-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "ing it with 300 \ufb02oats seems inef\ufb01cient compared to the Ox-\nford Language de\ufb01nition for the same word: \u201cthe female ruler\nof an independent state, especially one who inherits the posi-\ntion by right of birth.\u201d From this perspective, it appears ad-\nvantageous to create embeddings directly from words rather\nthan from arbitrary \ufb02oating-point values. Such interpretable\nembeddings would capture the multiple meanings of a word\nusing a few de\ufb01ning words, simplifying both computation\nand interpretation."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf963-934e-11ee-8757-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "and interpretation.\nIn this paper, we propose a Tsetlin Machine (TM)\n[Granmo, 2018 ]based autoencoder for creating interpretable\nembeddings. The autoencoder builds propositional logic ex-\npressions with context words that identify each target word.\nThe term \u201ccoffee\u201d can, for instance, be represented by \u201cone,\u201d\n\u201chot,\u201d \u201ccup,\u201d \u201ctable,\u201d and \u201cblack.\u201d In this manner, the TM\nbuilds contextual representations from a vast text corpus,\nwhich model the semantics of each word. In contrast to neu-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf964-934e-11ee-a909-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 0, "chroma:document": "ral network-based embedding, the logical TM embedding is\nsparse. The embedding space consists of, e.g., 500truth val-arXiv:2301.00709v1  [cs.CL]  2 Jan 2023"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf965-934e-11ee-aa55-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "ues, where each truth value is a logical expression over words.\nFor contextual representation, each target word links to less\nthan ten percent of these expressions. Despite the sparsity and\ncrispness of this representation, it is competitive with neural\nnetwork-based embedding.\nThe contributions of our work are summarized below:\n\u2022 We propose the \ufb01rst TM-based Autoencoder to learn ef-\n\ufb01cient encodings in a self-supervised manner.\n\u2022 We introduce TM-based word embedding that builds"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf966-934e-11ee-a9dc-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "human-comprehensible contextual representations from\nunlabeled data.\n\u2022 We compare our embedding with state-of-the-art ap-\nproaches on several intrinsic and extrinsic benchmarks,\noutperforming GloVe on six downstream classi\ufb01cation\ntasks.\n2 Related Work\nThe majority of self-supervised embedding approaches pro-\nduce dense word representations based on the distributional\nhypothesis [Harris, 1954 ], which states that words that oc-\ncur in the same context are likely to have similar meaning."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf967-934e-11ee-b164-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "Word2Vec [Mikolov et al. , 2013 ]is one of the best-known\nmodels. It builds embeddings from word co-occurrence using\na neural network, leveraging the hidden layer output weights.\nGloVe [Pennington et al. , 2014 ], on the other hand, em-\nbeds by factorizing a word co-occurrence matrix. Similarly,\ncanonical correlation analysis (CCA) is used in [Dhillon et\nal., 2015 ]for embedding words to maximise context corre-\nlation. In [Levy et al. , 2015 ], it is demonstrated how pre-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf968-934e-11ee-b85e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "cise factorization-based SVD can compete with neural em-\nbedding. However, all of these methods are challenging to\ntrain because they involve tweaking algorithms and hyperpa-\nrameters toward particular applications [Lample et al. , 2016 ],\nlimiting their wider applicability.\nBuilding upon word embedding, several studies focus on\nsentence embedding [Arora et al. , 2017; Logeswaran and\nLee, 2018 ]. Recent advances in sentence embedding in-\nclude supervised data inference [Reimers and Gurevych,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf969-934e-11ee-a609-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "2019 ], multitask learning [Ceret al. , 2018 ], contrastive learn-\ning[Zhang et al. , 2020 ], and pretrained large language mod-\nels[Liet al. , 2020 ]. However, the majority of sentence\nembedding techniques overlook intrinsic evaluations such as\nsimilarity tasks, and instead largely focus on extrinsic eval-\nuations involving downstream performance. The most re-\ncent building block for embedding originates from the trans-\nformer approach [Vaswani et al. , 2017 ]. Transformers pro-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf96a-934e-11ee-a054-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "vide context awareness by utilizing stacks of self-attention\nlayers. BERT [Kenton and Toutanova, 2019 ], for instance,\nemploys the transformer architecture to carry out extensive\nself-supervised training, making it capable of producing text\nembedding. Other embedding models use a contrastive loss\nfunction to perform supervised \ufb01ne-tuning on positive and\nnegative text pairs [Wang et al. , 2021 ]. Despite the large\nvariety of text embedding models, they all share three main"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf96b-934e-11ee-8bc4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "drawbacks: i) they are computationally demanding to train;\nii) they are intrinsically complex because they are trained on\na large amount of data to tune a huge amount of parameters;\nClause Pool+4 -5Input Output\n+1 +2 -7 +6Figure 1: Tsetlin Machine Autoencoder. In this illustration, x1is\nmasked by replacing it with value 1 for inferring \u02c6x1.\nand iii) the embeddings produced from these models are not\neasily interpreted by humans.\nTo improve interpretability, Faruqui et al. introduced"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf96c-934e-11ee-85ad-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "\u201cSparse Overcomplete Word Vectors\u201d (SPOWV) which cre-\nate a sparse non-negative projection of word embedding us-\ning dictionary learning [Faruqui et al. , 2015 ]. Similarly,\nSParse Interpretable Neural Embeddings (SPINE) employs\na k-sparse denoising autoencoder to generate sparse embed-\ndings [Subramanian et al. , 2018 ]. However, these methods\nare unable to distinguish between multiple context-dependent\nword meanings. To address this problem, another avenue of"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf96d-934e-11ee-9804-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "research focuses on composing linear combinations of dense\nvectors from Word2Vec and GloVe [Arora et al. , 2018 ]. How-\never, the assumption of linearity does not hold for real-world\ndata, yielding linear coef\ufb01cients that are dif\ufb01cult to compre-\nhend [Muet al. , 2017 ].\nThe logical embedding approach we present here is most\nclosely related to Naive Bayes word sense induction and topic\nmodeling [Charniak and others, 2013; Lau et al. , 2014 ]. This"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf96e-934e-11ee-b059-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "approach learns word meanings from local contexts by con-\nsidering each instance of the word in a document as a pseudo-\ndocument. However, the approach is not scalable because\nit requires training a single topic per target word. Our ap-\nproach, on the other hand, is scalable and builds non-linear\n(non-naive) logical embeddings that capture word composi-\ntions. To build the logical embeddings, we propose a novel\nhuman-interpretable algorithm based on the TM that provides"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf96f-934e-11ee-8900-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "logical rules describing contexts. The TM has recently per-\nformed competitively with other deep learning techniques in\nmany NLP tasks, including novelty detection [Bhattarai et\nal., 2022a ], sentiment analysis [Yadav et al. , 2021 ], and fake\nnews detection [Bhattarai et al. , 2022b ]. Furthermore, the\nlocal and global interpretability of TMs have been explored\nthrough direct manipulation of the logical rules [Blakely and\nGranmo, 2021 ].\n3 Tsetlin Machine Autoencoder"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf970-934e-11ee-850f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 1, "chroma:document": "Granmo, 2021 ].\n3 Tsetlin Machine Autoencoder\nWe here detail the TM Autoencoder based on the Coalesced\nTM [Glimsdal and Granmo, 2021 ], extended with input\nmasking and freezing of masked variables. For ease of ex-\nplanation, we use three inputs. Adding more inputs follows\ntrivially."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf971-934e-11ee-9ad2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "3.1 Architecture\nInput and Output. As seen in Figure 1, the TM Autoen-\ncoder digests and outputs propositional values: (x1,x2,x3)\u2208\n{0,1}3\u2192(\u02c6x1,\u02c6x2,\u02c6x3)\u2208 {0,1}3. For our purposes, the\npropositional variables x1,x2, andx3each represents a word,\nfor example, \u201cBrilliant,\u201d \u201cActor,\u201d and \u201cAwful.\u201d The value 1\nmeans that the word occurs in the input text, while the value\n0means that it does not. I.e., we represent natural language\ntext as a set of words . Notice also that the input variables have"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf972-934e-11ee-a9ec-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "corresponding output variables \u02c6x1,\u02c6x2, and\u02c6x3. In short,\u02c6x1\nis to be predicted from x2andx3,\u02c6x2fromx1andx3, and\nso on. Continuing our example, \u02c6x1predicts the presence of\n\u201cBrilliant\u201d based on knowing the occurrence of \u201cActor\u201d and\n\u201cAwful.\u201d\nClause Pool. A pool ofnconjunctive clauses, denoted\nCj,j\u2208 {1,2,...,n},encodes the input in order to pre-\ndict the output. A conjunctive clause Cjis simply an And-\nexpression over a given subset Lj\u2286{x1,x2,x3}of the input"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf973-934e-11ee-b740-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "(our autoencoder does not use the input negations \u00acx1,\u00acx2,\nand\u00acx3):\nCj(x1,x2,x3) =\u22c0\nxk\u2208Ljxk. (1)\nFor example, the input subset L1={x1,x2}gives the clause\nC1(x1,x2,x3) =x1\u2227x2in the \ufb01gure. This clause matches\nthe input ifx1andx2both are 1. In our example, the clause\naccordingly encodes the concept \u201cBrilliant Actor\u201d.\nWeights. An integer weight matrix Wconnects each of the\nnclauses to the three outputs \u02c6x1,\u02c6x2, and\u02c6x3:\nW=[w11\u00b7\u00b7\u00b7w1n\nw21\u00b7\u00b7\u00b7w2n\nw31\u00b7\u00b7\u00b7w3n]\n\u2208Z3\u00d7n. (2)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf974-934e-11ee-8c6b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "W=[w11\u00b7\u00b7\u00b7w1n\nw21\u00b7\u00b7\u00b7w2n\nw31\u00b7\u00b7\u00b7w3n]\n\u2208Z3\u00d7n. (2)\nThe row index is an output while the column index is a clause.\nThe weight w12, for instance, connects output \u02c6x1to clause\nC2. In Figure 1, six weights connect two clauses and three\noutputs:[+4\u22125\n+1 +2\n\u22127 +6]\n. (3)\nConsider, for example, the weights (+4,\u22125)of output\u02c6x1in\nthe \ufb01gure. The weight +4states that clause C1(x1,x2,x3) =\nx1\u2227x2favours\u02c6x1being 1, while clause C2(x1,x2,x3) =\nx2\u2227x3opposes it. For example, the concept \u201cAwful Actor\u201d"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf975-934e-11ee-ab2c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "opposes output \u201cBrilliant.\u201d\n3.2 Inference\nLet us consider the prediction of \u02c6x1\ufb01rst. The autoencoder\npredicts\u02c6x1from the clauses and weights:\n\u02c6x1= 0\u2264n\u2211\nj=1wj1Cj(1,x2,x3). (4)\nThat is, each clause Cjis multiplied by its weight wj1for\noutput\u02c6x1. The outcomes are then summed up to decide the\noutput. If the sum is larger than or equal to zero, the output\nis\u02c6x1= 1. Otherwise, it is \u02c6x1= 0. Clauses with positive\nMaximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten 4 \n \n3 \n \n2"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf976-934e-11ee-9cda-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "ForgottenMemorized\nForgotten 4 \n \n3 \n \n2 \n \n1Figure 2: Tsetlin Machine memory for single clause.\nweight thus promote output \u02c6x1= 1 while clauses with neg-\native weight encourage \u02c6x1= 0. Notice that x1is masked by\nreplacing it with value 1. Accordingly, the autoencoder infers\noutput\u02c6x1from the remaining inputs x2andx3.\nCorrespondingly, \u02c6x2and\u02c6x3are calculated by respectively\nmaskingx2andx3:\n\u02c6x2= 0\u2264n\u2211\nj=1wj2Cj(x1,1,x3), (5)\n\u02c6x3= 0\u2264n\u2211\nj=1wj3Cj(x1,x2,1). (6)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf977-934e-11ee-96bc-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "\u02c6x3= 0\u2264n\u2211\nj=1wj3Cj(x1,x2,1). (6)\nExample. Assume that the input is always either (1,1,0)\nor(0,1,1). The input (1,1,0)could for instance represent\n\u201cBrilliant Actor\u201d and (0,1,1)\u201cAwful Actor.\u201d Then notice\nhow Eq. (4) correctly determines the masked input x1with\noutput\u02c6x1in Figure 1, both for input (1,1,0)and(0,1,1).\n3.3 Learning\nWe next consider how to learn the variable subsets Ljfor the\nclausesCj,j\u2208{1,2,...,n},as well as how to determine\nthe weightswjiof the weight matrix W."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf978-934e-11ee-86e4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "the weightswjiof the weight matrix W.\nClause Memory. Each clause Cjhas a graded memory that\ncontains the input variables, shown in Figure 2. The graded\nmemory enables incremental learning of the variable subsets\nfrom data. Observe how each variable is in one of four mem-\nory positions (the number of memory positions is a user-\ncon\ufb01gurable parameter). Positions 1\u22122means Forgotten.\nPositions 3\u22124means Memorized. Memorized variables take\npart in the clause, while Forgotten ones do not. The memory"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf979-934e-11ee-a3eb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "in Figure 2 thus gives the clause Cj(x1,x2,x3) =x1\u2227x2.\nLearning Step. The TM Autoencoder learns incrementally\nusing three kinds of memory and weight updates: Type Ia,\nType Ib, and Type II. Each training example has the form\n[k,(x1,x2,x3),xk],1\u2264k\u22643. The \ufb01rst element is an index\nthat identi\ufb01es which input to mask and which output to pre-\ndict. The second element is an input vector (x1,x2,x3)and\nthe third element is the target value for output \u02c6xk, which is"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf97a-934e-11ee-8415-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 2, "chroma:document": "xk. We describe the update procedure step-by-step below for\nindex 1examples (output \u02c6x1prediction). The update proce-\ndure for\u02c6x2and\u02c6x3follows trivially.\nClause Update Probability. First, we calculate the\nweighted clause sum for \u02c6x1from Eqn. (4): v1=\u2211n\nj=1wj1Cj(1,x2,x3). The sum is then compared with a\nmarginT(hyper-parameter) to calculate a summation error \u03f5.\nThe error depends on the x1-value:\n\u03f5={T\u2212clip(v1,\u2212T,T), x 1= 1,\nT+ clip(v1,\u2212T,T), x 1= 0.(7)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf97b-934e-11ee-8eb7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "Maximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten\n 4 \n \n3 \n \n2 \n \n1Figure 3: Type Ia (Recognize) Feedback for input (1,1,0). The\nmasked variable x1is frozen.\nMaximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten\n 4 \n \n3 \n \n2 \n \n1\nFigure 4: Type Ib (Erase) Feedback for input (0,0,1). The masked\nvariable x1is frozen.\nThat is, for x1-value 1the weighted clause sum should be-\ncomeT, while forx1-value 0the sum should become \u2212T."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf97c-934e-11ee-803e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "The goal of the learning is thus to reach the margin for all\ninputs (x1,x2,x3), ensuring correct output from Eqn. (4).\nTo reach this goal, each clause Cjis updated randomly with\nprobability\u03f5\n2Tin each round. In other words, the update\nprobability drops with the error towards zero.\nUpdate Types. The kind of update depends on the values\nofx1,Cj(1,x2,x3), andwj1. We \ufb01rst consider clauses with\npositive weight, wj1\u22650. According to Eqn. 4, they are to"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf97d-934e-11ee-9bd6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "recognize patterns for x1= 1. Note that in all of the below\nupdates, the masked variable x1is frozen, leaving it unaf-\nfected by the update.\n\u2022Type Ia (Recognize) Feedback occurs when x1=\n1andCj(1,x2,x3) = 1 . Then one can say that\nCj(1,x2,x3) = 1 is atrue positive because it correctly\npredicts the masked x1-value. The Type Ia feedback re-\ninforces this successful match by updating the memory\nofCjto further mimic the input (see Figure 3). That is,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf97e-934e-11ee-afa8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "1-valued variables move one step upwards in memory,\nwith probability 1.0.2Conversely, 0-valued inputs move\none step downwards, however, randomly with probabil-\nity1\ns. Here,sis a hyperparameter called speci\ufb01city ,\nmeaning that a larger smakes the clauses more speci\ufb01c.\nThe clause overall is also reinforced by incrementing its\nweightwj1by1.\n\u2022Type Ib (Erase) Feedback occurs when x1= 1 and\nCj(1,x2,x3) = 0 . Then we call Cj(1,x2,x3) = 0\nafalse negative because it fails to promote x1= 1 ."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf97f-934e-11ee-b46f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "In that case, all inputs randomly move one step down-\nwards in memory (see Figure 4). Again, each downward\nmove happens with probability1\ns. Here, the purpose is\nto eliminate the false negative outcome by erasing vari-\nables from the clause.\n2Originally, the increment probability iss\u22121\ns, which can be\nboosted to 1.0to enhance learning of true positive patterns [Granmo,\n2018 ].\nMaximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten\n 4 \n \n3 \n \n2"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf980-934e-11ee-8c7e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "ForgottenMemorized\nForgotten\n 4 \n \n3 \n \n2 \n \n1Figure 5: Type II (Reject) Feedback for input (0,1,0). The masked\nvariable x1is frozen.\n\u2022Type II (Reject) Feedback occurs when x1= 0 and\nCj(1,x2,x3) = 1 . Then, one can say that Cj(1,x2) = 1\nis afalse positive because it promotes x1= 1 when in\nfact we have x1= 0. Then all Forgotten 0-valued in-\nputs move one step upwards in memory. The purpose\nis to eventually eliminate the current false positive out-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf981-934e-11ee-8ff8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "come by injecting 0-valued variables into the clause.\nThe clause is further diminished by decrementing its\nweightwj1by1. Note that the latter decrement can\nswitch the weight from positive to negative. In effect,\nthe clause then changes role, now training to recognize\nx1= 0instead.\nClausesCjwith negative weights, wj1<0, are updated\nthe same way. However, they are to recognize patterns for\nx1= 0. To achieve this, x1= 0 is treated as x1= 1 and\nx1= 1 is treated as x1= 0 when updating the memories."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf982-934e-11ee-a471-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "Furthermore, the weight updates are reversed. Increments be-\ncomes decrements, and vice versa.\nAlgorithm 1 TM word embedding\nRequire: V ocabularyV; DocumentsD\u2208G ,D\u2286V ; Accumula-\ntionu; Clauses n; Margin T; Speci\ufb01city s; Rounds r\n1: TMCreate (n, T, s ) \u22bfCreate TM with nclauses.\n2:forrrounds do\n3: forword k\u2208Vdo \u22bfCreate one example per word.\n4: qk\u2190Select({0,1}) \u22bfRandom target value.\n5: ifqk= 1then\n6:Gk\u2190{D| word k\u2208D,D\u2208G} \u22bfDocuments with\nword k.\n7: else\n8:Gk\u2190{D| word k/\u2208D,D\u2208G} \u22bfDocuments"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf983-934e-11ee-8a33-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "7: else\n8:Gk\u2190{D| word k/\u2208D,D\u2208G} \u22bfDocuments\nwithout word k.\n9:Sk\u2190SelectN(Gk, u) \u22bfRandom subset of size u.\n10:Uk\u2190\u22c3\nD\u2208S kD \u22bfUnion of selected documents.\n11: xk\u2190(x1, x2, . . . , x m), xi={\n1,word i\u2208Uk\n0,word i/\u2208Uk\n12: TMUpdate( k,xk, qk)\u22bfUpdate TM Autoencoder for\noutput index k, input xk, and target value \u02c6xk=qk.\n13:C,W\u2190TMGetState() \u22bfClauses Cj\u2208Cwith weights W.\n14:E\u2190clip(W,0, T) \u22bfElementwise clip of negative values\nproduces weighted logical word embeddings."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf984-934e-11ee-9c10-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 3, "chroma:document": "produces weighted logical word embeddings.\n15:B\u2190(W>0) \u22bfElementwise comparison with zero\nproduces purely logical word embeddings.\n16:returnC,E,B"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf985-934e-11ee-ad63-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "4 Logical Embedding Procedure\nWe now use the TM Autoencoder to build logical embed-\ndings. LetV={word 1,word 2,..., wordm}be the target\nvocabulary consisting of munique words.\nPre-processing. The \ufb01rst step is to pre-process the docu-\nment corpus. To this end, each document is represented by a\nsubset of wordsD\u2286V . For example, the document \u201cThe ac-\ntor was brilliant\u201d becomes the set D={\u201cactor\u201d, \u201cbrilliant\u201d,\n\u201cthe\u201d, \u201cwas\u201d}. The setG, in turn, contains all the documents,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf986-934e-11ee-bbee-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "D\u2208G . Finally, in propositional vector form, the word set D\nbecomes:\nx= (x1,x2,...,xt),xi={1,wordi\u2208D,\n0,wordi/\u2208D.(8)\nEmbedding. Algorithm 1 speci\ufb01es the procedure for em-\nbedding the mvocabulary words from Vby usingnclauses,\nCj,1\u2264j\u2264n, forming a clause set C. Each round of\ntraining produces a training example [k,(x1,x2,...,xm),qk]\nperwordkinV. First, a target value qkfor the word is set\nrandomly to either 0or1. This random selection balances\nthe dataset. If qkbecomes 1, we randomly select udoc-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf987-934e-11ee-b13c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "uments that contain wordkand assign them to the set Sk\n(positive examples). Otherwise, we randomly select udoc-\numents that does notcontain the word (negative examples).\nNext, the randomly selected documents are merged by OR-\ning them together, yielding the uni\ufb01ed document Uk. The\npurpose of ORing multiple documents is to increase the fre-\nquency of rare context words. Then, picking up characteristic\nones becomes easier. After that, the propositional vector form"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf988-934e-11ee-ae60-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "(x1,x2,...,xm)ofUkis obtained. Finally, the TM Autoen-\ncoder is updated with [k,(x1,x2,...,xm),qk]following the\ntraining procedure in Section 3.\nVector Space Representation. The weighted logical em-\nbedding of wordk\u2208V can now be obtained from row kof\nmatrix E(returned from Algorithm 1), while the the purely\nlogical embedding is found in row kof matrix B. Letekde-\nnote thek\u2019th row of E, and let eldenote thel\u2019th row. We can\nthen compare the similarity of two words wordkand wordl"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf989-934e-11ee-9643-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "using cosine similarity (CS) between their E-embedding:\nCS(wordk,wordl) =ek\u00b7el\n||ek||||el||. (9)\n5 Empirical Evaluation\nWe here evaluate our logical embedding scheme, comparing\nit with neural network approaches.\n5.1 Datasets and Setup\nWe \ufb01rst evaluate our logical embedding intrinsically, fol-\nlowed by an extrinsic evaluation using classi\ufb01cation tasks.\nIntrinsic Evaluation. We use word similarity and catego-\nrization benchmarks for intrinsic evaluation. That is, we ex-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf98a-934e-11ee-9299-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "amine to what degree our approach retains semantic word\nrelations. To this end, we measure how semantic relations\nmanifest in vector space using six datasets: SimLex-999,\nWordSim-353, MEN, MTurk-287, Mturk-771, and RG-65.\nEach dataset consists of human-scored word pairs, which arecompared with the corresponding vector space similarities.\nThe categorization tasks evaluate how well we can group\nwords into distinct word categories, only based on their em-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf98b-934e-11ee-8b1d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "bedding. We here use three datasets: AP, BLESS, and ESS-\nLLI.3As baselines, we chose Word2Vec, GloVe, and FastText\nbecause of their wide use.\nExtrinsic Evaluation. In our extrinsic evaluation, we in-\nvestigate how well our logical embedding supports down-\nstream NLP classi\ufb01cation tasks. Using the word embeddings\nas feature vectors, the performance of supervised classi\ufb01ca-\ntion models gives insight into the embedding quality. We\nemploy six standard text classi\ufb01cation datasets from SentE-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf98c-934e-11ee-9e05-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "val[Conneau and Kiela, 2018 ]: R8, R52, TREC, SUBJ, SST-\n2, and SST-5. For supervised learning, we use the standard\nattention-based BiLSTM model with the Adam optimizer and\ncross-entropy loss function. In this manner, we directly con-\ntrast GloVe embedding against the logical TM approach.\nEmbedding Datasets. For extrinsic evaluation with BiL-\nSTM, we use standard 300-dimensional GloVe embeddings,\npre-trained on the Wikipedia 2014 + Gigaword 5 datasets (6B"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf98d-934e-11ee-afa2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "tokens).4The purpose is to compare the TM embedding per-\nformance against widely used and successful GloVe embed-\ndings on downstream tasks. To directly compare the intrinsic\nproperties of Word2Vec, GloVe, FastText, and TM embed-\nding, we also train them from scratch using the One Billion\nWord dataset [Chelba et al. , 2014 ]. For training the TM, we\nuse Algorithm 1 with r= 2000 training rounds, produc-\ning2000 examples per word by accumulating u= 25 con-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf98e-934e-11ee-a2c9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "texts per example. We use the following hyperparameters: a\npool ofn= 600 clauses, margin T= 1200 , and speci\ufb01city\ns= 5.0.5Word2Vec Skip-Gram is trained with 10passes\nover the data, using separated embeddings for the input and\noutput contexts. The window size is 5and we use \ufb01ve nega-\ntive samples per example. Similarly, GloVe is trained for 30\nepochs with a window size of 10and a learning rate of 0.05.6\n5.2 Results and Discussion\nAs presented in Section 5.1, we employ two kinds of evalu-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf98f-934e-11ee-914a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "ation: intrinsic and extrinsic. Table 1 contains the intrinsic\nevaluation results from the six word similarity tasks. We here\ncompute the Spearman correlation, the Kendall coef\ufb01cient,\nand the cosine similarity between the human-set similarity\nscores and the predicted similarity scores per dataset. Con-\nsidering Spearman and Kendall score, Word2Vec and GloVe\nare marginally better than the comparable FastText and TM\nembedding. However, as reported in [Rastogi et al. , 2015 ],"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf990-934e-11ee-9e5e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "3To obtain the categorization accuracy, we use KMeans\nclustering from sklearn on the word embeddings and exam-\nine the cluster quality by calculating the purity score from\n(https://github.com/purity).\n4The pre-trained GloVe embeddings can be found here:\nhttps://nlp.stanford.edu/projects/glove/\n5The TM Autoencoder and logical word embedding implemen-\ntation can be found here: https://github.com/cair/tmu.\n6Word2Vec and FastText have been trained using"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf991-934e-11ee-a688-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 4, "chroma:document": "6Word2Vec and FastText have been trained using\nthe standard gensim library (https://github.com/RaRe-\nTechnologies/gensim/tree/develop/gensi). GloVe has been trained\nusing https://github.com/maciejkula/glove-python."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf992-934e-11ee-a2d0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "DatasetW2V FastText TM GloVe\nSpearman Kendall Cosine Spearman Kendall Cosine Spearman Kendall Cosine Spearman Kendall Cosine\nWS-353 0.53 0.37 0.87 0.46 0.32 0.79 0.45 0.31 0.90 0.41 0.28 0.90\nSIM999 0.26 0.18 0.79 0.23 0.16 0.79 0.14 0.10 0.76 0.25 0.17 0.80\nMEN 0.71 0.50 0.91 0.71 0.51 0.94 0.64 0.45 0.94 0.73 0.53 0.95\nMTURK287 0.66 0.47 0.77 0.63 0.44 0.93 0.63 0.44 0.92 0.66 0.47 0.86\nMTURK717 0.57 0.39 0.86 0.52 0.36 0.93 0.48 0.32 0.91 0.58 0.40 0.94"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf993-934e-11ee-a469-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "RG65 0.72 0.58 0.89 0.67 0.49 0.88 0.75 0.63 0.92 0.78 0.62 0.93\nAverage 0.58 0.42 0.85 0.54 0.38 0.88 0.52 0.38 0.89 0.57 0.42 0.90\nTable 1: Performance comparison of TM embedding with baseline algorithms on the similarity task.\nTarget words ClausesClauses\n student\nsurgery\nheart\nbaseball\nfootball\nqueen\nprincess\ncoffee\ntea\n \nFigure 6: Interpretability of clauses capturing distinct meanings of target words in the TM embedding.\nDataset W2V FastText TM GloVe\nAP 0.50 0.35 0.41 0.41"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf994-934e-11ee-b366-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "AP 0.50 0.35 0.41 0.41\nBLESS 0.64 0.66 0.62 0.66\nESSLI 0.63 0.60 0.57 0.56\nAverage 0.59 0.54 0.53 0.54\nTable 2: Performance comparison of TM embedding with baseline\nembeddings on the categorization task.\nDatasetGloVe TM TM hybrid\nAcc. F1 Acc. F1 Acc. F1\nR8 96.31 0.88 96.10 0.88 97.80 0.94\nTREC 95.20 0.95 96.40 0.96 96.80 0.96\nR52 90.34 0.58 91.23 0.62 94.23 0.68\nSUBJ 86.20 0.86 85.80 0.85 86.70 0.87\nSST-2 76.38 0.75 75.61 0.74 79.30 0.78\nSST-5 47.47 0.46 47.80 0.43 49.75 0.44"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf995-934e-11ee-a6d4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "SST-5 47.47 0.46 47.80 0.43 49.75 0.44\nTable 3: Performance comparison of our embedding with standard\nGloVe embedding on the classi\ufb01cation task.\nsmall differences in correlation-based measures are not nec-\nessarily signi\ufb01cant for smaller datasets. To more robustly\nassess performance, we therefore also use cosine similarity\nto compare predicted word similarities with the human-set\nsimilarities. In terms of cosine score, our model outperforms"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf996-934e-11ee-a553-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "Word2Vec and FastText on the majority of the datasets, whileperforming competitively with GloVe. This means that the\nangles between the human-set similarities and the GloVe/TM-\npredicted similarities are quite similar. Finally, Table 2 shows\nthe outcome for the word categorization tasks. As seen, the\nperformance of the selected embedding techniques are com-\nparable, with Word2Vec being slightly ahead.\nPrevious research indicates that intrinsic word similarity"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf997-934e-11ee-97cd-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "performance is minimally or even negatively correlated with\ndownstream NLP performance [Wang et al. , 2021 ]. There-\nfore, we also include an extrinsic evaluation with six down-\nstream classi\ufb01cation tasks. To avoid over\ufb01tting and robustly\nassess downstream properties, we keep our experimental\nsetup from above. Table 3 reports the outcome of the eval-\nuation, where the embeddings have been fed to an attention-\nbased BiLSTM model. The \ufb01rst con\ufb01guration (GloVe) uses"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf998-934e-11ee-8557-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "the pre-trained GloVe embeddings from the Wikipedia 2014\n+ Gigaword 5 datasets. The second con\ufb01guration consists\nof our purely logical TM embedding from One Billion Word\n(embedding Bfrom Algorithm 1). Being \ufb01ve times smaller,\nthe One Billion Word dataset only provides about 80percent\nof the vocabulary required for the classi\ufb01cation tasks. We em-\nbed the remaining 20percent of the words randomly. Hence,\nthe TM approach can potentially have a disadvantage in the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf999-934e-11ee-8a7b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 5, "chroma:document": "evaluation. In the third con\ufb01guration (TM hybrid ), we replace\nthe20percent random embeddings with the corresponding\nGloVe embeddings (approximately 80% TM + 20% GloVe).\nWe note that the downstream accuracy of BiLSTM is simi-\nlar for both TM and GloVe. Speci\ufb01cally, the TM embedding"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf99a-934e-11ee-8f96-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "exceeds GloVe by a small margin on TREC, R52, and SST-\n5. The hybrid embedding, on the other hand, clearly outper-\nforms the other two. In particular, for R52, SST-2, and SST-5,\nthe hybrid embedding is able to surpass GloVe by a substan-\ntial margin of roughly 2\u22124%. Given that the datasets are\nnot completely balanced, we also compute F1 macro scores.\nWe again observe that the TM embedding either outperforms\nor is competive with GloVe. For R8 and R52, the hybrid em-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf99b-934e-11ee-9992-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "bedding surpasses GloVe by a large margin, respectively by\naround 6%and10%. Based on these results, we conjecture\nthat logical TM embedding can successfully replace neural\nnetwork embedding. Even with 20% of the vocabulary miss-\ning, trained on \ufb01ve times smaller data, the logical embedding\nperform competitively with GloVe. Interestingly, the hybrid\napproach performed even better. One possible explanation of\nthis higher performance can be the extra information added"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf99c-934e-11ee-88d9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "by the larger vocabulary. Additionally, there may be synergy\nbetween the neural and logical representations that manifest\nin the hybrid approach.\n5.3 Interpretability and Visualization\nIn this section, we investigate the nature of the TM embed-\ndings in more detail, focusing on interpretability. Our em-\nbedding consists of the positive clause weights E, or, alter-\nnatively, the propositional version B, explained by the set\nof clausesC. As demonstrated in Figure 6, each clause in C"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf99d-934e-11ee-834b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "captures a facet of a context. The dotted lines in the \ufb01gure\nshowcase the connection between the target words and their\nclauses from matrix B(and, accordingly, E). Each target\nword gets its own color to more easily discern the connec-\ntions. In the \ufb01gure, we provide an excerpt of 18connections\nfromB, involving 8target words and the 11most triggered\nclauses for these words. Consider for example the target\nwords surgery andheart . These two target words share two"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf99e-934e-11ee-8f8a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "clauses: [ went\u2227hospital ] and [ old\u2227disease\u2227patient ].\nThe two clauses capture two joint contexts, both related to\nhealth. The clauses thus represent commonality between the\ntarget words, providing information on one particular mean-\ning of the words.\nThe two target words are also semantically different. The\ndifferences are captured by the clauses they do not share. The\ntarget word heart , for example, also relates to the meaning\n[woman\u2227love], which surgery does not. Surgery , on the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf99f-934e-11ee-bed4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "other hand, connects with [ injury\u2227game\u2227racing ]. In this\nmanner, the unique meanings and relations between words\nare represented through sharing of logical expressions. Ac-\ncordingly, it is feasible to capture a wide range of possible\ncontextual representations with concise logical expressions.\nAs such, the logical embedding provides a sparse representa-\ntion of words and their relations. Indeed, at most 10% of the\nclauses connect to each word in our experiments. As shown in"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a0-934e-11ee-b65b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "the intrinsic evaluations from the previous subsection, these\ncontextual representations are effective for measuring word\nsimilarity and categorizing words. Similarly, we observed\nthat the logical embedding is boosting downstream NLP clas-\nsi\ufb01cation tasks.\nTo cast further light on the TM embedding approach, we\nvisualize the embedding of 400words from the SimLex-999\ndataset in Figure 7, plotted using t-SNE. The \ufb01gure indicates\nFigure 7: TM embedding visualization plotted using t-SNE."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a1-934e-11ee-a94c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "that we are able to cluster contextually similar words in vec-\ntor space. To scrutinize the clusters, we zoom in on two\nof them. Consider the upper right cluster \ufb01rst. Notice how\nthe words in the cluster relate to hospital , such as heart and\ndiseases . As seen, the word embeddings are closely located\nin vector space. Similarly, we can observe that terminology\nconnected to weather and geography are grouped together in\nthe bottom cluster. From these two examples, it seems clear"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a2-934e-11ee-b99c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "that the TM embedding incorporates semantic relationships\namong words.\n6 Conclusion and Future Work\nIn this work, we \ufb01rst discussed the challenge and necessity\nof \ufb01nding computationally simpler and more interpretable\nword embedding approaches. We then motivated an ef\ufb01cient\nself-supervised approach, namely, a TM-based autoencoder,\nfor producing sparse and interpretable logical word embed-\ndings. We evaluated our approach on a wide range of intrin-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a3-934e-11ee-8563-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "sic and extrinsic tasks, demonstrating that it is competitive\nwith dense neural network-based embedding schemes such\nas Word2Vec, GloVe, and FastText. Further, we investigated\nthe interpretability our embedding through visualization and\na case study. Our conclusion from the study is that the log-\nical embedding is able to represent words with logical ex-\npressions. This structure makes the representation sparse, en-\nabling a clear-cut decomposition of each word into sets of\nsemantic concepts."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a4-934e-11ee-8eff-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 6, "chroma:document": "semantic concepts.\nFuture work includes scaling up our implementation using\nGPUs to support building of large scale vocabularies from\nmore massive datasets. Also, we intend to investigate how\nsentence-level and document-level embedding can be created\nusing clauses, for instance applicable for downstream sen-\ntence similarity tasks."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a5-934e-11ee-99ba-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "References\n[Angelov, 2020 ]Dimo Angelov. Top2vec: Distributed rep-\nresentations of topics. arXiv preprint arXiv:2008.09470 ,\n2020.\n[Arora et al. , 2017 ]Sanjeev Arora, Yingyu Liang, and\nTengyu Ma. A simple but tough-to-beat baseline for sen-\ntence embeddings. In International conference on learning\nrepresentations , 2017.\n[Arora et al. , 2018 ]Sanjeev Arora, Yuanzhi Li, Yingyu\nLiang, Tengyu Ma, and Andrej Risteski. Linear algebraic\nstructure of word senses, with applications to polysemy."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a6-934e-11ee-93fe-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "Transactions of the Association for Computational Lin-\nguistics , 6:483\u2013495, 2018.\n[Bengio et al. , 2000 ]Yoshua Bengio, R \u00b4ejean Ducharme, and\nPascal Vincent. A neural probabilistic language model.\nAdvances in neural information processing systems , 13,\n2000.\n[Bhattarai et al. , 2022a ]Bimal Bhattarai, Ole-Christoffer\nGranmo, and Lei Jiao. A Tsetlin Machine Framework for\nUniversal Outlier and Novelty Detection. In International\nConference on Agents and Arti\ufb01cial Intelligence , pages"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a7-934e-11ee-815b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "250\u2013268. Springer, 2022.\n[Bhattarai et al. , 2022b ]Bimal Bhattarai, Ole-Christoffer\nGranmo, and Lei Jiao. Explainable Tsetlin Machine\nframework for fake news detection with credibility score\nassessment. In LREC , 2022.\n[Blakely and Granmo, 2021 ]Christian D. Blakely and Ole-\nChristoffer Granmo. Closed-Form Expressions for Global\nand Local Interpretation of Tsetlin Machines. In\n34th International Conference on Industrial, Engineer-\ning and Other Applications of Applied Intelligent Systems"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a8-934e-11ee-9a0e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "(IEA/AIE 2021) . Springer, 2021.\n[Bojanowski et al. , 2017 ]Piotr Bojanowski, Edouard Grave,\nArmand Joulin, and Tomas Mikolov. Enriching word vec-\ntors with subword information. Transactions of the asso-\nciation for computational linguistics , 5:135\u2013146, 2017.\n[Borgeaud et al. , 2022 ]Sebastian Borgeaud, Arthur Men-\nsch, Jordan Hoffmann, Trevor Cai, Eliza Rutherford, Katie\nMillican, George Bm Van Den Driessche, Jean-Baptiste\nLespiau, Bogdan Damoc, Aidan Clark, et al. Improving"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9a9-934e-11ee-b72e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "language models by retrieving from trillions of tokens.\nInInternational conference on machine learning , pages\n2206\u20132240. PMLR, 2022.\n[Ceret al. , 2018 ]Daniel Cer, Yinfei Yang, Sheng-yi Kong,\nNan Hua, Nicole Limtiaco, Rhomni St John, Noah Con-\nstant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar,\net al. Universal sentence encoder for English. In Proceed-\nings of the 2018 conference on empirical methods in nat-\nural language processing: system demonstrations , pages\n169\u2013174, 2018."}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9aa-934e-11ee-b687-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "169\u2013174, 2018.\n[Charniak and others, 2013 ]Eugene Charniak et al. Naive\nBayes word sense induction. In Proceedings of the 2013\nConference on Empirical Methods in Natural Language\nProcessing , pages 1433\u20131437, 2013.[Chelba et al. , 2014 ]Ciprian Chelba, Tomas Mikolov, Mike\nSchuster, Qi Ge, T. Brants, Phillip Todd Koehn, and Tony\nRobinson. One billion word benchmark for measuring\nprogress in statistical language modeling. In Interspeech ,\n2014.\n[Conneau and Kiela, 2018 ]Alexis Conneau and Douwe"}


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id: 152cf9ab-934e-11ee-a783-ac198e74a6f7
encoding: FLOAT32
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{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "Kiela. SentEval: An Evaluation Toolkit for Universal Sen-\ntence Representations. In Proceedings of the Eleventh In-\nternational Conference on Language Resources and Eval-\nuation (LREC 2018) , 2018.\n[Devlin et al. , 2019 ]Jacob Devlin, Ming-Wei Chang, Ken-\nton Lee, and Kristina Toutanova. BERT: Pre-training\nof Deep Bidirectional Transformers for Language Under-\nstanding. In NAACL , 2019.\n[Dhillon et al. , 2015 ]Paramveer S Dhillon, Dean P Foster,"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9ac-934e-11ee-9d52-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "and Lyle H Ungar. Eigenwords: spectral word embed-\ndings. J. Mach. Learn. Res. , 16:3035\u20133078, 2015.\n[Faruqui et al. , 2015 ]Manaal Faruqui, Yulia Tsvetkov, Dani\nYogatama, Chris Dyer, and Noah A Smith. Sparse Over-\ncomplete Word Vector Representations. In Proceedings of\nthe 53rd Annual Meeting of the Association for Computa-\ntional Linguistics and the 7th International Joint Confer-\nence on Natural Language Processing (Volume 1: Long\nPapers) , pages 1491\u20131500, 2015."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9ad-934e-11ee-a1db-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "Papers) , pages 1491\u20131500, 2015.\n[Glimsdal and Granmo, 2021 ]Sondre Glimsdal and Ole-\nChristoffer Granmo. Coalesced Multi-Output Tsetlin Ma-\nchines with Clause Sharing. Submitted to ICML\u201923. Avail-\nable as an arXiv preprint, arXiv:2108.07594 , 2021.\n[Granmo, 2018 ]Ole-Christoffer Granmo. The Tsetlin Ma-\nchine - A Game Theoretic Bandit Driven Approach to Op-\ntimal Pattern Recognition with Propositional Logic. arXiv\npreprint arXiv:1804.01508 , 2018."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9ae-934e-11ee-97f6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "preprint arXiv:1804.01508 , 2018.\n[Harris, 1954 ]Zellig S Harris. Distributional structure.\nWord , 10(2-3):146\u2013162, 1954.\n[Huang et al. , 2020 ]Jui-Ting Huang, Ashish Sharma, Shuy-\ning Sun, Li Xia, David Zhang, Philip Pronin, Janani\nPadmanabhan, Giuseppe Ottaviano, and Linjun Yang.\nEmbedding-based retrieval in Facebook search. In Pro-\nceedings of the 26th ACM SIGKDD International Con-\nference on Knowledge Discovery & Data Mining , pages\n2553\u20132561, 2020."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9af-934e-11ee-ad0a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "2553\u20132561, 2020.\n[Kenton and Toutanova, 2019 ]Jacob Devlin Ming-\nWei Chang Kenton and Lee Kristina Toutanova. Bert:\nPre-training of deep bidirectional transformers for lan-\nguage understanding. In Proceedings of NAACL-HLT ,\npages 4171\u20134186, 2019.\n[Lample et al. , 2016 ]Guillaume Lample, Miguel Balles-\nteros, Sandeep Subramanian, Kazuya Kawakami, and\nChris Dyer. Neural Architectures for Named Entity\nRecognition. In Proceedings of NAACL-HLT , pages 260\u2013\n270, 2016."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b0-934e-11ee-b2fc-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 7, "chroma:document": "270, 2016.\n[Lauet al. , 2014 ]Jey Han Lau, Paul Cook, Diana McCarthy,\nSpandana Gella, and Timothy Baldwin. Learning word\nsense distributions, detecting unattested senses and identi-\nfying novel senses using topic models. In Proceedings of"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b1-934e-11ee-8666-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "the 52nd Annual Meeting of the Association for Compu-\ntational Linguistics (Volume 1: Long Papers) , pages 259\u2013\n270, 2014.\n[Levy et al. , 2015 ]Omer Levy, Yoav Goldberg, and Ido Da-\ngan. Improving distributional similarity with lessons\nlearned from word embeddings. Transactions of the as-\nsociation for computational linguistics , 3:211\u2013225, 2015.\n[Liet al. , 2020 ]Bohan Li, Hao Zhou, Junxian He, Mingx-\nuan Wang, Yiming Yang, and Lei Li. On the Sentence"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b2-934e-11ee-a91b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "Embeddings from Pre-trained Language Models. In Pro-\nceedings of the 2020 Conference on Empirical Methods\nin Natural Language Processing (EMNLP) , pages 9119\u2013\n9130, 2020.\n[Logeswaran and Lee, 2018 ]Lajanugen Logeswaran and\nHonglak Lee. An ef\ufb01cient framework for learning\nsentence representations. In International Conference on\nLearning Representations , 2018.\n[Mikolov et al. , 2013 ]Tomas Mikolov, Kai Chen, Gre-\ngory S. Corrado, and Jeffrey Dean. Ef\ufb01cient Estimation"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b3-934e-11ee-93ae-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "of Word Representations in Vector Space. In International\nConference on Learning Representations , 2013.\n[Muet al. , 2017 ]Jiaqi Mu, Suma Bhat, and Pramod\nViswanath. Geometry of polysemy. In 5th International\nConference on Learning Representations, ICLR 2017 ,\n2017.\n[Pennington et al. , 2014 ]Jeffrey Pennington, Richard\nSocher, and Christopher D Manning. Glove: Global vec-\ntors for word representation. In Proceedings of the 2014\nconference on empirical methods in natural language"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b4-934e-11ee-92f1-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "processing (EMNLP) , pages 1532\u20131543, 2014.\n[Peters et al. , 2018 ]Matthew E Peters, Mark Neumann, Mo-\nhit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee,\nand Luke Zettlemoyer. Deep contextualized word repre-\nsentations. arXiv preprint arXiv:1802.05365 , 2018.\n[Rastogi et al. , 2015 ]Pushpendre Rastogi, Benjamin\nVan Durme, and Raman Arora. Multiview LSA: Repre-\nsentation learning via generalized CCA. In Proceedings\nof the 2015 conference of the North American chapter of"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b5-934e-11ee-94ab-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "the Association for Computational Linguistics: human\nlanguage technologies , pages 556\u2013566, 2015.\n[Reimers and Gurevych, 2019 ]Nils Reimers and Iryna\nGurevych. Sentence-BERT: Sentence Embeddings using\nSiamese BERT-Networks. In Proceedings of the 2019\nConference on Empirical Methods in Natural Language\nProcessing and the 9th International Joint Conference on\nNatural Language Processing (EMNLP-IJCNLP) , pages\n3982\u20133992, 2019.\n[Schwartz et al. , 2020 ]Roy Schwartz, Jesse Dodge, Noah"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b6-934e-11ee-a458-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "Smith, and Oren Etzioni. Green AI. Communications of\nthe ACM , 63:54 \u2013 63, 2020.\n[Subramanian et al. , 2018 ]Anant Subramanian, Danish\nPruthi, Harsh Jhamtani, Taylor Berg-Kirkpatrick, and\nEduard Hovy. Spine: Sparse interpretable neural em-\nbeddings. In Proceedings of the AAAI Conference on\nArti\ufb01cial Intelligence , volume 32, 2018.[Vaswani et al. , 2017 ]Ashish Vaswani, Noam Shazeer, Niki\nParmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b7-934e-11ee-a9cc-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "\u0141ukasz Kaiser, and Illia Polosukhin. Attention is all you\nneed. Advances in neural information processing systems ,\n30, 2017.\n[Wang et al. , 2021 ]Kexin Wang, Nils Reimers, and Iryna\nGurevych. Tsdae: Using transformer-based sequential de-\nnoising auto-encoder for unsupervised sentence embed-\nding learning. arXiv preprint arXiv:2104.06979 , 2021.\n[Yadav et al. , 2021 ]Rohan Yadav, Lei Jiao, Ole-Christoffer\nGranmo, and Morten Goodwin. Human-Level Inter-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152cf9b8-934e-11ee-903f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\2301.00709.pdf", "page": 8, "chroma:document": "Granmo, and Morten Goodwin. Human-Level Inter-\npretable Learning for Aspect-Based Sentiment Analysis.\nInProceedings of AAAI , 2021.\n[Zhang et al. , 2020 ]Yan Zhang, Ruidan He, Zuozhu Liu,\nKwan Hui Lim, and Lidong Bing. An Unsupervised Sen-\ntence Embedding Method by Mutual Information Max-\nimization. In Proceedings of the 2020 Conference\non Empirical Methods in Natural Language Processing\n(EMNLP) , pages 1601\u20131610, 2020."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d209a-934e-11ee-8e03-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 0, "chroma:document": "Tsetlin Machine for Solving Contextual Bandit\nProblems\nRaihan Seraj\nElectrical and Computer Engineering\nMcGill University\nCanada\nraihan.seraj@mail.mcgill.caJivitesh Sharma\nCenter for Artificial Intelligence Research (CAIR)\nUniversity of Agder\nNorway\njivitesh.sharma@uia.no\nOle-Christoffer Granmo\nCenter for Artificial Intelligence Research (CAIR)\nUniversity of Agder\nNorway\nole.granmo@uia.no\nAbstract\nThis paper introduces an interpretable contextual bandit algorithm using Tsetlin"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d209b-934e-11ee-bbb7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 0, "chroma:document": "Machines, which is an inherently interpretable machine learning method that\nsolves complex pattern recognition tasks using propositional (Boolean) logic. The\nproposed bandit learning algorithm relies on straightforward bit manipulation, thus\nsimplifying computation and interpretation. We then present a mechanism for\nperforming Thompson sampling with Tsetlin Machine, given its non-parametric\nnature. Our empirical analysis shows that Tsetlin Machine as a base contextual"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d209c-934e-11ee-975a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 0, "chroma:document": "bandit learner outperforms other popular comparable base learners on eight out of\nnine datasets. We further analyze the interpretability of our learner, investigating\nhow arms are selected based on propositional expressions that model the context1.\n1 Introduction\nContextual bandits play a fundamental role in many applications involving sequential decision making,\nranging from personalized recommendations [ 19] to designing effective treatment allocation strategies"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d209d-934e-11ee-8207-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 0, "chroma:document": "in clinical trials [ 11,7]. Algorithms for contextual bandits have additionally gained significant interest\nbecause of their theoretical elegance. In brief, a decision-maker selects one of multiple bandit arms\nover a sequence of rounds, taking into account an observed context. Each round, the arm chosen\nelicits feedback in the form of a reward signal associated with the success of selecting that arm\n(such as a user purchasing a recommended product). The contextual bandit problem is particularly"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d209e-934e-11ee-91ac-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 0, "chroma:document": "intriguing because the decision-maker must maximize the expected reward in as few rounds as\npossible, trading exploitation against exploration to identify the optimal arm.\nIn this paper, we recast the Tsetlin Machine ( TM) [14] as a contextual bandit algorithm and study\nthe resulting scheme empirically. TMis a recent machine learning approach to pattern recognition\nthat employs a team of non-contextual bandit algorithms, in the form of Tsetlin automata [ 27], to"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d209f-934e-11ee-8879-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 0, "chroma:document": "learn patterns expressed in propositional logic. TMs have been shown to obtain competitive accuracy,\nmemory footprint, and learning speed on several benchmark datasets [ 3,18]. They have been\nparticularly successful in natural language processing, including explainable aspect-based sentiment\nanalysis [ 28] and robust text classification with ANDed word negations [ 29]. Being based on finite\nstate automata, they further support Markov chain-based convergence analysis [30]. Leveraging the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a0-934e-11ee-b6a2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 0, "chroma:document": "non-linear pattern recognition capability of TMs, our proposed scheme thus addresses the contextual\n1The code is available online on: github\n36th Conference on Neural Information Processing Systems (NeurIPS 2022)."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a1-934e-11ee-893c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 1, "chroma:document": "bandit problem using a team of non-contextual bandit algorithms.\nAlthough existing algorithms for contextual bandits provide theoretical guarantees, they are either\ndifficult to interpret or make assumptions that limit their usability in real-world settings. Motivated\nby these limitations, our work contributes as follows:\n\u2022We investigate TMas a base learner for contextual bandit problems and empirically demon-\nstrate its effectiveness compared to other popular comparable base learners."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a2-934e-11ee-856a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 1, "chroma:document": "\u2022We propose how the Thompson sampling scheme can be leveraged by TMs through boot-\nstrapping [12].\n\u2022We provide interpretability analysis that shows how algorithms using TMs select arms based\non propositional expressions of context features.\nTo the best of our knowledge, this is the first reported work on learning arm selection strategies for\ncontextual bandits expressed in propositional logic. The paper is organized as follows. Section 2 and"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a3-934e-11ee-b7ce-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 1, "chroma:document": "3 present the problem formulation and introduces the TMalgorithm, respectively. In Section 4, we\nshow how TMcan be used as a base learner for contextual bandits. The numerical results and the\ninterpretability analysis are then presented in Section 5 and 7, respectively. We conclude the paper\nand provide directions for future work in Section 9. A literature overview on contextual bandits and\ndifferent algorithms for solving them can be found in Appendix A.\n2 Problem Formulation"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a4-934e-11ee-baac-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 1, "chroma:document": "2 Problem Formulation\nWe consider an online stochastic contextual bandit setup where at time ta context-reward pair denoted\nby(st, rt)is sampled independently from past data distribution D. Here, st\u2208 S represents an M\ndimensional context vector and rt= ((rt(1), . . . , r t(K))\u2208 {0,1}Kis the reward vector for K\npossible actions. The learner chooses an arm ut\u2208 {1, . . . , K }after observing the context stand"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a5-934e-11ee-9cbe-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 1, "chroma:document": "receives a reward rt(ut)for the chosen arm. The objective of the learner is to perform a sequence of\nactions in order to minimize the cumulative expected regret given by\nRegret =EhTX\nt=1\u0010\nrt(\u03c0\u2217(st))\u2212rt(ut)\u0011i\n. (1)\nHere, \u03c0\u2217= arg max\n\u03c0\u2208\u03a0E(s,r)\u223cD[r(\u03c0(s))], where \u03a0denotes the set of large (possibly infinite) policies\nand\u03c0:S 7\u2192 { 1, . . . , K }. For our analysis we consider maximizing the expected total reward which\nis equivalent to minimizing regret.\nAction 1 Action 2\nPenatly Reward"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a6-934e-11ee-a2ac-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 1, "chroma:document": "Action 1 Action 2\nPenatly Reward\nFigure 1: A two-action Tsetlin Automaton with 2Nstates.\n3 Tsetlin Machine\nStructure. ATMin its simplest form takes a feature vector x= [x1, x2, . . . , x o]\u2208 {0,1}oofo\npropositional values as input and assigns the vector a class \u02c6y\u2208 {0,1}. To minimize classification\nerror, the TM produces nself-contained patterns. In brief, the input vector xprovides the literal set"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a7-934e-11ee-acc4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 1, "chroma:document": "L={l1, l2, . . . , l 2o}={x1, x2, . . . , x o,\u00acx1,\u00acx2, . . . ,\u00acxo}, consisting of the input features and\ntheir negations. By selecting subsets Lj\u2286Lof the literals, the TM can build arbitrarily complex\npatterns, ANDing the selected literals to form conjunctive clauses:\nCj(x) =^\nlk\u2208Ljlk. (2)\n2"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a8-934e-11ee-9e75-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 2, "chroma:document": "Input  \nType I feedback Type I feedbackinclude exclude include exclude\n1 N N+1 2 N1 N N+1 2 N\nTAstransition\nStep 1Step tFigure 2: TM learning dynamics for an XOR-gate training sample, with input ( x1= 0, x2= 1) and\noutput target y= 1.\nAbove, j\u2208 {1,2, . . . , n }refers to a particular clause Cjandk\u2208 {1,2, . . . , 2o}refers to a particular\nliteral lk. As an example, the clause Cj(x) =x1\u2227 \u00acx2consists of the literals Lj={x1,\u00acx2}and\nevaluates to 1when x1= 1andx2= 0."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20a9-934e-11ee-abc4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 2, "chroma:document": "evaluates to 1when x1= 1andx2= 0.\nThe TM assigns one Tsetlin Automata ( TA) per literal lkper clause Cjto build the clauses. The TA\nassigned to literal lkof clause Cjdecides whether lkisExcluded orIncluded inCj. Figure 1 depicts\na two-action TAwith2Nstates. For states 1toN, the TAperforms action Exclude (Action 1), while\nfor states N+ 1to2Nit performs action Include (Action 2). As feedback to the action performed,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20aa-934e-11ee-86af-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 2, "chroma:document": "the environment responds with either a Reward or a Penalty. If the TA receives a Reward, it moves\ndeeper into the side of the action. If it receives a Penalty, it moves towards the middle and eventually\nswitches action.\nWith nclauses and 2oliterals, we get n\u00d72oTAs. We organize the states of these in an n\u00d72o\nmatrix A= [aj\nk]\u2208 {1,2, . . . , 2N}n\u00d72o. We will use the function g(\u00b7)to map the automaton\nstateaj\nkto Action 0(Exclude ) for states 1toNand to Action 1(Include ) for states N+ 1to2N:\ng(aj"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ab-934e-11ee-abc6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 2, "chroma:document": "g(aj\nk) =aj\nk> N . We can connect the states aj\nkof the TAs assigned to clause Cjwith its composition\nas follows:\nCj(x) =^\nlk\u2208Ljlk=2o^\nk=1h\ng(aj\nk)\u21d2lki\n. (3)\nHere, lkis one of the literals and aj\nkis the state of its TAin clause Cj. The logical imply operator \u21d2\nimplements the Exclude /Include action. That is, the imply operator is always 1ifg(aj\nk) = 0 (Exclude ),\nwhile if g(aj\nk) = 1 (Include ) the truth value is decided by the truth value of the literal. The complexity"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ac-934e-11ee-93b7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 2, "chroma:document": "of the algorithm is proportional (linear) to the number of features (input bits). E.g., doubling the\nnumber of features doubles the number of computations[1].\nClassification. Classification is performed as a majority vote. The odd-numbered half of the clauses\nvote for class \u02c6y= 1and the even-numbered half vote for \u02c6y= 0:\n\u02c6y= 0\u2264n\u22121X\nj=1,3,...2o^\nk=1h\ng(aj\nk)\u21d2lki\n\u2212nX\nj=2,4,...2o^\nk=1h\ng(aj\nk)\u21d2lki\n. (4)\nAs such, the odd-numbered clauses have positive polarity, while the even-numbered ones have"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ad-934e-11ee-9c79-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 2, "chroma:document": "negative polarity. As an example, consider the input vector x= [0,1]in the lower part of Figure 2.\n3"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ae-934e-11ee-968c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "The figure depicts two clauses of positive polarity, C1(x) =x1\u2227 \u00acx2andC3(x) =\u00acx1\u2227 \u00acx2\n(the negative polarity clauses are not shown). Both of the clauses evaluate to zero, leading to class\nprediction \u02c6y= 1.\nINPUTCLAUSE 1 0\nLITERAL 1 0 1 0\nINCLUDE LITERALP(R EWARD )s\u22121\nsNA 0 0\nP(INACTION )1\nsNAs\u22121\nss\u22121\ns\nP(P ENALTY ) 0 NA1\ns1\ns\nEXCLUDE LITERALP(R EWARD ) 01\ns1\ns1\ns\nP(INACTION )1\nss\u22121\nss\u22121\nss\u22121\ns\nP(P ENALTY )s\u22121\ns0 0 0\nTable 1: Type I FeedbackINPUTCLAUSE 1 0\nLITERAL 1 0 1 0"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20af-934e-11ee-8a34-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "LITERAL 1 0 1 0\nINCLUDE LITERALP(R EWARD ) 0 NA 0 0\nP(INACTION )1.0 NA 1.0 1.0\nP(P ENALTY ) 0 NA 0 0\nEXCLUDE LITERALP(R EWARD ) 0 0 0 0\nP(INACTION )1.0 0 1.0 1.0\nP(P ENALTY ) 0 1.0 0 0\nTable 2: Type II Feedback\nLearning. The upper part of Figure 2 illustrates learning. A TM learns online, processing one\ntraining example (x, y)at a time. Based on (x, y), the TM rewards and penalizes its TAs, which\namounts to incrementing and decrementing their states. There are two kinds of feedback: Type"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b0-934e-11ee-a456-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "I Feedback produces frequent patterns and Type II Feedback increases the discrimination power\nof the patterns. Type I feedback is given stochastically to clauses with positive polarity when\ny= 1 and to clauses with negative polarity when y= 0. Conversely, Type II Feedback is\ngiven stochastically to clauses with positive polarity when y= 0 and to clauses with negative\npolarity when y= 1. The probability of a clause being updated is based on the vote sum v:\nv=Pn\u22121\nj=1,3,...V2o\nk=1h\ng(aj\nk)\u21d2lki"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b1-934e-11ee-a504-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "v=Pn\u22121\nj=1,3,...V2o\nk=1h\ng(aj\nk)\u21d2lki\n\u2212Pn\nj=2,4,...V2o\nk=1h\ng(aj\nk)\u21d2lki\n. The voting error is calculated\nas:\n\u03f5=\u001aT\u2212v y = 1\nT+v y = 0.(5)\nHere, Tis a user-configurable voting margin yielding an ensemble effect. The probability of updating\neach clause is P(Feedback) =\u03f5\n2T. After random sampling from P(Feedback) has decided which\nclauses to update, the following TAstate updates can be formulated as matrix additions, subdividing\nType I Feedback into feedback Type Ia and Type Ib:\nA\u2217"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b2-934e-11ee-a242-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "A\u2217\nt+1=At+FII+FIa\u2212FIb. (6)\nHere, At= [aj\nk]\u2208 {1,2, . . . , 2N}n\u00d72ocontains the states of the TAs at time step tandA\u2217\nt+1\ncontains the updated state for time step t+ 1(before clipping). The matrices FIa\u2208 {0,1}n\u00d72oand\nFIb\u2208 {0,1}n\u00d72ocontains Type I Feedback. A zero-element means no feedback and a one-element\nmeans feedback. As shown in Table 1, two rules govern Type I feedback:\n\u2022Type Ia Feedback is given with probabilitys\u22121\nswhenever both clause and literal are"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b3-934e-11ee-a17c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "swhenever both clause and literal are\n1-valued.2It penalizes Exclude actions and rewards Include actions. The purpose is to\nremember and refine the patterns manifested in the current input x. This is achieved by\nincreasing selected TAstates. The user-configurable parameter scontrols pattern frequency,\ni.e., a higher sproduces less frequent patterns.\n\u2022Type Ib Feedback is given with probability1\nswhenever either clause or literal is 0-valued."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b4-934e-11ee-bed8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "swhenever either clause or literal is 0-valued.\nThis feedback rewards Exclude actions and penalizes Include actions to coarsen patterns,\ncombating overfitting. Thus, the selected TA states are decreased.\nThe matrix FII\u2208 {0,1}n\u00d72ocontains Type II Feedback to the TAs, given per Table 2.\n\u2022Type II Feedback penalizes Exclude actions to make the clauses more discriminative,\ncombating false positives. That is, if the literal is 0-valued and the clause is 1-valued, TA"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b5-934e-11ee-96c5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 3, "chroma:document": "states below N+ 1are increased. Eventually the clause becomes 0-valued for that particular\ninput, upon inclusion of the 0-valued literal.\nThe final updating step for training example (x, y)is to clip the state values to make sure that they\nstay within value 1and2N:\nAt+1=clip\u0000\nA\u2217\nt+1,1,2N\u0001\n. (7)\n2Note that the probabilitys\u22121\nsis replaced by 1when boosting true positives.\n4"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b6-934e-11ee-9154-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "For example, both of the clauses in Figure 2 receives Type I Feedback over several training examples,\nmaking them resemble the input associated with y= 1.\nWeighted Tsetlin Machine: The learning of weights is based on increasing the weight of clauses\nthat receive Type Ia feedback (due to true positive output) and decreasing the weight of clauses that\nreceive Type II feedback (due to false positive output). The overall rationale is to determine which"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b7-934e-11ee-b76b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "clauses are inaccurate and thus must team up to obtain high accuracy as a team (low weight clauses),\nand which clauses are sufficiently accurate to operate more independently (high weight clauses). The\nweight updating procedure is summarized in Algorithm 1. Here, wiis the weight of clause Ciat the\nnthtraining round (ignoring polarity to simplify notation). The first step of a training round is to\ncalculate the clause output as per Equation 2. The weight of a clause is only updated if the clause"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b8-934e-11ee-b4c3-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "output Ciis 1 and the clause has been selected for feedback ( Pi= 1). Then the polarity of the clause\nand the class label ydecide the type of feedback given. That is, like a regular TM, positive polarity\nclauses receive Type Ia feedback if the clause output is a true positive and Type II feedback if the\nclause output is a false positive. For clauses with negative polarity, the feedback types switch roles."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20b9-934e-11ee-9178-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "When clauses receive Type Ia or Type II feedback, their weights are updated accordingly. We use the\nstochastic searching on the line (SSL) automaton to learn appropriate weights. SSL is an optimization\nscheme for unknown stochastic environments pioneered by Oommen [ 22]. The goal is to find an\nunknown location \u03bb\u2217within a search interval [0,1]. In order to find \u03bb\u2217, the only available information\nfor the Learning Mechanism (LM) is the possibly faulty feedback from its attached environment E."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ba-934e-11ee-812a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "In SSL, the search space \u03bbis discretized into Npoints, {0,1/N,2/N, ..., (N\u22121)/N,1}with N being\nthe discretization resolution. During the search, the LM has a location \u03bb\u2208 {0,1/N,2/N, ..., (N\u2212\n1)/N,1}, and can freely move to the left or to the right from its current location. The environment E\nprovides two types of feedback: E= 1is the environment suggestion to increase the value of \u03bbby\none step, and E= 0is the environment suggestion to decrease the value of \u03bbby one step. The next"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20bb-934e-11ee-aef9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "location of \u03bb, i.e.\u03bbn+1, can thus be expressed as follows:\n\u03bbn+1=\u001a\u03bbn+ 1/N, ifEn= 1,\n\u03bbn\u22121/N, ifEn= 0.(8)\n\u03bbn+1=\u001a\u03bbn,if\u03bbn= 1andEn= 1,\n\u03bbn,if\u03bbn= 0andEn= 0.(9)\nAsymptotically, the learning mechanics is able to find a value arbitrarily close to \u03bb\u2217when N\u2192 \u221e\nandn\u2192 \u221e . In our case, the search space of clause weights is [0,\u221e], so we use resolution N= 1,\nwith no upper bound for \u03bb. Accordingly, we operate with integer weights. As described in Algorithm"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20bc-934e-11ee-8cce-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "1, if the clause output is a true positive, we simply increase the weight by 1. Conversely, if the clause\noutput is a false positive, we decrease the weight by 1.\nBy following the above procedure, the goal is to make low precision clauses team up by giving\nthem low weights, so that they together can reach the summation target T. By teaming up, precision\nincreases due to the resulting ensemble effect. Clauses with high precision, however, gets a higher"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20bd-934e-11ee-8cf7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "weight, allowing them to operate more independently.\nThe above weighting scheme has several advantages. First of all, increment and decrement operations\non integers are computationally less costly than multiplication based updates of real-valued weights.\nAdditionally, a clause with an integer weight can be seen as multiple copies of the same clause,\nmaking it more interpretable than real-valued weighting, as studied in the next section. Additionally,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20be-934e-11ee-92a2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "clauses can be turned completely off by setting their weights to 0if they do not contribute positively\nto the classification task.\n4 Contextual Bandits with Tsetlin Machines\nWe use TMas a contextual bandit learner that learns a mapping from context to actions incrementally\nwith streaming data. Since both inputs, patterns and outputs of TM are represented as bits [ 14],\neachMdimensional context stis binarized with appropriate number of bits. This results in a B"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20bf-934e-11ee-ac9f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "dimensional binarized context with B\u2265Mwhich is fed to each TMlearner. We outline appropriate\nchoices of bits for binarization for different datasets in the next section. The TMlearner, being\nnon-parametric, has the advantage that it makes few or no assumption about the underlying functions\nto be learned; hence it is adaptive and exhibit high degree of flexibility. Another popular non-\nparametric learner for contextual bandits is the decision tree, which has been thoroughly studied in"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c0-934e-11ee-9600-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 4, "chroma:document": "the literature [ 12,26,13]. The TM, in contrast, learns a linear combination of conjunctive clauses\n5"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c1-934e-11ee-995d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 5, "chroma:document": "Algorithm 1 Complete WTM learning process\n1:Input: Training data batch (B, x, y )\u25b7B\u22651\n2:Initialize: Random initialization of TAs\n3:Begin: nthtraining round\n4:fori= 1, ..., m do ifpi= 1\n5: if(y= 1andiis odd) or(y= 0andiis even) then\n6: ifci= 1then\n7: wi\u2190wi+ 1\n8: forfeature k= 1, ...,2odo\n9: iflk= 1then\n10: Type Ia Feedback\n11: else:\n12: Type Ib Feedback\n13: end if\n14: end for\n15: else:\n16: wi\u2190wi\u25b7[No Change]\n17: Type Ib Feedback\n18: end if\n19: else: (y= 1andiis even) or(y= 0andiis odd)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c2-934e-11ee-9663-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 5, "chroma:document": "19: else: (y= 1andiis even) or(y= 0andiis odd)\n20: ifci= 1then\n21: ifwi>0then\n22: wi\u2190wi\u22121\n23: end if\n24: forfeature k= 1, ...,2odo\n25: iflk= 0then\n26: Type II Feedback\n27: else:\n28: Inaction\n29: end if\n30: end for\n31: else:\n32: wi\u2190wi\u25b7[No Change]\n33: Inaction\n34: end if\n35: end if\n36:end for\nin propositional logic by producing decision rules similar to the branches in decision trees [ 1], with\nthe added advantage of being memory and energy efficient, computationally simple and not having"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c3-934e-11ee-b237-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 5, "chroma:document": "a tendency to overfit the training data [ 18]. We consider two contextual bandit learning algorithms\nusing TM: (i) TM with epsilon greedy arm selection; and (ii) TM with Thompson sampling.\nEpsilon greedy TM:The exploration-exploitation trade-off is a fundamental problem in learning to\nmake decisions under uncertainty. In a multi-armed bandit setting, the \u03b5\u2212greedy algorithm is one\nof the simplest ones. The learner either chooses the empirically best arm with probability (1\u2212\u03b5)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c4-934e-11ee-9290-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 5, "chroma:document": "(exploitation) or a random arm with probability \u03b5(exploration). In the contextual bandit setting,\ngiven a set of contexts, the learner chooses to select the current empirically best arm with some high\nprobability, maximizing immediate rewards. Otherwise, it selects a randomized arm with the hope\nof improving future rewards given the context. Variations of epsilon greedy algorithms have been\nexplored in the literature, including decaying the \u03b5parameter and eventually dropping the probability"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c5-934e-11ee-805a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 5, "chroma:document": "of choosing a random arm to zero. In our setting, we consider a fixed small value of \u03b5for each TM\nlearner associated with the arm.\nTMwith Thompson sampling: We now show how Thompson sampling can be achieved with TM.\nOur approach is similar to that presented in the Tree Bootstrap Algorithm in [ 12]. Since a TM\nlearner is non-parametric, we use bootstrapping to simulate the behavior of sampling from a posterior"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c6-934e-11ee-85bb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 5, "chroma:document": "distribution. At each time instant t,Ncontext reward pairs are bootstrapped from the Dt,uwith\nreplacement, where Dt,urepresents the set of observations (context reward pairs) for arm uand\nN=|Dt,u|. ATMlearner is fitted to each of these bootstrapped datasets and at each time t, the\n6"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c7-934e-11ee-bfb0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "bootstrapping algorithm selects the arm utthat has the maximum probability of success \u02c6p. For ease\nof exposition, we outline Thompson Sampling with bootstrapping for TMs in Algorithm 2.\nAlgorithm 2 Thompson Sampling with TM\nfort= 1 toTdo\nget context st\nforu= 1, . . . , K do\nSample bootstrapped dataset \u02dcDt,uof size Nfrom Dt,uwith replacement.\nFit Tsetlin Machine TM t,uto\u02dcDt,u\nend for\nChoose action ut= arg max\nu(\u02c6p(TM t,u, st))\nUpdate Dt,utwith(st, rt,ut)\nend for\n5 Empirical Analysis"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c8-934e-11ee-81fe-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "end for\n5 Empirical Analysis\nGiven the interactive aspect of contextual bandit algorithms, it is often difficult to evaluate them on\nreal datasets except for a handful number of tasks [ 6]. Therefore, we use standard supervised learning\nclassification datasets for our evaluation. We consider both binary as well as multiclass classification\ndatasets. Each unique label in these datasets is considered as an arm of the equivalent contextual"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20c9-934e-11ee-8936-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "bandit problem, where a separate learner associated with each arm is trained independently. Our\nnumerical analysis contrasts the performance of TMwith\u03b5\u2212greedy arm selection and TMwith\nThompson sampling against Tree Bootstrap [ 12], Linear UCB [ 8], Neutral Network ( \u03b5\u2212greedy) and\nLogistic regression ( \u03b5\u2212greedy). For our analysis we consider two scenarios described as follows.\nScenario 1: In this scenario, we consider five standard supervised learning classification datasets from"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ca-934e-11ee-9790-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "UCI machine learning repository [ 10]:Iris, Breast Cancer Wisconsin (Diagnostic), Adult, Covertype ,\nandStatlog (Shuttle) . The response variables for each of these datasets are, Iris types, Diagnosis,\nOccupation , and Covertypes , respectively, where as for the Statlog(Shuttle) dataset, we consider the\nlast column as the response variable. Additionally we considered two other classification datasets:"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20cb-934e-11ee-a92e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "MNIST [9] and Noisy XOR , where the respective response variables are digits and the XOR output .\nThe noisy XOR dataset contains the XOR operation of 12bit input, where the output is flipped with\nprobability 0.4. These classification datasets are converted into a contextual bandit problem where\nthe learner receives a reward of +1for correctly identifying the target value and 0for incorrect\nclassification. The datasets are processed by removing entries with missing values."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20cc-934e-11ee-b948-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "Scenario 2: In this scenario, we consider two datasets: Movielens 100 K [15] and Simulated\nArticle [23]. Both these datasets simulate a recommender system. The reward function for the\nMovielens dataset is the user\u2019s rating for a particular movie while for Simulated Article dataset, a\nreward of +1is received if the recommended article is clicked by the user. The Movielens dataset\nis preprocessed, so that top 10 rated movies are selected. One important takeaway from these two"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20cd-934e-11ee-89fa-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "datasets is that only the reward function for correct recommendation is provided. For instance the\nSimulated Article dataset provides recommended articles that has been clicked by the user. Similarly,\nthe Movielens dataset provides user movie ratings, where the users only rated the movies they have\nwatched. In such circumstances, the rewards are partial and sparse since there are no information on"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ce-934e-11ee-8b44-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "the ratings of all movies or user click information for each article. To circumvent this, we perform\nsingular value decomposition (SVD) for both the datasets which is a popular approach in collaborative\nfiltering [ 17]. For the Movielens dataset, let Si,jbe the rating of user ion movie j. Low rank SVD\nof this matrix results in two matrices WandX, where S\u2248W\u2217X. The ithrow of Wrepresents\nthe context features for each user iand the rows of Xrepresents the actions or the movie to be"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20cf-934e-11ee-8948-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "recommended. The reward for recommending a movie jto user iis then the dot product of the\ncorresponding row of WiandXj. We perform SVD for Simulated Article dataset in a similar manner.\nIn order to obtain binarized rewards for the actions, the maximum reward corresponding to an arm\nthat yields a reward of 1, and the rest a reward of 0. For our analysis we consider rank 10for both the\nMovielens and the Simulated Article dataset."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d0-934e-11ee-ac16-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 6, "chroma:document": "Movielens and the Simulated Article dataset.\nFor both the scenarios we use appropriate binarization before fitting the TMlearners. Details of\nbinarization and parameter choices are provided in Appendix B. The results for both the scenarios\nare presented in Figures 3. These results show that for eight datasets, TMwith Thompson sampling\noutperforms all other algorithms except for Linear UCB on Covertype. Further, TMwith\u03b5\u2212greedy\n7"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d1-934e-11ee-b3a1-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "Figure 3: Performance comparison of TM with different base learners.\n(a) Iris\n0 200 400 600 800 1000\nRounds0.30.40.50.60.70.80.91.0Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n (b) Breast Cancer\n0 200 400 600 800 1000\nRounds0.40.50.60.70.80.91.0Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n (c) MNIST\n0 200 400 600 800 1000"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d2-934e-11ee-95a4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "(c) MNIST\n0 200 400 600 800 1000\nRounds0.00.10.20.30.40.50.60.7Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n(d) Adult\n0 500 1000 1500 2000\nRounds0.00.20.40.60.8Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n (e) Covertype\n0 500 1000 1500 2000\nRounds0.10.20.30.40.50.6Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d3-934e-11ee-8a62-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "Tree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n (f) MovieLens\n0 200 400 600 800 1000\nRounds0.00.10.20.30.40.50.60.70.8Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n(g) Statlog\n0 500 1000 1500 2000\nRounds0.20.40.60.81.0Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n (h) Noisy XOR"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d4-934e-11ee-a967-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "Neural Network -greedy\n (h) Noisy XOR\n0 200 400 600 800 1000\nRounds0.30.40.50.60.70.80.91.0Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\n (i) Simulated Article\n0 500 1000 1500 2000\nRounds0.00.20.40.60.81.0Mean RewardLinear UCB\nTree bootstrap\nTM Thompson Sampling\nTM -greedy\nLogistic Regression -greedy\nNeural Network -greedy\nprovides competitive performance when compared with popular contextual bandit algorithms in the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d5-934e-11ee-bfa2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "literature. Also, the TMlearner learns at a faster rate compared to other learners. These experiments\nwere performed with 10independent runs for each dataset. The average result across the independent\nruns are reported.\n6 Choice of Binarization Parameters\nThe working principle of TM relies on binarization of the context features. We use gap-based\nthresholding as described in [ 2], each threshold yielding one bit. The binarization is performed with"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d6-934e-11ee-9440-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "the parameter which controls the maximum number of bits allowed to represent each feature. We\nvaried the number of bits allowed per feature for binarization and performed an ablation study across\nthese datasets, in order to observe the effects of sub-optimal binarization. Our analysis shows that\nTM as a base learner is robust to the choice of maximum bits per feature allowable for binarization.\nDetailed evaluation results of TM with epsilon greedy and TM with Thompson sampling are provided"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d7-934e-11ee-96f7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "in Appendix C.\n7 Interpretability\nUnlike neural networks and some other complex machine learning approaches, one of the advantages\nofTMs is that they produce propositional logic expressions. These are in flat AND-form, which have\nproven to be human interpretable [ 21]. As explained in Section 3, each propositional expression is a\nconjunctive clause, consisting of features, in their original or negated forms, interacting with each"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d8-934e-11ee-91ac-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 7, "chroma:document": "other using logical AND operations. These clauses can form a simplified representation of the arm\nselection policy by combining them into a single Disjunctive Normal Form (DNF) expression. Since\n8"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20d9-934e-11ee-94c4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "clauses are assigned to each arm of the multi-armed contextual bandit problem, we produce a single\nDNF expression for each arm. These DNF expressions are propositional logic expressions made up\nof the context. The TMis able to produce these interpretations demonstrating how it interprets the\ncontext with respect to each arm.\nHere, we show the simplified propositional expressions for each arm, obtained from TMs trained\nwith Thompson Sampling on the Iris dataset:\nArm-1: x10\u2228x14\u2228x15\u2228x3"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20da-934e-11ee-a3b8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "Arm-1: x10\u2228x14\u2228x15\u2228x3\nArm-2: \u00acx1\u2228x12\u2228 \u00acx13\u2228 \u00acx14\u2228x16\u2228x8\u2228 \u00acx9\u2228(x10\u2227x11\u2227x15\u2227x2\u2227 \u00acx4\u2227x5\u2227x6\u2227\n\u00acx7)\u2228(x10\u2227x11\u2227x15\u2227x3\u2227 \u00acx4\u2227x5\u2227 \u00acx7)\nArm-3: \u00acx10\u2228 \u00acx11\u2228 \u00acx15\u2228(x1\u2227 \u00acx12\u2227x13\u2227x14\u2227 \u00acx16\u2227x2\u2227x3\u2227x5\u2227 \u00acx8\u2227x9)\u2228\n(\u00acx12\u2227 \u00acx16\u2227x4)\nThe next set of propositional expressions represent the arms for the Simulated Article dataset:\nArm-1: \u00acx4\u2228(x1\u2227 \u00acx10\u2227 \u00acx11\u2227x12\u2227x13\u2227x14\u2227x15\u2227x16\u2227x17\u2227x18\u2227x19\u2227x2\u2227x20\u2227\n\u00acx21\u2227x3\u2227x5\u2227x6\u2227x7\u2227x8\u2227x9)\nArm-2: (x1\u2227 \u00acx10\u2227 \u00acx11\u2227x12\u2227x13\u2227x14\u2227x15\u2227x16\u2227x17\u2227x18\u2227x19\u2227x2\u2227x20\u2227 \u00acx21\u2227"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20db-934e-11ee-857d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "x3\u2227x4\u2227x5\u2227x6\u2227x7\u2227x8\u2227x9)\u2228(\u00acx1\u2227x10\u2227x11\u2227x12\u2227x13\u2227x14\u2227x15\u2227x16\u2227\nx17\u2227x18\u2227x19\u2227x2\u2227 \u00acx20\u2227 \u00acx21\u2227x3\u2227x4\u2227x5\u2227x6\u2227 \u00acx7\u2227x8\u2227x9)\nArm-3: (x1\u2227 \u00acx10\u2227 \u00acx11\u2227x12\u2227x13\u2227x14\u2227x15\u2227x16\u2227x17\u2227x19\u2227x2\u2227x20\u2227 \u00acx21\u2227x3\u2227\nx4\u2227x5\u2227x6\u2227x7\u2227x9)\u2228(\u00acx11\u2227x12\u2227x13\u2227x14\u2227x16\u2227x17\u2227x18\u2227x19\u2227x2\u2227x20\u2227\n\u00acx21\u2227x3\u2227x4\u2227x5\u2227x8\u2227x9)\nArm-4: x10\u2227x11\u2227x12\u2227x13\u2227x14\u2227x15\u2227x16\u2227x17\u2227x18\u2227x19\u2227x2\u2227x3\u2227x4\u2227x5\u2227x6\u2227\nx8\u2227x9\u2227 \u00acx1\u2227 \u00acx20\u2227 \u00acx21\u2227 \u00acx7\nThe propositional expressions shown above exhibit how the TMlearns interactions between proposi-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20dc-934e-11ee-a8ad-ac198e74a6f7
encoding: FLOAT32
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "tional input features (literals xi). As seen, the interactions are learned with logical AND operations\nbetween original and negated features. These learnt interactions (clauses) are combined with the\nlogical ORoperation (or by addition for T > 1). From the propositional expressions, we can see\nwhich arm is selected by TMby simply plugging in the input values. An arm is selected if its\npropositional expression evaluates to True (or it obtains the largest net sum for T > 1). This is"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20dd-934e-11ee-8789-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "arguably an important advantage of TMs \u2014 a trained TMmodel can be reduced down to simple\npropositional expressions. Of course, as such, any machine learning algorithm can be converted into\nlogical form. However, the complexity of such representations can be immense for the competitive\nclassic machine learning models, which the TMstill outperforms by a large margin [ 24]. Note that,\nalthough the length of expressions can increase with the increase in dimensionality of data, there are"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20de-934e-11ee-b2d5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "simple ways of reducing the length of expressions in O(n)time [20].\n8 Limitations\nOne of the limitations of using TMis that it requires the contexts to be binarized. While such\nbinarization leads to a loss of information, our empirical analysis shows that TMwith Thompson\nsampling performs substantially better than the other evaluated learners on the majority of the datasets\nand TM learners are quite robust to sub-optimal binarization. While our work does not provide any"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20df-934e-11ee-95bd-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "theoretical guarantees on performance or regret bounds for TM learners, we believe the latter will\nform an interesting direction of research.\n9 Conclusion\nIn this paper, we presented an interpretable and practical contextual bandit learner using Tsetlin\nMachine (TM). Our analysis showed that TMas a contextual bandit learner provides competitive\nperformance compared to other popular contextual bandit algorithms. We then presented how"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e0-934e-11ee-b620-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 8, "chroma:document": "Thompson sampling can be implemented using TM, where our approach is derived from the Tree\nBootstrap algorithm. Finally, we perform interpretability analysis where the arm selection strategy\ncan be characterized by a propositional function of the contexts.. Having such promising empirical\nresults, we aim at providing theoretical performance guarantees for algorithms with TMas future\nwork.\n9"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e1-934e-11ee-b043-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "References\n[1]K Darshana Abeyrathna, Ole-Christoffer Granmo, and Morten Goodwin. Extending the tsetlin\nmachine with integer-weighted clauses for increased interpretability. IEEE Access , 9:8233\u20138248,\n2021.\n[2]K Darshana Abeyrathna, Ole-Christoffer Granmo, Xuan Zhang, and Morten Goodwin. Adaptive\ncontinuous feature binarization for tsetlin machines applied to forecasting dengue incidences in\nthe philippines. In 2020 IEEE Symposium Series on Computational Intelligence (SSCI) , pages"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e2-934e-11ee-ac0b-ac198e74a6f7
encoding: FLOAT32
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "2084\u20132092. IEEE, 2020.\n[3]Kuruge Darshana Abeyrathna, Bimal Bhattarai, Morten Goodwin, Saeed Rahimi Gorji, Ole-\nChristoffer Granmo, Lei Jiao, Rupsa Saha, and Rohan K Yadav. Massively parallel and asyn-\nchronous tsetlin machine architecture supporting almost constant-time scaling. In International\nConference on Machine Learning , pages 10\u201320. PMLR, 2021.\n[4]Shipra Agrawal and Navin Goyal. Thompson sampling for contextual bandits with linear"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e3-934e-11ee-9c2b-ac198e74a6f7
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metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "payoffs. In International Conference on Machine Learning , pages 127\u2013135. PMLR, 2013.\n[5]Robin Allesiardo, Rapha \u00a8el F\u00b4eraud, and Djallel Bouneffouf. A neural networks committee for\nthe contextual bandit problem. In International Conference on Neural Information Processing ,\npages 374\u2013381. Springer, 2014.\n[6]Alberto Bietti, Alekh Agarwal, and John Langford. A contextual bandit bake-off. Journal of\nMachine Learning Research , 22(133):1\u201349, 2021."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e4-934e-11ee-8989-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "Machine Learning Research , 22(133):1\u201349, 2021.\n[7]Djallel Bouneffouf, Irina Rish, and Charu Aggarwal. Survey on applications of multi-armed\nand contextual bandits. In 2020 IEEE Congress on Evolutionary Computation (CEC) , pages\n1\u20138. IEEE, 2020.\n[8]Wei Chu, Lihong Li, Lev Reyzin, and Robert Schapire. Contextual bandits with linear payoff\nfunctions. In Proceedings of the Fourteenth International Conference on Artificial Intelligence"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e5-934e-11ee-901d-ac198e74a6f7
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metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "and Statistics , pages 208\u2013214. JMLR Workshop and Conference Proceedings, 2011.\n[9]Li Deng. The mnist database of handwritten digit images for machine learning research. IEEE\nSignal Processing Magazine , 29(6):141\u2013142, 2012.\n[10] Dheeru Dua and Casey Graff. UCI machine learning repository, 2017.\n[11] Audrey Durand, Charis Achilleos, Demetris Iacovides, Katerina Strati, Georgios D Mitsis,\nand Joelle Pineau. Contextual bandits for adapting treatment in a mouse model of de novo"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e6-934e-11ee-a39c-ac198e74a6f7
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metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "carcinogenesis. In Machine learning for healthcare conference , pages 67\u201382. PMLR, 2018.\n[12] Adam N. Elmachtoub, Ryan McNellis, Sechan Oh, and Marek Petrik. A practical method\nfor solving contextual bandit problems using decision trees. In Gal Elidan, Kristian Kersting,\nand Alexander T. Ihler, editors, Proceedings of the Thirty-Third Conference on Uncertainty\nin Artificial Intelligence, @articleTsetlin1961, author = Tsetlin, Michael Lvovitch, journal ="}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e7-934e-11ee-a6a8-ac198e74a6f7
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metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "Avtomat. i Telemekh, number = 10, pages = 1345\u20131354, title = On behaviour of finite automata\nin random medium, volume = 22, year = 1961 UAI 2017, Sydney, Australia, August 11-15, 2017 .\nAUAI Press, 2017.\n[13] Rapha \u00a8el F\u00b4eraud, Robin Allesiardo, Tanguy Urvoy, and Fabrice Cl \u00b4erot. Random forest for the\ncontextual bandit problem. In Artificial intelligence and statistics , pages 93\u2013101. PMLR, 2016.\n[14] Ole-Christoffer Granmo. The tsetlin machine\u2013a game theoretic bandit driven approach to"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e8-934e-11ee-8d45-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "optimal pattern recognition with propositional logic. arXiv preprint arXiv:1804.01508 , 2018.\n[15] F Maxwell Harper and Joseph A Konstan. The movielens datasets: History and context. Acm\ntransactions on interactive intelligent systems (tiis) , 5(4):1\u201319, 2015.\n[16] Insaf Ismath, KB Shashika Manosha, Samad Ali, Nandana Rajatheva, and Matti Latva-aho. Deep\ncontextual bandits for fast initial access in mmwave based user-centric ultra-dense networks. In"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20e9-934e-11ee-9d72-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 9, "chroma:document": "2021 IEEE 93rd Vehicular Technology Conference (VTC2021-Spring) , pages 1\u20137. IEEE, 2021.\n10"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ea-934e-11ee-9c0f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "[17] Yehuda Koren, Robert Bell, and Chris V olinsky. Matrix factorization techniques for recom-\nmender systems. Computer , 42(8):30\u201337, 2009.\n[18] Jie Lei, Adrian Wheeldon, Rishad Shafik, Alex Yakovlev, and Ole-Christoffer Granmo. From\narithmetic to logic based ai: A comparative analysis of neural networks and tsetlin machine. In\n2020 27th IEEE International Conference on Electronics, Circuits and Systems (ICECS) , pages\n1\u20134, 2020."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20eb-934e-11ee-961d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "1\u20134, 2020.\n[19] Lihong Li, Wei Chu, John Langford, and Robert E Schapire. A contextual-bandit approach to\npersonalized news article recommendation. In Proceedings of the 19th international conference\non World wide web , pages 661\u2013670, 2010.\n[20] Aaron Meurer, Christopher P. Smith, Mateusz Paprocki, Ond \u02c7rej\u02c7Cert\u00b4\u0131k, Sergey B. Kirpichev,\nMatthew Rocklin, AMiT Kumar, Sergiu Ivanov, Jason K. Moore, Sartaj Singh, Thilina Rath-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ec-934e-11ee-b1eb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "nayake, Sean Vig, Brian E. Granger, Richard P. Muller, Francesco Bonazzi, Harsh Gupta,\nShivam Vats, Fredrik Johansson, Fabian Pedregosa, Matthew J. Curry, Andy R. Terrel, \u02c7St\u02c7ep\u00b4an\nRou\u02c7cka, Ashutosh Saboo, Isuru Fernando, Sumith Kulal, Robert Cimrman, and Anthony Sco-\npatz. Sympy: symbolic computing in python. PeerJ Computer Science , 3:e103, January\n2017.\n[21] I. Noveck, R. B. Lea, George M. Davidson, and D. O\u2019brien. Human reasoning is both logical"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ed-934e-11ee-b9a5-ac198e74a6f7
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metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "and pragmatic. Intellectica , 11:81\u2013109, 1991.\n[22] B.J. Oommen. Stochastic searching on the line and its applications to parameter learning\nin nonlinear optimization. IEEE Transactions on Systems, Man, and Cybernetics, Part B\n(Cybernetics) , 27(4):733\u2013739, 1997.\n[23] Dattaraj Rao. Contextual bandits for adapting to changing user preferences over time. arXiv\npreprint arXiv:2009.10073 , 2020.\n[24] Jivitesh Sharma, Rohan Kumar Yadav, Ole-Christoffer Granmo, and Lei Jiao. Drop clause:"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ee-934e-11ee-bb60-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "Enhancing performance, interpretability and robustness of the tsetlin machine. CoRR ,\nabs/2105.14506, 2022.\n[25] Yilin Shen, Yue Deng, Avik Ray, and Hongxia Jin. Interactive recommendation via deep\nneural memory augmented contextual bandits. In Proceedings of the 12th ACM Conference on\nRecommender Systems , pages 122\u2013130, 2018.\n[26] Dennis Soemers, Tim Brys, Kurt Driessens, Mark Winands, and Ann Now \u00b4e. Adapting to"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ef-934e-11ee-8b2c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "concept drift in credit card transaction data streams using contextual bandits and decision trees.\nInProceedings of the AAAI Conference on Artificial Intelligence , volume 32, 2018.\n[27] Michael Lvovitch Tsetlin. On behaviour of finite automata in random medium. Avtomat. i\nTelemekh , 22(10):1345\u20131354, 1961.\n[28] Rohan Kumar Yadav, Lei Jiao, Ole-Christoffer Granmo, and Morten Goodwin. Human-level\ninterpretable learning for aspect-based sentiment analysis. In The Thirty-Fifth AAAI Conference"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f0-934e-11ee-882a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "on Artificial Intelligence (AAAI-21). AAAI , 2021.\n[29] Rohan Kumar Yadav, Lei Jiao, Ole Christoffer Granmo, and Morten Goodwin. Robust Inter-\npretable Text Classification against Spurious Correlations Using AND-rules with Negation. In\nThe 31st International Joint Conference on Artificial Intelligence (IJCAI) , 2022.\n[30] Xuan Zhang, Lei Jiao, Ole-Christoffer Granmo, and Morten Goodwin. On the Convergence\nof Tsetlin Machines for the IDENTITY- and NOT Operators. IEEE Transactions on Pattern"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f1-934e-11ee-8e2b-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 10, "chroma:document": "Analysis and Machine Intelligence , 2021.\n[31] Dongruo Zhou, Lihong Li, and Quanquan Gu. Neural contextual bandits with ucb-based\nexploration. In International Conference on Machine Learning , pages 11492\u201311502. PMLR,\n2020.\n11"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f2-934e-11ee-a99c-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 11, "chroma:document": "Checklist\n1. For all authors...\n(a)Do the main claims made in the abstract and introduction accurately reflect the paper\u2019s\ncontributions and scope? [Yes]\n(b) Did you describe the limitations of your work? [Yes] See Section 8\n(c) Did you discuss any potential negative societal impacts of your work? [N/A]\n(d)Have you read the ethics review guidelines and ensured that your paper conforms to\nthem? [Yes]\n2. If you are including theoretical results..."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f3-934e-11ee-a496-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 11, "chroma:document": "2. If you are including theoretical results...\n(a) Did you state the full set of assumptions of all theoretical results? [N/A]\n(b) Did you include complete proofs of all theoretical results? [N/A]\n3. If you ran experiments...\n(a)Did you include the code, data, and instructions needed to reproduce the main experi-\nmental results (either in the supplemental material or as a URL)? [Yes] See footnote on\nfirst page."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f4-934e-11ee-97ba-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 11, "chroma:document": "first page.\n(b)Did you specify all the training details (e.g., data splits, hyperparameters, how they\nwere chosen)? [Yes]\n(c)Did you report error bars (e.g., with respect to the random seed after running experi-\nments multiple times)? [No]\n(d)Did you include the total amount of compute and the type of resources used (e.g., type\nof GPUs, internal cluster, or cloud provider)? [No]\n4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets..."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f5-934e-11ee-8df6-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 11, "chroma:document": "(a) If your work uses existing assets, did you cite the creators? [Yes]\n(b) Did you mention the license of the assets? [N/A]\n(c)Did you include any new assets either in the supplemental material or as a URL? [N/A]\n(d)Did you discuss whether and how consent was obtained from people whose data you\u2019re\nusing/curating? [N/A]\n(e)Did you discuss whether the data you are using/curating contains personally identifiable\ninformation or offensive content? [N/A]"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f6-934e-11ee-8b74-ac198e74a6f7
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{"source": "source_documents\\NeurIPS-2022-tsetlin-machine-for-solving-contextual-bandit-problems-Paper-Conference.pdf", "page": 11, "chroma:document": "information or offensive content? [N/A]\n5. If you used crowdsourcing or conducted research with human subjects...\n(a)Did you include the full text of instructions given to participants and screenshots, if\napplicable? [N/A]\n(b)Did you describe any potential participant risks, with links to Institutional Review\nBoard (IRB) approvals, if applicable? [N/A]\n(c)Did you include the estimated hourly wage paid to participants and the total amount\nspent on participant compensation? [N/A]\n12"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
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encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "Tsetlin Machine Embedding: Representing Words Using Logical\nExpressions\nAnonymous ACL submission\nAbstract\nEmbedding words in vector space is a fun- 001\ndamental first step in state-of-the-art natural 002\nlanguage processing (NLP). Typical NLP so- 003\nlutions employ pre-defined vector representa- 004\ntions to improve generalization by co-locating 005\nsimilar words in vector space. For instance, 006\nWord2Vec is a self-supervised predictive model 007"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f8-934e-11ee-aa05-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "that captures the context of words using a neu- 008\nral network. Similarly, GLoVe is a popular 009\nunsupervised model incorporating corpus-wide 010\nword co-occurrence statistics. Such word em- 011\nbedding has significantly boosted important 012\nNLP tasks, including sentiment analysis, doc- 013\nument classification, and machine translation. 014\nHowever, the embeddings are dense floating- 015\npoint vectors, making them expensive to com- 016\npute and difficult to interpret. In this paper, we 017"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20f9-934e-11ee-8b8d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "instead propose to represent the semantics of 018\nwords with a few defining words that are related 019\nusing propositional logic. To produce such 020\nlogical embeddings, we introduce a Tsetlin 021\nMachine-based autoencoder that learns logi- 022\ncal clauses self-supervised. The clauses consist 023\nof contextual words like \u201cblack\u201d, \u201ccup\u201d, and 024\n\u201chot\u201d to define other words like \u201ccoffee\u201d, thus 025\nbeing human-understandable. We evaluate our 026\nembedding approach on several intrinsic and 027"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20fa-934e-11ee-9e4b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "embedding approach on several intrinsic and 027\nextrinsic benchmarks, outperforming GLoVe 028\non six classification tasks. Furthermore, we in- 029\nvestigate the interpretability of our embedding 030\nusing the logical representations acquired dur- 031\ning training. We also visualize word clusters 032\nin vector space, demonstrating how our logical 033\nembedding co-locate similar words.1034\n1 Introduction 035\nThe success of natural language processing (NLP) 036"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20fb-934e-11ee-8d6f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "relies on advances in word, sentence, and docu- 037\nment representation. By capturing word semantics 038\n1The Tsetlin Machine Autoencoder and logical word em-\nbedding implementation is available here: XXXX (Removed\nfor anonymity).and similarities, such representations boost the per- 039\nformance of downstream tasks (Borgeaud et al., 040\n2022), including clustering, topic modelling (An- 041\ngelov, 2020), searching, and text mining (Huang 042\net al., 2020). 043"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20fc-934e-11ee-b643-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "et al., 2020). 043\nWhile straightforward, the traditional bag-of- 044\nwords encoding does not consider the words\u2019 posi- 045\ntion, semantics, and context within a document. 046\nDistributed word representation (Bengio et al., 047\n2000; Bojanowski et al., 2017) addresses this lack 048\nby encoding words as low-dimensional vectors, 049\nreferred to as embeddings . The purpose is to 050\nco-locate similar or contextually relevant words 051\nin vector space. There are many algorithms for 052"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20fd-934e-11ee-9a6d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "learning word embeddings. Contemporary self- 053\nsupervised techniques like Word2Vec (Mikolov 054\net al., 2013), FastText (Bojanowski et al., 2017), 055\nand GloVe (Pennington et al., 2014) have demon- 056\nstrated how to build embeddings from word co- 057\noccurrence, utilizing massive training data. Intro- 058\nducing context-dependent embeddings, the more 059\nsophisticated language models BERT (Devlin et al., 060\n2019) and ELMO (Peters et al., 2018) now perform 061"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20fe-934e-11ee-9042-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "remarkably well in downstream tasks (Reimers and 062\nGurevych, 2019). However, they require significant 063\ncomputation power (Schwartz et al., 2020). 064\nThe above approaches represent words as dense 065\nfloating point vectors. Word2Vec, for instance, typ- 066\nically builds a 300-dimensional vector per word. 067\nThe size and density of these vectors make them ex- 068\npensive to compute and difficult to interpret. Con- 069\nsider, for example, the word \u201cqueen.\u201d Representing 070"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d20ff-934e-11ee-938e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 0, "chroma:document": "it with 300 floats seems inefficient compared to 071\nthe Oxford Language definition for the same word: 072\n\u201cthe female ruler of an independent state, especially 073\none who inherits the position by right of birth.\u201d 074\nFrom this perspective, it appears advantageous to 075\ncreate embeddings directly from words rather than 076\nfrom arbitrary floating-point values. Such inter- 077\npretable embeddings would capture the multiple 078\nmeanings of a word using a few defining words, 079\n1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2100-934e-11ee-9b79-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "simplifying both computation and interpretation. 080\nIn this paper, we propose a Tsetlin Ma- 081\nchine (TM) (Granmo, 2018) based autoencoder 082\nfor creating interpretable embeddings. The autoen- 083\ncoder builds propositional logic expressions with 084\ncontext words that identify each target word. The 085\nterm \u201ccoffee\u201d can, for instance, be represented by 086\n\u201cone,\u201d \u201chot,\u201d \u201ccup,\u201d \u201ctable,\u201d and \u201cblack.\u201d In this 087\nmanner, the TM builds contextual representations 088"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2101-934e-11ee-9b3a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "from a vast text corpus, which model the seman- 089\ntics of each word. In contrast to neural network- 090\nbased embedding, the logical TM embedding is 091\nsparse and energy efficient (Maheshwari et al., 092\n2023; Abeyrathna et al., 2023). The embedding 093\nspace consists of, e.g., 500truth values, where each 094\ntruth value is a logical expression over words. For 095\ncontextual representation, each target word links to 096\nless than ten percent of these expressions. Despite 097"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2102-934e-11ee-80e5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "the sparsity and crispness of this representation, it 098\nis competitive with neural network-based embed- 099\nding. 100\nThe contributions of our work are summarized 101\nbelow: 102\n\u2022We propose the TM-based Autoencoder to 103\nlearn efficient encodings in a self-supervised 104\nmanner. To the best of our knowledge, it is 105\nthe first logic-based word embedding. 106\n\u2022We introduce TM-based word embedding that 107\nbuilds human-comprehensible contextual rep- 108\nresentations from unlabeled data. 109"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2103-934e-11ee-8e64-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "resentations from unlabeled data. 109\n\u2022We compare our embedding with state-of-the- 110\nart approaches on several intrinsic and extrin- 111\nsic benchmarks, outperforming GloVe on six 112\ndownstream classification tasks. 113\n2 Related Work 114\nThe majority of self-supervised embedding ap- 115\nproaches produce dense word representations 116\nbased on the distributional hypothesis (Harris, 117\n1954), which states that words that occur in the 118\nsame context are likely to have similar meanings. 119"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2104-934e-11ee-8c77-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "Word2Vec (Mikolov et al., 2013) is one of the best- 120\nknown models. It builds embeddings from word co- 121\noccurrence using a neural network, leveraging the 122\nhidden layer output weights. GloVe (Pennington 123\net al., 2014), on the other hand, embeds by factoriz- 124\ning a word co-occurrence matrix. Similarly, canon- 125\nical correlation analysis (CCA) is used in (Dhillon 126\net al., 2015) for embedding words to maximize con- 127"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2105-934e-11ee-afc8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "text correlation. In (Levy et al., 2015), it is demon- 128strated how precise factorization-based SVD can 129\ncompete with neural embedding. However, all of 130\nthese methods are challenging to train because they 131\ninvolve tweaking algorithms and hyperparameters 132\ntoward particular applications (Lample et al., 2016), 133\nlimiting their wider applicability. 134\nBuilding upon word embedding, several stud- 135\nies focus on sentence embedding (Arora et al., 136"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2106-934e-11ee-9528-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "2017; Logeswaran and Lee, 2018). Recent ad- 137\nvances in sentence embedding include supervised 138\ndata inference (Reimers and Gurevych, 2019), mul- 139\ntitask learning (Cer et al., 2018), contrastive learn- 140\ning (Zhang et al., 2020), and pretrained large lan- 141\nguage models (Li et al., 2020). However, the ma- 142\njority of sentence embedding techniques overlook 143\nintrinsic evaluations such as similarity tasks, and 144\ninstead largely focus on extrinsic evaluations in- 145"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2107-934e-11ee-a5b4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "volving downstream performance. The most re- 146\ncent building block for embedding originates from 147\nthe transformer approach (Vaswani et al., 2017). 148\nTransformers provide context awareness by utiliz- 149\ning stacks of self-attention layers. BERT (Kenton 150\nand Toutanova, 2019), for instance, employs the 151\ntransformer architecture to carry out extensive self- 152\nsupervised training, making it capable of producing 153\ntext embedding. Other embedding models use a 154"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2108-934e-11ee-b742-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "text embedding. Other embedding models use a 154\ncontrastive loss function to perform supervised fine- 155\ntuning on positive and negative text pairs (Wang 156\net al., 2021). Despite the large variety of text em- 157\nbedding models, they all share three main draw- 158\nbacks: i) they are computationally demanding to 159\ntrain; ii) they are intrinsically complex because they 160\nare trained on a large amount of data to tune a huge 161\namount of parameters; and iii) the embeddings pro- 162"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2109-934e-11ee-813a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "duced from these models are not easily interpreted 163\nby humans. 164\nTo improve interpretability, Faruqui et al. 165\nintroduced \u201cSparse Overcomplete Word Vec- 166\ntors\" (SPOWV) which create a sparse non-negative 167\nprojection of word embedding using dictionary 168\nlearning (Faruqui et al., 2015). Similarly, SParse In- 169\nterpretable Neural Embeddings (SPINE) employs a 170\nk-sparse denoising autoencoder to generate sparse 171\nembeddings (Subramanian et al., 2018). However, 172"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d210a-934e-11ee-abd6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 1, "chroma:document": "these methods are unable to distinguish between 173\nmultiple context-dependent word meanings. To 174\naddress this problem, another avenue of research 175\nfocuses on composing linear combinations of dense 176\nvectors from Word2Vec and GloVe (Arora et al., 177\n2018). However, the assumption of linearity does 178\nnot hold for real-world data, yielding linear coeffi- 179\ncients that are difficult to comprehend (Mu et al., 180\n2"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d210b-934e-11ee-9781-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "Clause Pool+4 -5Input Output\n+1 +2 -7 +6Figure 1: Tsetlin Machine Autoencoder. In this illus-\ntration, x1is masked by replacing it with value 1 for\ninferring \u02c6x1.\n2017). 181\nThe logical embedding approach we present 182\nhere is most closely related to Naive Bayes word 183\nsense induction and topic modeling (Charniak et al., 184\n2013; Lau et al., 2014). This approach learns word 185\nmeanings from local contexts by considering each 186\ninstance of the word in a document as a pseudo- 187"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d210c-934e-11ee-8f28-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "document. However, the approach is not scalable 188\nbecause it requires training a single topic per target 189\nword. Our approach, on the other hand, is scalable 190\nand builds non-linear (non-naive) logical embed- 191\ndings that capture word compositions. To build the 192\nlogical embeddings, we propose a novel human- 193\ninterpretable algorithm based on the TM that pro- 194\nvides logical rules describing contexts. The TM 195\nhas recently performed competitively with other 196"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d210d-934e-11ee-aa41-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "deep learning techniques in many NLP tasks, in- 197\ncluding novelty detection (Bhattarai et al., 2022a,c), 198\nsentiment analysis (Abeyrathna et al., 2023; Yadav 199\net al., 2021), knowledge representation (Bhattarai 200\net al., 2023), and fake news detection (Bhattarai 201\net al., 2022b). Furthermore, the local and global 202\ninterpretability of TMs have been explored through 203\ndirect manipulation of the logical rules (Blakely 204\nand Granmo, 2021). In addition, TM has been 205"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d210e-934e-11ee-816b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "and Granmo, 2021). In addition, TM has been 205\nshown to be hardware-friendly for low-power IoT 206\ndevices (Maheshwari et al., 2023). 207\n3 Tsetlin Machine Autoencoder 208\nWe here detail the TM Autoencoder based on the 209\nCoalesced TM (Glimsdal and Granmo, 2021), ex- 210\ntended with input masking and freezing of masked 211\nvariables. For ease of explanation, we use three 212\ninputs. Adding more inputs follows trivially. 213\n3.1 Architecture 214\nInput and Output. As seen in Figure 1, the 215"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d210f-934e-11ee-86bd-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "Input and Output. As seen in Figure 1, the 215\nTM Autoencoder digests and outputs propositional 216values: (x1, x2, x3)\u2208 {0,1}3\u2192(bx1,bx2,bx3)\u2208 217\n{0,1}3. For our purposes, the propositional vari- 218\nables x1,x2, andx3each represent a word, for ex- 219\nample, \u201cBrilliant,\u201d \u201cActor,\u201d and \u201cAwful.\u201d The value 220\n1means that the word occurs in the input text, while 221\nthe value 0means that it does not. I.e., we repre- 222\nsent natural language text as a set of words . Notice 223"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2110-934e-11ee-8dd8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "also that the input variables have corresponding 224\noutput variables bx1,bx2, andbx3. In short, bx1is to 225\nbe predicted from x2andx3,bx2from x1andx3, 226\nand so on. Continuing our example, bx1predicts 227\nthe presence of \u201cBrilliant\u201d based on knowing the 228\noccurrence of \u201cActor\u201d and \u201cAwful.\u201d 229\nClause Pool. A pool of nconjunctive clauses, 230\ndenoted Cj, j\u2208 {1,2, . . . , n },encodes the input 231\nin order to predict the output. A conjunctive clause 232"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2111-934e-11ee-8fa6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "Cjis simply an And-expression over a given subset 233\nLj\u2286 {x1, x2, x3}of the input (our autoencoder 234\ndoes not use the input negations \u00acx1,\u00acx2, and 235\n\u00acx3): 236\nCj(x1, x2, x3) =^\nxk\u2208Ljxk. (1) 237\nFor example, the input subset L1={x1, x2}gives 238\nthe clause C1(x1, x2, x3) =x1\u2227x2in the figure. 239\nThis clause matches the input if x1andx2both 240\nare1. In our example, the clause accordingly en- 241\ncodes the concept \u201cBrilliant Actor\u201d. 242\nWeights. An integer weight matrix Wconnects 243"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2112-934e-11ee-9eae-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "Weights. An integer weight matrix Wconnects 243\neach of the nclauses to the three outputs bx1,bx2, 244\nandbx3: 245\nW=\uf8ee\n\uf8f0w11\u00b7\u00b7\u00b7w1n\nw21\u00b7\u00b7\u00b7w2n\nw31\u00b7\u00b7\u00b7w3n\uf8f9\n\uf8fb\u2208Z3\u00d7n. (2) 246\nThe row index is an output while the column index 247\nis a clause. The weight w12, for instance, connects 248\noutputbx1to clause C2. In Figure 1, six weights 249\nconnect two clauses and three outputs: 250\n\uf8ee\n\uf8f0+4\u22125\n+1 +2\n\u22127 +6\uf8f9\n\uf8fb. (3) 251\nConsider, for example, the weights (+4,\u22125)of 252"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2113-934e-11ee-994e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 2, "chroma:document": "Consider, for example, the weights (+4,\u22125)of 252\noutputbx1in the figure. The weight +4states that 253\nclause C1(x1, x2, x3) =x1\u2227x2favoursbx1being 254\n1, while clause C2(x1, x2, x3) =x2\u2227x3opposes it. 255\nFor example, the concept \u201cAwful Actor\u201d opposes 256\noutput \u201cBrilliant.\u201d 257\n3"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2114-934e-11ee-9424-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "Maximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten 4 \n \n3 \n \n2 \n \n1Figure 2: Tsetlin Machine memory for single clause.\n3.2 Inference 258\nLet us consider the prediction of bx1first. The au- 259\ntoencoder predicts bx1from the clauses and weights: 260\nbx1= 0\u2264nX\nj=1w1jCj(1, x2, x3). (4) 261\nThat is, each clause Cjis multiplied by its weight 262\nw1jfor output bx1. The outcomes are then summed 263\nup to decide the output. If the sum is larger than or 264"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2115-934e-11ee-9a94-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "equal to zero, the output is bx1= 1. Otherwise, it is 265\nbx1= 0. Clauses with positive weight thus promote 266\noutputbx1= 1while clauses with negative weight 267\nencourage bx1= 0. Notice that x1is masked by 268\nreplacing it with value 1. Accordingly, the autoen- 269\ncoder infers output bx1from the remaining inputs 270\nx2andx3. 271\nCorrespondingly, bx2andbx3are calculated by 272\nrespectively masking x2andx3: 273\nbx2= 0\u2264nX\nj=1w2jCj(x1,1, x3), (5) 274\nbx3= 0\u2264nX\nj=1w3jCj(x1, x2,1). (6) 275"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2116-934e-11ee-b822-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "bx3= 0\u2264nX\nj=1w3jCj(x1, x2,1). (6) 275\nExample. Assume that the input is always either 276\n(1,1,0)or(0,1,1). The input (1,1,0)could for 277\ninstance represent \u201cBrilliant Actor\u201d and (0,1,1) 278\n\u201cAwful Actor.\u201d Then notice how Eq. (4) correctly 279\ndetermines the masked input x1with output bx1in 280\nFigure 1, both for input (1,1,0)and(0,1,1). 281\n3.3 Learning 282\nWe next consider how to learn the variable subsets 283\nLjfor the clauses Cj, j\u2208 {1,2, . . . , n },as well 284"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2117-934e-11ee-a898-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "as how to determine the weights wijof the weight 285\nmatrix W. 286\nClause Memory. Each clause Cjhas a graded 287\nmemory that contains the input variables, shown 288\nin Figure 2. The graded memory enables incre- 289\nmental learning of the variable subsets from data. 290Observe how each variable is in one of four mem- 291\nory positions (the number of memory positions is 292\na user-configurable parameter). Positions 1\u22122 293\nmeans Forgotten. Positions 3\u22124means Memo- 294"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2118-934e-11ee-92e0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "means Forgotten. Positions 3\u22124means Memo- 294\nrized. Memorized variables take part in the clause, 295\nwhile Forgotten ones do not. The memory in Figure 296\n2 thus gives the clause Cj(x1, x2, x3) =x1\u2227x2. 297\nLearning Step. The TM Autoencoder learns 298\nincrementally using three kinds of memory 299\nand weight updates: Type Ia, Type Ib, and 300\nType II. Each training example has the form 301\n[k,(x1, x2, x3), xk],1\u2264k\u22643. The first element 302\nis an index that identifies which input to mask and 303"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2119-934e-11ee-a45b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "which output to predict. The second element is an 304\ninput vector (x1, x2, x3)and the third element is 305\nthe target value for output bxk, which is xk. We 306\ndescribe the update procedure step-by-step below 307\nfor index 1examples (output bx1prediction). The 308\nupdate procedure for bx2andbx3follows trivially. 309\nClause Update Probability. First, we calculate 310\nthe weighted clause sum for bx1from Eqn. (4): 311\nv1=Pn\nj=1w1jCj(1, x2, x3). The sum is then 312"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d211a-934e-11ee-adb7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "v1=Pn\nj=1w1jCj(1, x2, x3). The sum is then 312\ncompared with a margin T(hyper-parameter) to 313\ncalculate a summation error \u03f5. The error depends 314\non the x1-value: 315\n\u03f5=(\nT\u2212clip(v1,\u2212T, T), x 1= 1,\nT+ clip( v1,\u2212T, T), x 1= 0.(7) 316\nThat is, for x1-value 1the weighted clause sum 317\nshould become T, while for x1-value 0the sum 318\nshould become \u2212T. The goal of the learning is 319\nthus to reach the margin for all inputs (x1, x2, x3), 320\nensuring correct output from Eqn. (4). To reach 321"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d211b-934e-11ee-b927-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "this goal, each clause Cjis updated randomly with 322\nprobability\u03f5\n2Tin each round. In other words, the 323\nupdate probability drops with the error toward zero. 324\nUpdate Types. The kind of update depends on 325\nthe values of x1,Cj(1, x2, x3), and w1j. We first 326\nconsider clauses with positive weight, w1j\u22650. 327\nAccording to Eqn. 4, they are to recognize pat- 328\nterns for x1= 1. Note that in all of the below 329\nupdates, the masked variable x1is frozen, leaving 330"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d211c-934e-11ee-8497-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 3, "chroma:document": "it unaffected by the update. 331\n\u2022Type Ia (Recognize) Feedback occurs when 332\nx1= 1 andCj(1, x2, x3) = 1 . Then one 333\ncan say that Cj(1, x2, x3) = 1 is atrue posi- 334\ntivebecause it correctly predicts the masked 335\nx1-value. The Type Ia feedback reinforces 336\nthis successful match by updating the memory 337\n4"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d211d-934e-11ee-a5b4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "Maximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten\n 4 \n \n3 \n \n2 \n \n1Figure 3: Type Ia (Recognize) Feedback for input\n(1,1,0). The masked variable x1is frozen.\nMaximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten\n 4 \n \n3 \n \n2 \n \n1\nFigure 4: Type Ib (Erase) Feedback for input (0,0,1).\nThe masked variable x1is frozen.\nofCjto further mimic the input (see Figure 338\n3). That is, 1-valued variables move one step 339\nupwards in memory, with probability 1.0.2340"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d211e-934e-11ee-9ce0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "upwards in memory, with probability 1.0.2340\nConversely, 0-valued inputs move one step 341\ndownwards, however, randomly with proba- 342\nbility1\ns. Here, sis a hyperparameter called 343\nspecificity , meaning that a larger smakes the 344\nclauses more specific (Zhang et al., 2022). 345\nThe clause overall is also reinforced by in- 346\ncrementing its weight wj1by1. 347\n\u2022Type Ib (Erase) Feedback occurs when x1= 348\n1andCj(1, x2, x3) = 0 . Then we call 349\nCj(1, x2, x3) = 0 afalse negative because 350"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d211f-934e-11ee-85fd-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "Cj(1, x2, x3) = 0 afalse negative because 350\nit fails to promote x1= 1. In that case, all 351\ninputs randomly move one step downwards 352\nin memory (see Figure 4). Again, each down- 353\nward move happens with probability1\ns. Here, 354\nthe purpose is to eliminate the false negative 355\noutcome by erasing variables from the clause. 356\n2Originally, the increment probability iss\u22121\ns, which can\nbe boosted to 1.0to enhance the learning of true positive\npatterns (Granmo, 2018).\nMaximally\nMemorized"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2120-934e-11ee-993f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "patterns (Granmo, 2018).\nMaximally\nMemorized\nMaximally\nForgottenMemorized\nForgotten\n 4 \n \n3 \n \n2 \n \n1\nFigure 5: Type II (Reject) Feedback for input (0,1,0).\nThe masked variable x1is frozen.\u2022Type II (Reject) Feedback occurs when 357\nx1= 0 andCj(1, x2, x3) = 1 . Then, one 358\ncan say that Cj(1, x2) = 1 is afalse positive 359\nbecause it promotes x1= 1when in fact we 360\nhavex1= 0. Then all Forgotten 0-valued in- 361\nputs move one step upwards in memory. The 362"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2121-934e-11ee-8b27-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "puts move one step upwards in memory. The 362\npurpose is to eventually eliminate the current 363\nfalse positive outcome by injecting 0-valued 364\nvariables into the clause. The clause is further 365\ndiminished by decrementing its weight w1jby 366\n1. Note that the latter decrement can switch 367\nthe weight from positive to negative. In effect, 368\nthe clause then changes role, now training to 369\nrecognize x1= 0instead. 370\nClauses Cjwith negative weights, w1j<0, are 371"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2122-934e-11ee-871a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "Clauses Cjwith negative weights, w1j<0, are 371\nupdated the same way. However, they are to recog- 372\nnize patterns for x1= 0. To achieve this, x1= 0is 373\ntreated as x1= 1andx1= 1is treated as x1= 0 374\nwhen updating the memories. Furthermore, the 375\nweight updates are reversed. Increments become 376\ndecrements and vice versa. 377\nAlgorithm 1 TM word embedding\nRequire: V ocabulary V; Documents D \u2208 G ,D \u2286 V ; Accu-\nmulation u; Clauses n; Margin T; Specificity s; Rounds\nr"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2123-934e-11ee-862f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "r\n1: TMCreate (n, T, s ) \u25b7Create TM with nclauses.\n2:forrrounds do\n3: forword k\u2208 Vdo\u25b7Create one example per word.\n4: qk\u2190Select( {0,1}) \u25b7Random target value.\n5: ifqk= 1then\n6: Gk\u2190 {D| word k\u2208 D,D \u2208 G} \u25b7\nDocuments with word k.\n7: else\n8: Gk\u2190 {D| word k/\u2208 D,D \u2208 G} \u25b7\nDocuments without word k.\n9: Sk\u2190SelectN( Gk, u)\u25b7Random subset of size\nu.\n10: Uk\u2190S\nD\u2208S kD\u25b7Union of selected documents.\n11: xk \u2190 (x1, x2, . . . , x m), xi =(\n1,word i\u2208 Uk\n0,word i/\u2208 Uk\n12: TMUpdate( k,xk, qk) \u25b7Update"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2124-934e-11ee-ba96-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 4, "chroma:document": "0,word i/\u2208 Uk\n12: TMUpdate( k,xk, qk) \u25b7Update\nTM Autoencoder for output index k, input xk, and target\nvaluebxk=qk.\n13:C,W\u2190TMGetState() \u25b7Clauses Cj\u2208 Cwith\nweights W.\n14:E\u2190clip(W,0, T) \u25b7Elementwise clip of negative\nvalues produces weighted logical word embeddings.\n15:B\u2190(W>0)\u25b7Elementwise comparison with zero\nproduces purely logical word embeddings.\n16:return C,E,B\n4 Logical Embedding Procedure 378\nWe now use the TM Autoencoder to 379\nbuild logical embeddings. Let V = 380\n5"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2125-934e-11ee-9169-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "DatasetW2V FastText TM GloVe\nSpearman Kendall Cosine Spearman Kendall Cosine Spearman Kendall Cosine Spearman Kendall Cosine\nWS-353 0.53 0.37 0.87 0.46 0.32 0.79 0.45 0.31 0.90 0.41 0.28 0.90\nSIM999 0.26 0.18 0.79 0.23 0.16 0.79 0.14 0.10 0.76 0.25 0.17 0.80\nMEN 0.71 0.50 0.91 0.71 0.51 0.94 0.64 0.45 0.94 0.73 0.53 0.95\nMTURK287 0.66 0.47 0.77 0.63 0.44 0.93 0.63 0.44 0.92 0.66 0.47 0.86\nMTURK717 0.57 0.39 0.86 0.52 0.36 0.93 0.48 0.32 0.91 0.58 0.40 0.94"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2126-934e-11ee-afa2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "RG65 0.72 0.58 0.89 0.67 0.49 0.88 0.75 0.63 0.92 0.78 0.62 0.93\nAverage 0.58 0.42 0.85 0.54 0.38 0.88 0.52 0.38 0.89 0.57 0.42 0.90\nTable 1: Performance comparison of TM embedding with baseline algorithms on the similarity task.\n{word 1,word 2, . . . , word m}be the target 381\nvocabulary consisting of munique words. 382\nPre-processing. The first step is to pre-process 383\nthe document corpus. To this end, each document 384\nis represented by a subset of words D \u2286 V . For 385"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2127-934e-11ee-81ed-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "example, the document \u201cThe actor was brilliant\u201d 386\nbecomes the set D={\u201cactor\u201d, \u201cbrilliant\u201d, \u201cthe\u201d, 387\n\u201cwas\u201d}. The set G, in turn, contains all the doc- 388\numents, D \u2208 G . Finally, in propositional vector 389\nform, the word set Dbecomes: 390\nx= (x1, x2, . . . , x t), xi=(\n1,word i\u2208D,\n0,word i/\u2208D.\n(8) 391\nEmbedding. Algorithm 1 specifies the procedure 392\nfor embedding the mvocabulary words from Vby 393\nusing nclauses, Cj,1\u2264j\u2264n, forming a clause 394"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2128-934e-11ee-b182-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "using nclauses, Cj,1\u2264j\u2264n, forming a clause 394\nsetC. Each round of training produces a training 395\nexample [k,(x1, x2, . . . , x m), qk]perword kinV. 396\nFirst, a target value qkfor the word is set randomly 397\nto either 0or1. This random selection balances 398\nthe dataset. If qkbecomes 1, we randomly select 399\nudocuments that contain word kand assign them 400\nto the set Sk(positive examples). Otherwise, we 401\nrandomly select udocuments that do notcontain 402"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2129-934e-11ee-a525-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "randomly select udocuments that do notcontain 402\nthe word (negative examples). Next, the randomly 403\nselected documents are merged by ORing them to- 404\ngether, yielding the unified document Uk. The pur- 405\npose of ORing multiple documents is to increase 406\nthe frequency of rare context words. Then, picking 407\nup characteristic ones becomes easier. After that, 408\nthe propositional vector form (x1, x2, . . . , x m)of 409\nUkis obtained. Finally, the TM Autoencoder is up- 410"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d212a-934e-11ee-9627-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "dated with [k,(x1, x2, . . . , x m), qk]following the 411\ntraining procedure in Section 3. 412\nVector Space Representation. The weighted 413\nlogical embedding of word k\u2208 V can now be ob- 414\ntained from row kof a matrix E(returned from 415\nAlgorithm 1), while the purely logical embedding 416\nis found in row kof the matrix B. Letekdenote 417\nthek\u2019th row of E, and let eldenote the l\u2019th row. 418\nWe can then compare the similarity of two words 419Dataset W2V FastText TM GloVe\nAP 0.50 0.35 0.41 0.41"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d212b-934e-11ee-b241-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "AP 0.50 0.35 0.41 0.41\nBLESS 0.64 0.66 0.62 0.66\nESSLI 0.63 0.60 0.57 0.56\nAverage 0.59 0.54 0.53 0.54\nTable 2: Performance comparison of TM embedding\nwith baseline embeddings on the categorization task.\nword kandword lusing cosine similarity (CS) be- 420\ntween their E-embedding: 421\nCS(word k,word l) =ek\u00b7el\n||ek|| ||el||. (9) 422\n5 Empirical Evaluation 423\nWe here evaluate our logical embedding scheme, 424\ncomparing it with neural network approaches. The 425"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d212c-934e-11ee-9b14-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "details of the evaluation and training setup are pro- 426\nvided in the appendix. 427\nDatasetGloVe TM TM hybrid\nAcc. F1 Acc. F1 Acc. F1\nR8 96.31 0.88 96.10 0.88 97.80 0.94\nTREC 95.20 0.95 96.40 0.96 96.80 0.96\nR52 90.34 0.58 91.23 0.62 94.23 0.68\nSUBJ 86.20 0.86 85.80 0.85 86.70 0.87\nSST-2 76.38 0.75 75.61 0.74 79.30 0.78\nSST-5 47.47 0.46 47.80 0.43 49.75 0.44\nTable 3: Performance comparison of our embedding\nwith standard GloVe embedding on the classification\ntask.\n5.1 Results and Discussion 428"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d212d-934e-11ee-aa0e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "task.\n5.1 Results and Discussion 428\nWe employ two kinds of evaluation: intrinsic and 429\nextrinsic. Table 1 contains the intrinsic evalu- 430\nation results from the six-word similarity tasks. 431\nWe here compute the Spearman correlation, the 432\nKendall coefficient, and the cosine similarity be- 433\ntween the human-set similarity scores and the pre- 434\ndicted similarity scores per dataset. Considering 435\nSpearman and Kendall score, Word2Vec and GloVe 436"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d212e-934e-11ee-bf97-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 5, "chroma:document": "are marginally better than the comparable FastText 437\nand TM embedding. However, as reported in (Ras- 438\ntogi et al., 2015), small differences in correlation- 439\nbased measures are not necessarily significant for 440\n6"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d212f-934e-11ee-8d4d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 6, "chroma:document": "Target words ClausesClauses\n student\nsurgery\nheart\nbaseball\nfootball\nqueen\nprincess\ncoffee\ntea\n Figure 6: Interpretability of clauses capturing distinct meanings of target words in the TM embedding.\nFigure 7: TM embedding visualization plotted using\nt-SNE.\nsmaller datasets. To more robustly assess perfor- 441\nmance, we therefore also use cosine similarity 442\nto compare predicted word similarities with the 443\nhuman-set similarities. In terms of cosine score, 444"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2130-934e-11ee-b5c8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 6, "chroma:document": "our model outperforms Word2Vec and FastText 445\non the majority of the datasets, while performing 446\ncompetitively with GloVe. This means that the 447\nangles between the human-set similarities and the 448\nGloVe/TM-predicted similarities are quite similar. 449\nFinally, Table 2 shows the outcome for the word 450\ncategorization tasks. As seen, the performance of 451\nthe selected embedding techniques are comparable, 452\nwith Word2Vec being slightly ahead. 453"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2131-934e-11ee-8d95-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 6, "chroma:document": "with Word2Vec being slightly ahead. 453\nPrevious research indicates that intrinsic word 454\nsimilarity performance is minimally or even neg- 455\natively correlated with downstream NLP perfor- 456\nmance (Wang et al., 2021). Therefore, we also in- 457\nclude an extrinsic evaluation with six downstream 458\nclassification tasks. To avoid overfitting and ro- 459bustly assess downstream properties, we keep our 460\nexperimental setup from above. Table 3 reports 461"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2132-934e-11ee-8188-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 6, "chroma:document": "the outcome of the evaluation, where the embed- 462\ndings have been fed to an attention-based BiLSTM 463\nmodel. The first configuration (GloVe) uses the 464\npre-trained GloVe embeddings from the Wikipedia 465\n2014 + Gigaword 5 datasets. The second configu- 466\nration consists of our purely logical TM embedding 467\nfrom One Billion Word (embedding Bfrom Algo- 468\nrithm 1). Being five times smaller, the One Billion 469\nWord dataset only provides about 80percent of the 470"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2133-934e-11ee-8375-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 6, "chroma:document": "vocabulary required for the classification tasks. We 471\nembed the remaining 20percent of the words ran- 472\ndomly. Hence, the TM approach can potentially 473\nhave a disadvantage in the evaluation. In the third 474\nconfiguration (TM hybrid ), we replace the 20per- 475\ncent random embeddings with the corresponding 476\nGloVe embeddings (approximately 80% TM + 20% 477\nGloVe). We note that the downstream accuracy of 478\nBiLSTM is similar for both TM and GloVe. Specifi- 479"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2134-934e-11ee-9b14-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 6, "chroma:document": "cally, the TM embedding exceeds GloVe by a small 480\nmargin on TREC, R52, and SST-5. The hybrid em- 481\nbedding, on the other hand, clearly outperforms the 482\nother two. In particular, for R52, SST-2, and SST-5, 483\nthe hybrid embedding is able to surpass GloVe by 484\na substantial margin of roughly 2\u22124%. Given that 485\nthe datasets are not completely balanced, we also 486\ncompute F1 macro scores. We again observe that 487\nthe TM embedding either outperforms or is com- 488"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2135-934e-11ee-a945-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 6, "chroma:document": "petitive with GloVe. For R8 and R52, the hybrid 489\nembedding surpasses GloVe by a large margin, re- 490\nspectively by around 6%and10%. Based on these 491\nresults, we conjecture that logical TM embedding 492\ncan successfully replace neural network embedding. 493\nEven with 20% of the vocabulary missing, trained 494\non five times smaller data, the logical embedding 495\nperforms competitively with GloVe. Interestingly, 496\n7"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2136-934e-11ee-9136-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "the hybrid approach performed even better. One 497\npossible explanation for this higher performance 498\ncan be the extra information added by the larger 499\nvocabulary. Additionally, there may be synergy 500\nbetween the neural and logical representations that 501\nmanifest in the hybrid approach. 502\n5.2 Interpretability and Visualization 503\nIn this section, we investigate the nature of the 504\nTM embeddings in more detail, focusing on inter- 505"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2137-934e-11ee-8f36-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "pretability. Our embedding consists of the positive 506\nclause weights E, or, alternatively, the proposi- 507\ntional version B, explained by the set of clauses 508\nC. As demonstrated in Figure 6, each clause in C 509\ncaptures a facet of a context. The dotted lines in the 510\nfigure showcase the connection between the target 511\nwords and their clauses from matrix B(and, ac- 512\ncordingly, E). Each target word gets its own color 513\nto more easily discern the connections. In the fig- 514"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2138-934e-11ee-82bf-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "ure, we provide an excerpt of 18connections from 515\nB, involving 8target words and the 11most trig- 516\ngered clauses for these words. Consider for exam- 517\nple the target words surgery andheart . These two 518\ntarget words share two clauses: [ went\u2227hospital ] 519\nand [ old\u2227disease \u2227patient ]. The two clauses 520\ncapture two joint contexts, both related to health. 521\nThe clauses thus represent commonality between 522\nthe target words, providing information on one par- 523"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2139-934e-11ee-ace3-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "ticular meaning of the words. 524\nThe two target words are also semantically dif- 525\nferent. The differences are captured by the clauses 526\nthey do not share. The target word heart , for exam- 527\nple, also relates to the meaning [ woman \u2227love], 528\nwhich surgery does not. Surgery , on the other hand, 529\nconnects with [ injury \u2227game \u2227racing ]. In 530\nthis manner, the unique meanings and relations be- 531\ntween words are represented through the sharing 532"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d213a-934e-11ee-8d20-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "of logical expressions. Accordingly, it is feasible 533\nto capture a wide range of possible contextual rep- 534\nresentations with concise logical expressions. As 535\nsuch, the logical embedding provides a sparse rep- 536\nresentation of words and their relations. Indeed, at 537\nmost 10% of the clauses connect to each word in 538\nour experiments. As shown in the intrinsic evalua- 539\ntions from the previous subsection, these contextual 540\nrepresentations are effective for measuring word 541"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d213b-934e-11ee-aa9a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "similarity and categorizing words. Similarly, we 542\nobserved that the logical embedding is boosting 543\ndownstream NLP classification tasks. 544\nTo cast further light on the TM embedding ap- 545\nproach, we visualize the embedding of 400words 546\nfrom the SimLex-999 dataset in Figure 7, plot- 547ted using t-SNE. The figure indicates that we are 548\nable to cluster contextually similar words in vec- 549\ntor space. To scrutinize the clusters, we zoom in 550"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d213c-934e-11ee-9262-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "on two of them. Consider the upper right cluster 551\nfirst. Notice how the words in the cluster relate 552\ntohospital , such as heart anddiseases . As seen, 553\nthe word embeddings are closely located in vector 554\nspace. Similarly, we can observe that terminology 555\nconnected to weather and geography are grouped 556\ntogether in the bottom cluster. From these two 557\nexamples, it seems clear that the TM embedding 558\nincorporates semantic relationships among words. 559"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d213d-934e-11ee-857c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "6 Conclusion and Future Work 560\nIn this work, we first discussed the challenge and 561\nnecessity of finding computationally simpler and 562\nmore interpretable word embedding approaches. 563\nWe then motivated an efficient self-supervised ap- 564\nproach, namely, a TM-based autoencoder, for pro- 565\nducing sparse and interpretable logical word em- 566\nbeddings. We evaluated our approach on a wide 567\nrange of intrinsic and extrinsic tasks, demonstrating 568"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d213e-934e-11ee-a235-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "that it is competitive with dense neural network- 569\nbased embedding schemes such as Word2Vec, 570\nGloVe, and FastText. Further, we investigated the 571\ninterpretability of our embedding through visual- 572\nization and a case study. Our conclusion from the 573\nstudy is that logical embedding is able to repre- 574\nsent words with logical expressions. This structure 575\nmakes the representation sparse, enabling a clear- 576\ncut decomposition of each word into sets of seman- 577"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d213f-934e-11ee-bb7d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "tic concepts. Future work includes scaling up our 578\nimplementation using GPUs to support the build- 579\ning of large-scale vocabularies from more massive 580\ndatasets. 581\n7 Limitations 582\nThe primary purpose of the conducted experiments 583\nin the context of the downstream classification task 584\nis to thoroughly analyze and comprehend the prac- 585\ntical implementation of our embedding approach. 586\nConsequently, the evaluation did not involve a com- 587"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2140-934e-11ee-adca-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 7, "chroma:document": "parison of performance against other contemporary 588\ntransformer-based large language models, such as 589\nBERT, which are considered the state of the art. 590\nFurther, we intend to investigate how sentence- 591\nlevel and document-level embedding can be created 592\nusing clauses, for instance, applicable for down- 593\nstream sentence similarity tasks. 594\n8"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2141-934e-11ee-88a6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 8, "chroma:document": "References 595\nK. Darshana Abeyrathna, Ahmed Abdulrahem Oth- 596\nman Abouzeid, Bimal Bhattarai, Charul Giri, Sondre 597\nGlimsdal, Ole-Christoffer Granmo, Lei Jiao, Rupsa 598\nSaha, Jivitesh Sharma, Svein Anders Tunheim, and 599\nXuan Zhang. 2023. Building concise logical patterns 600\nby constraining tsetlin machine clause size. In In- 601\nternational joint conference on artificial intelligence 602\n(IJCAI) . 603\nDimo Angelov. 2020. Top2vec: Distributed representa- 604"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2142-934e-11ee-bb7e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 8, "chroma:document": "tions of topics. arXiv preprint arXiv:2008.09470 . 605\nSanjeev Arora, Yuanzhi Li, Yingyu Liang, Tengyu Ma, 606\nand Andrej Risteski. 2018. Linear algebraic struc- 607\nture of word senses, with applications to polysemy. 608\nTransactions of the Association for Computational 609\nLinguistics , 6:483\u2013495. 610\nSanjeev Arora, Yingyu Liang, and Tengyu Ma. 2017. A 611\nsimple but tough-to-beat baseline for sentence em- 612\nbeddings. In Proceedings of ICLR . 613"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2143-934e-11ee-8654-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 8, "chroma:document": "beddings. In Proceedings of ICLR . 613\nYoshua Bengio, R\u00e9jean Ducharme, and Pascal Vincent. 614\n2000. A neural probabilistic language model. Ad- 615\nvances in neural information processing systems , 13. 616\nBimal Bhattarai, Ole-Christoffer Granmo, and Lei Jiao. 617\n2022a. A Tsetlin Machine Framework for Universal 618\nOutlier and Novelty Detection. In International Con- 619\nference on Agents and Artificial Intelligence , pages 620\n250\u2013268. Springer. 621"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2144-934e-11ee-b731-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 8, "chroma:document": "250\u2013268. Springer. 621\nBimal Bhattarai, Ole-Christoffer Granmo, and Lei Jiao. 622\n2022b. Explainable Tsetlin Machine framework for 623\nfake news detection with credibility score assessment. 624\nInLREC . 625\nBimal Bhattarai, Ole-Christoffer Granmo, and Lei Jiao. 626\n2022c. Word-level human interpretable scoring 627\nmechanism for novel text detection using tsetlin ma- 628\nchines. Applied Intelligence . 629\nBimal Bhattarai, Ole-Christoffer Granmo, and Lei Jiao. 630"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2145-934e-11ee-99af-ac198e74a6f7
encoding: FLOAT32
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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2146-934e-11ee-b4c0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 8, "chroma:document": "Conference on Industrial, Engineering and Other Ap- 640\nplications of Applied Intelligent Systems (IEA/AIE 641\n2021) . Springer. 642\nPiotr Bojanowski, Edouard Grave, Armand Joulin, and 643\nTomas Mikolov. 2017. Enriching word vectors with 644\nsubword information. Transactions of the associa- 645\ntion for computational linguistics , 5:135\u2013146. 646\nSebastian Borgeaud, Arthur Mensch, Jordan Hoff- 647\nmann, Trevor Cai, Eliza Rutherford, Katie Milli- 648"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2147-934e-11ee-b965-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 8, "chroma:document": "can, George Bm Van Den Driessche, Jean-Baptiste 649Lespiau, Bogdan Damoc, Aidan Clark, et al. 2022. 650\nImproving language models by retrieving from tril- 651\nlions of tokens. In Proceedings of ICML , pages 2206\u2013 652\n2240. PMLR. 653\nDaniel Cer, Yinfei Yang, Sheng-yi Kong, Nan Hua, 654\nNicole Limtiaco, Rhomni St John, Noah Constant, 655\nMario Guajardo-Cespedes, Steve Yuan, Chris Tar, 656\net al. 2018. Universal sentence encoder for English. 657\nInProceedings of EMNLP , pages 169\u2013174. 658"}


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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d214a-934e-11ee-b62f-ac198e74a6f7
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id: 152d214c-934e-11ee-9bcb-ac198e74a6f7
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topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d214d-934e-11ee-b3db-ac198e74a6f7
encoding: FLOAT32
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created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d214e-934e-11ee-9f3c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "Bohan Li, Hao Zhou, Junxian He, Mingxuan Wang, 713\nYiming Yang, and Lei Li. 2020. On the Sentence 714\nEmbeddings from Pre-trained Language Models. In 715\nProceedings of EMNLP , pages 9119\u20139130. 716\nLajanugen Logeswaran and Honglak Lee. 2018. An 717\nefficient framework for learning sentence representa- 718\ntions. In Proceedings of ICLR . 719\nSidharth Maheshwari, Tousif Rahman, Alex Yakovlev, 720\nAshur Rafiev, Lei Jiao, Ole-Christoffer Granmo, et al. 721"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d214f-934e-11ee-aa04-ac198e74a6f7
encoding: FLOAT32
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{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "2023. REDRESS: Generating compressed models 722\nfor edge inference using Tsetlin machines. IEEE 723\nTransactions on Pattern Analysis and Machine Intel- 724\nligence . 725\nTomas Mikolov, Kai Chen, Gregory S. Corrado, and 726\nJeffrey Dean. 2013. Efficient Estimation of Word 727\nRepresentations in Vector Space. In Proceedings of 728\nICLR . 729\nJiaqi Mu, Suma Bhat, and Pramod Viswanath. 2017. 730\nGeometry of polysemy. In Proceedings of ICLR . 731"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2150-934e-11ee-bf35-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "Jeffrey Pennington, Richard Socher, and Christopher D 732\nManning. 2014. Glove: Global vectors for word 733\nrepresentation. In Proceedings of EMNLP , pages 734\n1532\u20131543. 735\nMatthew E Peters, Mark Neumann, Mohit Iyyer, Matt 736\nGardner, Christopher Clark, Kenton Lee, and Luke 737\nZettlemoyer. 2018. Deep contextualized word repre- 738\nsentations. arXiv preprint arXiv:1802.05365 . 739\nPushpendre Rastogi, Benjamin Van Durme, and Raman 740\nArora. 2015. Multiview LSA: Representation learn- 741"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2151-934e-11ee-acb8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "ing via generalized CCA. In Proceedings of NAACL , 742\npages 556\u2013566. 743\nNils Reimers and Iryna Gurevych. 2019. Sentence- 744\nBERT: Sentence Embeddings using Siamese BERT- 745\nNetworks. In Proceedings of EMNLP-IJCNLP , 746\npages 3982\u20133992. 747\nRoy Schwartz, Jesse Dodge, Noah Smith, and Oren 748\nEtzioni. 2020. Green AI. Communications of the 749\nACM , 63:54 \u2013 63. 750\nAnant Subramanian, Danish Pruthi, Harsh Jhamtani, 751\nTaylor Berg-Kirkpatrick, and Eduard Hovy. 2018. 752"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2152-934e-11ee-8565-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "Spine: Sparse interpretable neural embeddings. In 753\nProceedings of AAAI , volume 32. 754Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob 755\nUszkoreit, Llion Jones, Aidan N Gomez, \u0141ukasz 756\nKaiser, and Illia Polosukhin. 2017. Attention is all 757\nyou need. Advances in neural information processing 758\nsystems , 30. 759\nKexin Wang, Nils Reimers, and Iryna Gurevych. 2021. 760\nTsdae: Using transformer-based sequential denoising 761\nauto-encoder for unsupervised sentence embedding 762"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2153-934e-11ee-965b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "learning. arXiv preprint arXiv:2104.06979 . 763\nRohan Yadav, Lei Jiao, Ole-Christoffer Granmo, and 764\nMorten Goodwin. 2021. Human-Level Interpretable 765\nLearning for Aspect-Based Sentiment Analysis. In 766\nProceedings of AAAI . 767\nXuan Zhang, Lei Jiao, Ole-Christoffer Granmo, and 768\nMorten Goodwin. 2022. On the Convergence of 769\nTsetlin Machines for the IDENTITY-and NOT Op- 770\nerators. IEEE Transactions on Pattern Analysis and 771\nMachine Intelligence , 44(10):6345\u20136359. 772"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2154-934e-11ee-b8d7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "Machine Intelligence , 44(10):6345\u20136359. 772\nYan Zhang, Ruidan He, Zuozhu Liu, Kwan Hui Lim, 773\nand Lidong Bing. 2020. An Unsupervised Sentence 774\nEmbedding Method by Mutual Information Maxi- 775\nmization. In Proceedings of EMNLP , pages 1601\u2013 776\n1610. 777\nA Appendix 778\nDatasets and Setup We first evaluate our logical 779\nembedding intrinsically, followed by an extrinsic 780\nevaluation using classification tasks. 781\nIntrinsic Evaluation. We use word similarity 782"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2155-934e-11ee-91fd-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "Intrinsic Evaluation. We use word similarity 782\nand categorization benchmarks for intrinsic eval- 783\nuation. That is, we examine to what degree our 784\napproach retains semantic word relations. To this 785\nend, we measure how semantic relations manifest 786\nin vector space using six datasets: SimLex-999, 787\nWordSim-353, MEN, MTurk-287, Mturk-771, and 788\nRG-65. Each dataset consists of human-scored 789\nword pairs, which are compared with the corre- 790"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2156-934e-11ee-94ed-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "sponding vector space similarities. The catego- 791\nrization tasks evaluate how well we can group 792\nwords into distinct word categories, only based 793\non their embedding. We here use three datasets: 794\nAP, BLESS, and ESSLLI.3As baselines, we chose 795\nWord2Vec, GloVe, and FastText because of their 796\nwide use. 797\nExtrinsic Evaluation. In our extrinsic evalua- 798\ntion, we investigate how well our logical embed- 799\nding supports downstream NLP classification tasks. 800"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2157-934e-11ee-9883-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 9, "chroma:document": "Using the word embeddings as feature vectors, the 801\n3To obtain the categorization accuracy, we use KMeans\nclustering from sklearn on the word embeddings and exam-\nine the cluster quality by calculating the purity score from\n(https://github.com/purity).\n10"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2158-934e-11ee-a21b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 10, "chroma:document": "performance of supervised classification models 802\ngives insight into the embedding quality. We em- 803\nploy six standard text classification datasets from 804\nSentEval (Conneau and Kiela, 2018): R8, R52, 805\nTREC, SUBJ, SST-2, and SST-5. For supervised 806\nlearning, we use the standard attention-based BiL- 807\nSTM model with the Adam optimizer and cross- 808\nentropy loss function. In this manner, we directly 809\ncontrast GloVe embedding against the logical TM 810\napproach. 811"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2159-934e-11ee-8a13-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 10, "chroma:document": "approach. 811\nEmbedding Datasets. For extrinsic evaluation 812\nwith BiLSTM, we use standard 300-dimensional 813\nGloVe embeddings, pre-trained on the Wikipedia 814\n2014 + Gigaword 5 datasets (6B tokens).4The pur- 815\npose is to compare the TM embedding performance 816\nagainst widely used and successful GloVe embed- 817\ndings on downstream tasks. To directly compare 818\nthe intrinsic properties of Word2Vec, GloVe, Fast- 819\nText, and TM embedding, we also train them from 820"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d215a-934e-11ee-b94e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 10, "chroma:document": "scratch using the One Billion Word dataset (Chelba 821\net al., 2014). For training the TM, we use r= 2000 822\ntraining rounds, producing 2000 examples per 823\nword by accumulating u= 25 contexts per exam- 824\nple. We use the following hyperparameters: a pool 825\nofn= 600 clauses, margin T= 1200 , and speci- 826\nficity s= 5.0.5Word2Vec Skip-Gram is trained 827\nwith 10passes over the data, using separated em- 828\nbeddings for the input and output contexts. The 829"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d215b-934e-11ee-a7ed-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 10, "chroma:document": "window size is 5and we use five negative samples 830\nper example. Similarly, GloVe is trained for 30 831\nepochs with a window size of 10and a learning 832\nrate of 0.05.6833\n4The pre-trained GloVe embeddings can be found here:\nhttps://nlp.stanford.edu/projects/glove/\n5The TM Autoencoder and logical word embedding im-\nplementation can be found here: XXXX (Removed for\nanonymity).\n6Word2Vec and FastText have been trained using\nthe standard gensim library (https://github.com/RaRe-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d215c-934e-11ee-a6d8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\TM_Embedding_EACL.pdf", "page": 10, "chroma:document": "Technologies/gensim/tree/develop/gensim). GloVe has been\ntrained using https://github.com/maciejkula/glove-python.\n11"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d215d-934e-11ee-ab1b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 0, "chroma:document": "The Tsetlin Machine \u2013 A Game Theoretic Bandit Driven\nApproach to Optimal Pattern Recognition with\nPropositional Logic\u2217\nOle-Christo\ufb00er Granmo\u2020\nAbstract\nAlthough simple individually, arti\ufb01cial neurons provide state-of-the-art performance\nwhen interconnected in deep networks. Arguably, the Tsetlin Automaton is an even simpler\nand more versatile learning mechanism, capable of solving the multi-armed bandit problem."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d215e-934e-11ee-920e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 0, "chroma:document": "Merely by means of a single integer as memory, it learns the optimal action in stochastic\nenvironments through increment and decrement operations. In this paper, we introduce\nthe Tsetlin Machine, which solves complex pattern recognition problems with propositional\nformulas, composed by a collective of Tsetlin Automata. To eliminate the longstanding\nproblem of vanishing signal-to-noise ratio, the Tsetlin Machine orchestrates the automata"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d215f-934e-11ee-8199-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 0, "chroma:document": "using a novel game. Further, both inputs, patterns, and outputs are expressed as bits, while\nrecognition and learning rely on bit manipulation, simplifying computation. Our theoret-\nical analysis establishes that the Nash equilibria of the game align with the propositional\nformulas that provide optimal pattern recognition accuracy. This translates to learning\nwithout local optima, only global ones. In \ufb01ve benchmarks, the Tsetlin Machine provides"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2160-934e-11ee-8994-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 0, "chroma:document": "competitive accuracy compared with SVMs, Decision Trees, Random Forests, Naive Bayes\nClassi\ufb01er, Logistic Regression, and Neural Networks. We further demonstrate how the\npropositional formulas facilitate interpretation. In conclusion, we believe the combination\nof high accuracy, interpretability, and computational simplicity makes the Tsetlin Machine\na promising tool for a wide range of domains.\nKeywords: Bandit Problem, Game Theory, Interpretable Pattern Recognition, Propo-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2161-934e-11ee-bc88-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 0, "chroma:document": "sitional Logic, Tsetlin Automata Games, Learning Automata, Frequent Pattern Mining,\nResource Allocation.\n1 Introduction\nAlthough simple individually, arti\ufb01cial neurons provide state-of-the-art performance when in-\nterconnected in deep networks [1]. However, deep neural networks often require huge amounts\nof training data and extensive computational resources. Unknown to many, there exists an\narguably even more fundamental and versatile learning mechanism than the arti\ufb01cial neuron,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2162-934e-11ee-b39d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 0, "chroma:document": "namely, the Tsetlin Automaton, developed by M.L. Tsetlin in the Soviet Union in the early\n1960s [2]. In this paper, we propose a novel technique for large-scale and complex pattern\nrecognition based on Tsetlin Automata.\n1.1 The Tsetlin Automaton\nTsetlin Automata have been used to model biological systems, and have attracted considerable\ninterest because they can learn the optimal action when operating in unknown stochastic"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2163-934e-11ee-b7cb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 0, "chroma:document": "environments [2, 3]. Furthermore, they combine rapid and accurate convergence with low\ncomputational complexity.\n\u2217Source code and datasets for this paper can be found at https://github.com/cair/TsetlinMachine and\nhttps://github.com/cair/pyTsetlinMachine.\n\u2020Author\u2019s status: Professor . The author can be contacted at: Centre for Arti\ufb01cial Intelligence Research\n(https://cair.uia.no), University of Agder, Grimstad, Norway. E-mail: ole.granmo@uia.no\n1arXiv:1804.01508v15  [cs.AI]  2 Jan 2021"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2164-934e-11ee-9ad2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 1, "chroma:document": "In all brevity, the Tsetlin Automaton is one of the pioneering solutions to the well-known\nmulti-armed bandit problem [4, 5] and the \ufb01rst Learning Automaton [3]. It performs actions\n\u03b1z,z\u2208{1,2}, sequentially in an environment. Each action \u03b1ztriggers either a reward or a\npenalty. That is, the action is rewarded with probability pz, otherwise it is penalized. The\nreward probabilities are unknown to the automaton and may change over time. Under such"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2165-934e-11ee-b112-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 1, "chroma:document": "challenging conditions, the goal is to identify the action with the highest reward probability\nusing as few attempts as possible.\n12NN+2N+12NAction 1Action 2\u2026\u2026RewardPenalty\nFigure 1: A Tsetlin Automaton for two-action environments.\nThe mechanism driving the Tsetlin Automaton is surprisingly simple. Informally, as illus-\ntrated in Figure 1, a Tsetlin Automaton is simply a \ufb01xed \ufb01nite-state automaton [6] with an\nunusual interpretation:"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2166-934e-11ee-a2eb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 1, "chroma:document": "unusual interpretation:\n\u2022The current state of the automaton decides which action to perform. The automaton in\nthe \ufb01gure has 2 Nstates. Action 1 ( \u03b11) is performed in the states with index 1 to N,\nwhile Action 2 ( \u03b12) is performed in the states with index N+ 1 to 2N.\n\u2022The state transitions of the automaton govern learning. One set of state transitions is\nactivated on reward (solid lines in the \ufb01gure), and one set of state transitions is activated"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2167-934e-11ee-9e63-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 1, "chroma:document": "on penalty (dotted lines in the \ufb01gure). As seen, rewards and penalties trigger speci\ufb01c\ntransitions from one state to another, designed to reinforce successful actions (those\neliciting rewards).\nFormally, a Two-Action Tsetlin Automaton can be de\ufb01ned as a quintuple [3]:\n{\u03a6,\u03b1,\u03b2,F(\u00b7,\u00b7),G(\u00b7)}.\n\u03a6={\u03c61,\u03c62,...,\u03c6 2N}is the set of internal states. \u03b1={\u03b11,\u03b12}is the set of automaton actions.\n\u03b2={\u03b2Penalty,\u03b2Reward}is the set of inputs that can be given to the automaton. An output"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2168-934e-11ee-985f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 1, "chroma:document": "functionG(\u03c6u) determines the next action performed by the automaton, given the current\nautomaton state \u03c6u:\nG(\u03c6u) ={\n\u03b11,if1\u2264u\u2264N\n\u03b12,ifN+ 1\u2264u\u22642N.(1)\nFinally, a transition function F(\u03c6u,\u03b2v) determines the new automaton state from: (i) the\ncurrent automaton state \u03c6uand (ii) the response \u03b2vof the environment to the action performed\nby the automaton:\nF(\u03c6u,\u03b2v) =\uf8f1\n\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\n\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3\u03c6u+1,if1\u2264u\u2264Nandv= Penalty\n\u03c6u\u22121,ifN+ 1\u2264u\u22642Nandv= Penalty\n\u03c6u\u22121,if1<u\u2264Nandv= Reward\n\u03c6u+1,ifN+ 1\u2264u<2Nandv= Reward\n\u03c6u,otherwise.(2)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2169-934e-11ee-a9cd-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 1, "chroma:document": "\u03c6u+1,ifN+ 1\u2264u<2Nandv= Reward\n\u03c6u,otherwise.(2)\n2"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d216a-934e-11ee-92f0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "Implementation-wise, a Tsetlin Automaton simply maintains an integer (the state index),\nand learning is performed through increment and decrement operations, according to the tran-\nsitions speci\ufb01ed by F(\u03c6u,\u03b2v) (and depicted in Figure 1). The Tsetlin Automaton is thus\nextremely simple computationally, with a very small memory footprint.\n1.2 State-of-the-art in the Field of Learning Automata\nLearning Automata have attracted considerable interest because they can learn the optimal"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d216b-934e-11ee-80be-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "action when operating in unknown stochastic environments [3]. As such, they have been used\nfor pattern classi\ufb01cation over more than four decades. Early work includes the stochastic\nLearning Automata-based classi\ufb01er of Barto and Anandan [7], as well as the games of Learning\nAutomata proposed by Narendra and Thathachar [3]. These approaches can learn the optimal\nclassi\ufb01er for speci\ufb01c forms of discriminant functions (e.g., linear classi\ufb01ers), also when feedback"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d216c-934e-11ee-b535-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "is noisy. Along the same lines, Sastry and Thathachar have provided several algorithms based\non cooperating systems of Learning Automata [8]. More recently, Zahiri proposed hyperplane-\nand rule-based classi\ufb01ers, which performed comparably to other well-known methods in the\nliterature [9, 10]. Recent research also includes the noise-tolerant learning algorithm by Sastry\net al., built upon a team of continuous-action Learning Automata [11]. Further, Goodwin et al."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d216d-934e-11ee-a5db-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "proposed a Learning Automata-guided random walk on a grid to construct robust discriminant\nfunctions [12], while Motieghader et al. introduced a hybrid scheme that combines a genetic\nalgorithm with Learning Automata for classi\ufb01cation [13].\nIn general, however, previous Learning Automata schemes have mainly addressed relatively\nsmall-scale pattern recognition tasks. The above solutions are further primarily based on so-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d216e-934e-11ee-81b4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "called variable structure learning automata [14]. Although still simple, these are signi\ufb01cantly\nmore complex than the Tsetlin Automaton because they need to maintain an action probability\nvector for sampling actions. Their success in pattern recognition is additionally limited by\nconstrained pattern representation capability (linearly separable classes and simple decision\ntrees).\nThe Tsetlin Automaton has formed the core of more advanced learning automata designs as"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d216f-934e-11ee-8c1c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "well. This includes decentralized control [15], searching on the line [16], equi-partitioning [17],\nstreaming sampling for social activity networks [18], faulty dichotomous search [19], learning\nin deceptive environments [20], and routing in telecommunication networks [21]. The strength\nof these \ufb01nite state learning automata solutions is that they have provided state-of-the-art\nperformance when problem properties are unknown and stochastic, while the problem must be"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2170-934e-11ee-b93b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "solved as quickly as possible through trial and error.\n1.3 The Vanishing Signal-to-Noise Ratio Problem\nThe ability to handle stochastic and unknown environments for a wide range of problems,\ncombined with their computational simplicity and small memory footprint, make Learning\nAutomata an attractive building block for complex machine learning tasks. However, a par-\nticularly adverse challenge has hindered the success of Learning Automata. Complex problem"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2171-934e-11ee-805f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "solving requires a team of interacting automata, and unfortunately, each team member in-\ntroduces noise. This challenge manifests sharply in the Goore Game, which employs a simple\nuni-modal stochastic objective function to be optimized by a Learning Automata team [22, 15].\nThe challenge manifests as follows. Consider a team of W+ 1 two-action variable structure\nLearning Automata. Each automaton maintains an action probability p. In any round of the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2172-934e-11ee-97a9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "Goore Game, an automaton performs its \ufb01rst action with probability pand its second action\nwith probability 1 \u2212p. Now, let \u00b5be the average value with which a single automaton can\nmodify the game\u2019s objective function by switching action. From the perspective of a single\nautomaton, each other automaton in the team then increases the objective function variance,\namounting to p(1\u2212p)\u00b52[22]. In particular, consider the case of a deterministic objective"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2173-934e-11ee-9687-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 2, "chroma:document": "function, with pbeing the action probability within the team that currently is farthest away\n3"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2174-934e-11ee-9764-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 3, "chroma:document": "from 0.5. Then the best-case signal-to-noise ratio (SNR) becomes:\nSNR =\u00b52\n\u03c32=\u00b52\nWp(1\u2212p)\u00b52=1\nWp(1\u2212p). (3)\nThat is, as the number of automata Wgrows, SNR drops to zero (vanishes).\nFor Tsetlin Automata, which are deterministic, the stochasticity of the action selection\narises from the automata changing state. These state changes are stochastic since they are\ndriven by stochastic feedback. Thus, again, as the number of automata grows, so does the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2175-934e-11ee-8bf4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 3, "chroma:document": "variance of the feedback. Even for a simple uni-modal objective function, such as the one used\nin the Goore Game, this e\ufb00ect is problematic. Indeed, as explored by Kleinrock and Tung in\n1996, a team of Tsetlin Automata solving the Goore Game needs to maintain an increasingly\nlarger state-space as the number of Tsetlin Automata grows, and in the end an in\ufb01nite number\nof states per Tsetlin Automaton [15].\nIn general, this problem is inherent to decision-making with Learning Automata because"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2176-934e-11ee-a35f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 3, "chroma:document": "of their decentralized and stochastic nature. The automata independently decide upon their\nactions, directly based on the feedback from the environment. This is on one hand a strength\nbecause it allows problems to be solved in a decentralized manner. On the other hand, as the\nnumber of automata grows, the level of noise increases. In the following, we will refer to this\ne\ufb00ect as the vanishing signal-to-noise ratio problem .\n0 0 * 1 * 0 0 0\n* 0 * 1 * 0 0 0\n0 * * 1 * * * 0\n0 * * * * 0 0 *"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2177-934e-11ee-9fc2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 3, "chroma:document": "* 0 * 1 * 0 0 0\n0 * * 1 * * * 0\n0 * * * * 0 0 *\n0 0 0 * * 0 0 0\n0 * 0 * * * 0 0\n0 0 * 1 * * * 0\n0 0 0 * 1 * * *\nTable 1: A bit pattern produced by the Tsetlin Machine for handwritten digits \u20191\u2019. The \u2019*\u2019\nsymbol can either take the value \u20190\u2019 or \u20191\u2019. The remaining bit values require strict matching.\nThe pattern is relatively easy to interpret for humans compared to, e.g., the weights of a\nneural network. It is also e\ufb03cient to evaluate for computers. Despite this simplicity, the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2178-934e-11ee-a3da-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 3, "chroma:document": "Tsetlin Machine produces bit patterns that deliver competitive pattern recognition accuracy\nfor several datasets, as reported in Section 5.\n1.4 Paper Contributions\nIn this paper, we attack the limited pattern expression capability and vanishing signal-to-noise\nratio of Learning Automata-based pattern recognition, introducing the Tsetlin Machine . The\ncontributions of the paper can be summarized as follows:\n\u2022We introduce the Tsetlin Machine, which solves complex pattern recognition problems"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2179-934e-11ee-8b80-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 3, "chroma:document": "with propositional formulas , composed by a collective of Tsetlin Automata.\n\u2022We eliminate the longstanding vanishing signal-to-noise ratio problem with a unique\ndecentralized learning scheme based on game theory [23, 2]. The game we have designed\nallows millions of Tsetlin Automata to successfully cooperate.\n\u2022The game orchestrated by the Tsetlin Machine is based on resource allocation principles\n[24], in inter-play with frequent pattern mining [25]. By allocating sparse pattern rep-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d217a-934e-11ee-a4e5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 3, "chroma:document": "resentation resources according to the frequency of the patterns, the Tsetlin Machine is\nable to capture intricate unlabelled sub-patterns, for instance addressing the so-called\nNoisy XOR-problem.\n4"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d217b-934e-11ee-97d3-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "\u2022Our theoretical analysis establishes that the Nash equilibria of the Tsetlin Machine game\nare aligned with the propositional formulas that provide optimal pattern recognition\naccuracy. This translates to learning without local optima, only global ones.\n\u2022The propositional formulas are represented as bit patterns. These bit patterns are rela-\ntively easy to interpret, compared to e.g. a neural network (see Table 1 for an example"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d217c-934e-11ee-b765-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "bit pattern). This facilitates human quality assurance and scrutiny, which for instance\ncan be important in safety-critical domains such as medicine.\n\u2022The Tsetlin Machine is a new approach to global construction of decision rules [26, 27].\nWe demonstrate that decision rules for large-scale pattern recognition can be learned\non-line, under particularly noisy conditions.\n\u2022The Tsetlin Machine is particularly suited for digital computers, being directly based on"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d217d-934e-11ee-83fc-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "bit manipulation with AND-, OR-, and NOT operators.\n\u2022In our empirical evaluation on \ufb01ve datasets, the Tsetlin Machine provides competitive per-\nformance in comparison with Multilayer Perceptron Networks, Support Vector Machines,\nDecision Trees, Random Forests, the Naive Bayes Classi\ufb01er, and Logistic Regression.\n\u2022We demonstrate how the Tsetlin Machine can be used as a building block to create more\nadvanced architectures."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d217e-934e-11ee-82a4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "advanced architectures.\nIn conclusion, we believe the combination of high accuracy, interpretability, and computational\nsimplicity makes the Tsetlin Machine a promising tool for a wide range of domains.\n1.5 Paper Organization\nThe paper is organized as follows. In Section 2, we de\ufb01ne the exact nature of the pattern\nrecognition problem we are going to solve, also introducing the crucial concept of sub-patterns."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d217f-934e-11ee-a4c5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "Then, in Section 3, we cover the Tsetlin Machine in detail. We \ufb01rst present the Tsestlin\nMachine inference structure, before we introduce the Tsetlin Automata teams that write for-\nmulas in propositional logic. These Tsetlin Automata teams are in turn organized to recognize\ncomplex patterns. We conclude the section by presenting the Tsetlin Machine game that we\nuse to coordinate millions of Tsetlin Automata, eliminating the vanishing signal-to-noise ratio\nproblem."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2180-934e-11ee-8898-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "problem.\nIn Section 4, we analyze pertinent properties of the Tsetlin Machine formally, and establish\nthat the Nash equilibria of the game are aligned with the propositional formulas that solve the\npattern recognition problem at hand. This allows the Tsetlin Machine as a whole to robustly\nand accurately uncover complex patterns with propositional logic.\nIn our empirical evaluation in Section 5, we evaluate the performance of the Tsetlin Machine"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2181-934e-11ee-b8ac-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "on \ufb01ve datasets: Flower categorization, digit recognition, board game planning, the Noisy XOR\nProblem with Non-informative Features, as well as MNIST.\nThe Tsetlin Machine has been designed to act as a building block in more advanced ar-\nchitectures, and in Section 6 we demonstrate how four distinct architectures can be built by\ninterconnecting multiple Tsetlin Machines.\nAs the \ufb01rst step in a new research direction, the Tsetlin Machine also opens up a range of"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2182-934e-11ee-8c07-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "new research questions. In Section 7, we summarize our main \ufb01ndings and provide pointers to\nsome of the open research problems.\n2 The Pattern Recognition Problem\nThe accuracy of a machine learning technique is bounded by its pattern representation capa-\nbility. The Naive Bayes Classi\ufb01er, for instance, assumes that input variables are independent\ngiven the output category. In general, compared to the representation capability of the underly-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2183-934e-11ee-a1fe-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 4, "chroma:document": "ing language of digital computers, namely, Boolean algebra, most machine learning techniques\n5"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2184-934e-11ee-bb06-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 5, "chroma:document": "P(y = 0 | X ) > P(y = 1 | X )Sub-pattern X1\nSub-pattern X0Sub-pattern X0\nSub-pattern X1P(y = 0 | X ) <= P(y = 1 | X )1\n21\n2Figure 2: A partitioning of the input space according to the posterior probability of the output\nvariabley, highlighting distinct sub-patterns in the input space, X\u2208X. Sub-patterns most\nlikely belonging to output y= 1 can be found on the right side, while sub-patterns most likely\nbelonging to y= 0 on the left."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2185-934e-11ee-8c61-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 5, "chroma:document": "belonging to y= 0 on the left.\nappear somewhat limited, neural networks being one of the exceptions. With propositional\nlogic/Boolean algebra1as the starting point, we here de\ufb01ne the pattern recognition problem\nto be solved by the Tsetlin Machine.\nInput and Output. Consider an input vector of opropositional variables: X= (x1,...,x o)\u2208\nX,X={0,1}o. Together with their negated counterparts, \u00af xk=\u00acxk= 1\u2212xk, the variables"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2186-934e-11ee-8abf-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 5, "chroma:document": "form a literal set L={l1,...,l 2o}={x1,...,x o,\u00afx1,..., \u00afxo}. From the literals, we are to pro-\nduce an output vector of mpropositional variables: Y= (y1,y2,...,ym)\u2208Y,Y={0,1}m.\nFor the sake of simple notation, we will in the remainder of the paper represent any particular\noutput variable yi,i= 1,...,m , as a single output variable y, without loss of generality.\nPatterns. In Tsetlin Machine learning, a pattern is expressed as a conjunctive clause in"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2187-934e-11ee-b889-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 5, "chroma:document": "propositional logic, denoted Cj, withjbeing the index of the clause. A conjunctive clause is\nformed by ANDing a subset Lj\u2286Lof the literal set:\nCj(X) =\u22c0\nlk\u2208Ljlk. (4)\nor, equivalently, as a product of the literals:\nCj(X) =\u220f\nlk\u2208Ljlk. (5)\nFor example, the clause Cj(X) =x1\u2227x2=x1x2consists of the literals Lj={x1,x2}and\noutputs 1 i\ufb00 x1=x2= 1.\nNote that any formula in propositional logic can be reformulated as a disjunction of con-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2188-934e-11ee-a901-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 5, "chroma:document": "junctive clauses (referred to as disjunctive normal form). With oinput variables, one can\nexpress no less than 22ounique propositional formulas, making conjunctive clauses a powerful\nbuilding block for expressing complex non-linear patterns. Arguably, they are also particularly\neasy for humans to comprehend [28].\n1We found the Tsetlin Machine on propositional logic, which can be mapped to Boolean algebra, and vice\nversa.\n6"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2189-934e-11ee-8601-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "Learning Problem and Sub-Patterns. As illustrated in Figure 2, the input space Xcan\nbe partitioned into two parts, X1={X|P(y= 0|X)\u2264P(y= 1|X)}andX0={X|P(y=\n0|X)>P(y= 1|X)}. For input vectors in X1, classi\ufb01cation accuracy is maximized by assigning\nythe value 1. Conversely, for input vectors in X0, accuracy is maximized by assigning ythe\nvalue 0. Accordingly, the learning problem that we address consists of \ufb01nding the partitioning\nof the input space that maximizes classi\ufb01cation accuracy ."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d218a-934e-11ee-886f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "We now come to the crucial concept of unlabelled sub-patterns. As Figure 2 exempli\ufb01es,\nwe decomposeX0andX1inton/2 sub-partitions each, respectively, X0\nj,j\u2208{1,...,n/ 2},\nandX1\nj,j\u2208{1,...,n/ 2}. Each sub-partition X\u03c9\nj,\u03c9\u2208{0,1},is de\ufb01ned by a corresponding\nconjunctive clause, containing all the inputs that make the clause evaluate to 1, and only those\ninputs. Consider for instance the XOR-relation. Then X0decomposes intoX0\n1={X|x1x2= 1}\nandX0\n2={X|\u00afx1\u00afx2= 1}, whileX1decomposes intoX1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d218b-934e-11ee-8103-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "andX0\n2={X|\u00afx1\u00afx2= 1}, whileX1decomposes intoX1\n1={X|x1\u00afx2= 1}andX1\n2={X|\u00afx1x2=\n1}.\nDuring pattern learning, however, we only observe training examples ( X,y) sampled from\nan unknown input-output distribution P(X,y). Which sub-partition Xwas sampled from is\nunavailable to us. What we know is that (i) each sub-partition can be described by a conjunctive\nclause, and (ii) the probability of obtaining a training example from a speci\ufb01c sub-partition\nis maximally close to a known value1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d218c-934e-11ee-b967-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "is maximally close to a known value1\ns,s\u22651, but not equal to or below. That is, we are\ntargeting sub-patterns with a certain frequency. Formally, the pattern recognition problem\nthat we address can be de\ufb01ned as follows:\nDe\ufb01nition 1. Letybe a random output variable taking values \u03c9\u2208 {0,1}and letX=\n(x1,...,x o)refer toorandom input variables with domain X={0,1}o. Given the joint input-\noutput distribution P(X,y), the input domain Xpartitions into two parts, X1={X|P(y="}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d218d-934e-11ee-808e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "0|X)\u2264P(y= 1|X)}andX0={X|P(y= 0|X)> P(y= 1|X)}. Assume that the joint\nprobability distribution P(X,y)relatesytonsub-patterns inX,n/2sub-patterns per y-\nvalue. Let each sub-pattern be de\ufb01ned by a corresponding conjunctive clause Q\u03c9\nj(X),j\u2208\n{1,...,n/ 2},\u03c9\u2208{0,1}. That is, clause Q\u03c9\nj(X)delineates a subset X\u03c9\njof inputs withinX:\nX\u03c9\nj={X|Q\u03c9\nj(X) = 1}. Let these subsets have the following properties:\n1. Conditioned on X\u03c9, the probability of input subset X\u03c9\njis larger than a given constant\n1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d218e-934e-11ee-af81-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "jis larger than a given constant\n1\ns,s\u2208[1,\u221e):P(X\u03c9\nj|X\u03c9)>1\ns.\n2. If clause Q\u03c9\nj(X)is extended with additional literals, the resulting input subset X\u03c9\u2032\njbecomes\na subset ofX\u03c9\nj,X\u03c9\u2032\nj\u2282X\u03c9\nj, and occurs with probability less than1\ns, conditioned onX\u03c9:\nP(X\u03c9\u2032\nj|X\u03c9)<1\ns.\n3. Input outside the X\u03c9\nj-subsets cannot satisfy property (2) above: P(X\\\u22c3\njX\u03c9\nj|X\u03c9)<1\ns.\n4. Each input Xfrom the subsetX\u03c9\njis a noisy predictor of the associated y-value\u03c9,P(y=\n\u03c9|X\u03c9\nj)>0.5, implyingX\u03c9\nj\u2286X\u03c9."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d218f-934e-11ee-8437-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "\u03c9|X\u03c9\nj)>0.5, implyingX\u03c9\nj\u2286X\u03c9.\nUnder these assumptions, given nands, the pattern recognition problem consists of recovering\nthe clauses Q\u03c9\nj(X),j\u2208{1,...n/ 2},\u03c9\u2208{0,1}, only by observing input-output samples (X,y)\nfrom the unknown distribution P(X,y).\nNote that in practice, for real-world data, nandsmay not be known. They can then be\nconsidered as hyper-parameters, optimized through a hyper-parameter search.\n3 The Tsetlin Machine"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2190-934e-11ee-b5bc-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "3 The Tsetlin Machine\nThe Tsetlin Machine decomposes problems into self-contained patterns that are expressed as\nconjunctive clauses in propositional logic. In this section, we \ufb01rst cover the clause-based infer-\nence structure, before we proceed to present the Tsetlin Automata teams, which are responsible\nfor composing the clauses. The teams are coordinated by means of a game that optimizes accu-\nracy using so-called Type I and Type II Feedback, combined with resource allocation dynamics."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2191-934e-11ee-849d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 6, "chroma:document": "We conclude the section with discussing the details of the game, including pseudo code.\n7"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2192-934e-11ee-b99e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 7, "chroma:document": "0/1\n\u2026\u2026\u2026+-+-\n\u2211\u2335\u2335\u2335\u23350 1 0 \u2026 1 0 10/1\u22110/10/1\n0/10/1Output\nInputThreshold FunctionSummationConjunctiveClausesFigure 3: The Tsetlin Machine inference structure, introducing clause polarity, a summation\noperator collecting \u201dvotes\u201d, and a threshold function arbitrating the \ufb01nal output.\n3.1 Tsetlin Machine Inference Structure\nThe Tsetlin Machine inference structure is shown in Figure 3. As seen, an input vector X=\n(x1,...,x o)\u2208X,X={0,1}o,ofopropositional variables is fed to several conjunctive clauses"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2193-934e-11ee-860e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 7, "chroma:document": "for evaluation.\nThe number of clauses employed is a user set parameter n. Half of the clauses are assigned\npositive polarity, denoted by upper index 1: C1\nj,j\u2208{1,...,n/ 2}. The other half is assigned\nnegative polarity, denoted by upper index 0: C0\nj,j\u2208{1,...,n/ 2}. In the \ufb01gure, polarity is\nindicated with a \u2019+\u2019 or \u2019-\u2019 sign, attached to each clause.\nThe role of the clauses is to capture sub-patterns in Xper Eq. 4, however, with a slight\nadjustment. We allow clauses without literals, i.e., L\u03c9"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2194-934e-11ee-bb72-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 7, "chroma:document": "j=\u2205,j\u2208{1,...,n/ 2},\u03c9\u2208{0,1}. These\noutput 1 during learning and 0 during classi\ufb01cation:\nC\u03c9\nj(X) =\uf8f1\n\uf8f4\uf8f4\uf8f2\n\uf8f4\uf8f4\uf8f31, ifL\u03c9\nj=\u2205during learning ,\n0, ifL\u03c9\nj=\u2205during classi\ufb01cation ,\u22c0\nlk\u2208L\u03c9\njlk,ifL\u03c9\nj\u0338=\u2205.(6)\nIn other words, clauses without literals play no role during classi\ufb01cation, and can be pruned\nfrom the inference structure. During learning, however, outputting 1 stimulates further updat-\ning of the clause, as explored in Section 3.3."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2195-934e-11ee-8e0c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 7, "chroma:document": "ing of the clause, as explored in Section 3.3.\nThe clause evaluations, in turn, are combined into a \ufb01nal output decision through summa-\ntion and unit step thresholding, u(v) = 1 ifv\u22650else 0:\n\u02c6y=u\uf8eb\n\uf8edn/2\u2211\nj=1C1\nj(X)\u2212n/2\u2211\nj=1C0\nj(X)\uf8f6\n\uf8f8. (7)\nThat is, output is decided based on a majority vote, with the positive polarity clauses voting\nfory= 1 and the negative ones for y= 0. The purpose is to reach a balanced output"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2196-934e-11ee-ba08-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 7, "chroma:document": "decision, weighting positive against negative evidence, found in the input. The classi\ufb01er \u02c6 y=\nu(x1\u00afx2+ \u00afx1x2\u2212x1x2\u2212\u00afx1\u00afx2), for instance, captures the XOR-relation.\n3.2 The Tsetlin Automata Team for Composing Clauses\nRecall that a Tsetlin Automaton can be represented as a quintuple {\u03a6,\u03b1,\u03b2,F(\u00b7,\u00b7),G(\u00b7)}. Each\nclauseC\u03c9\nj,\u03c9\u2208{0,1},j\u2208{1,...,n/ 2}, of a Tsetlin Machine is composed by a team of such\nTsetlin Automata. The team consists of 2 oautomata, one assigned per literal lk\u2208L. Let\n\u03a6\u03c9"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2197-934e-11ee-b7c4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 7, "chroma:document": "\u03a6\u03c9\njk\u2208{\u03c61,\u03c62...,\u03c6 2N}refer to the state of the Tsetlin Automaton in charge of literal lkfor\n8"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2198-934e-11ee-86e7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 8, "chroma:document": "IncludeExcludex2x1\u00acx2\u00acx1\u2335\u2335\u2335x2x1\u00acx2\u00acx1\u2335\u2335\u2335Clause +Clause \u2014\nX = (x1, x2)Majority votingFeedback Type I or IIClause outputClause output\nInput feature vectorDecided classFeedback Type I or II+\u2014\n123456\n123456\n123456\n123456\n123456\n123456\n123456\n123456Figure 4: Two Tsetlin Automata teams, each producing a conjunctive clause. The overall\noutput is based on majority voting.\nclauseC\u03c9\nj,\u03c9\u2208{0,1},j\u2208{1,...,n/ 2}. It is this state that decides whether the literal is"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d2199-934e-11ee-a7df-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 8, "chroma:document": "excluded (action \u03b11) or included (action \u03b12) in the clause.\nFigure 4 depicts two clauses, one with positive polarity ( C1) and one with negative polarity\n(C0). They are both assigned four Tsetlin Automata, each with six states, controlling the\ninclusion of literals L={x1,x2,\u00acx1,\u00acx2}. The leftmost Tsetlin Automaton in the \ufb01gure\nhas for example decided to include literal x1because it is in state \u03c64. The second Tsetlin"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d219a-934e-11ee-b0ef-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 8, "chroma:document": "Automaton, being in state \u03c63, has decided to exclude literal x2. Given the decisions of the four\nTsetlin Automata of C1, the clause has the form x1\u00afx2. Similarly, clause C0obtains the form\nx1x2.\n3.3 The Tsetlin Machine Game for Learning Conjunctive Clauses\nWe here introduce the game theoretic learning mechanism that guides the Tsetlin Automata\ntowards solving the pattern recognition problem from De\ufb01nition 1. The game is designed"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d219b-934e-11ee-8c41-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 8, "chroma:document": "to deal with the problem of vanishing signal-to-noise ratio for learning automata games (cf.\nSection 1.3).\n3.3.1 Tsetlin Automata Games\nA game of Tsetlin Automata involves multiple automata and is played over several rounds [3].\nIn each round of the game, the participating Tsetlin Automata independently decide upon\ntheir next action from \u03b1. Thus, if two actions are available to each automaton, they jointly\nselect among 2Wunique action con\ufb01gurations, with Wbeing the number of automata. After"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d219c-934e-11ee-91d0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 8, "chroma:document": "the Tsetlin Automata have made their decisions, the round ends with the automata being\nindividually penalized/rewarded based on the jointly selected action con\ufb01guration. The next\nround of the game then starts.\nTo fully specify the game, we need to assign a reward probability per Tsetlin Automaton for\neach unique con\ufb01guration of actions. We refer to these reward probabilities as the payo\ufb00 matrix\nof the game. With Wtwo-action automata, we accordingly need W2Wreward probabilities"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d219d-934e-11ee-aa1a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 8, "chroma:document": "for the payo\ufb00 matrix.\n9"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d219e-934e-11ee-b6b9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "3.3.2 Design of the Payo\ufb00 Matrix\nThe potential complexity of the Tsetlin Machine game is immense, because the decisions of\nevery single Tsetlin Automaton jointly decide the behaviour of the Tsetlin Machine as a whole.\nIndeed, under the right conditions, a single Tsetlin Automaton has the power to completely\ndisrupt a clause by introducing a contradiction. The payo\ufb00s of the game must therefore be\ncarefully designed so that the Tsetlin Automata are guided towards building the clauses that"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d219f-934e-11ee-8cf6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "solve the pattern recognition problem at hand. To complicate further, an explicit enumeration\nof the payo\ufb00s is impractical due to the potentially tremendous size of the payo\ufb00 matrix.\nTo address the above challenges, we decompose the payo\ufb00 matrix based on the notion of\ntrue positive, false negative, and false positive clause output, treating clauses as individual\nclassi\ufb01ers. True negative clause output is ignored to increase the freedom of the automata. By"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a0-934e-11ee-86ad-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "progressively suppressing false negative and false positive clause output, and reinforcing true\npositive clause output, we intend to guide the Tsetlin Automata towards maximizing pattern\nrecognition accuracy. This guiding is based on what we will refer to as Type I and Type II\nFeedback. Based on the principles exposed in this section, we will de\ufb01ne the resulting game\nformally in Section 4.\n3.3.3 Type I Feedback \u2013 Combating False Negative Output"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a1-934e-11ee-885c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "In each round of the game, the Tsetlin Machine receives an input-output sample ( X,y). Type I\nFeedback is then given to the Tsetlin Automata of clauses C\u03c9\njwheny=\u03c9. Consider a particular\nTsetlin Automaton with state \u03a6\u03c9\njk, controlling inclusion of literal lkin clauseC\u03c9\nj, i.e., thejth\nclause of polarity \u03c9. Feedback is then decided by three factors:\n1. The decisions of the Tsetlin Automata team for clause C\u03c9\nj(X) as a whole, manifested by\nthe output of the clause."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a2-934e-11ee-97ec-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "the output of the clause.\n2. The truth value of the literal lkassigned to the automaton.\n3. The action \u03b1z=G(\u03a6\u03c9\njk),z\u2208{1,2},decided by the automaton.\nTable 2 contains the probabilities that we use to generate Type I Feedback, looking up the\nabove three factors. For instance, assume that:\n1. ClauseC\u03c9\nj(X) evaluates to 1,\n2. Literallkis 1, and\n3. Automaton state \u03a6\u03c9\njkproduces the Include action (\u03b12).\nBy examining the corresponding cell in Table 2, we observe that the probability of receiving a"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a3-934e-11ee-9109-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "rewardP(Reward) iss\u22121\ns. Further, the probability of inaction P(Inaction) is1\ns. Finally, the\nprobability of receiving a penalty P(Penalty) is zero.\nNote that the Inaction feedback is a novel extension to the Tsetlin Automaton, which\ntraditionally receives either a Reward or a Penalty. When receiving the Inaction feedback, the\nTsetlin Automaton is simply left unchanged.\nBrief Analysis of Type I Feedback. Notice how the Type I Feedback probabilities"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a4-934e-11ee-806c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "force production of clauses that contain many literals, up to a certain point. We achieve this\nby reinforcing Include actions more strongly than Exclude actions, using reward- and penalty\nprobabilities favouring Include . As seen, the hyper-parameter s,s\u22651, controls how strongly\nwe favour Include . In e\ufb00ect, sdecides how \u201d\ufb01ne-grained\u201d patterns the clauses are going to\ncapture. The larger the value of s, the more the Tsetlin Automata are stimulated to include"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a5-934e-11ee-81ba-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 9, "chroma:document": "literals in their clauses. The countering force is non-matching input, i.e., input that makes the\nclause evaluate to 0. For such input, the Exclude action is rewarded and the Include action\nis penalized. Clearly, the probability of encountering non-matching input grows as sincreases\n(more literals are included). When the above two counter-acting forces are in balance, we have\n10"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a6-934e-11ee-abf8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 10, "chroma:document": "a Nash equilibrium as discussed further in Section 4. These dynamics are a critical part of the\nTsetlin Machine, allowing learning of any sub-pattern, no matter how infrequent, as decided\nbys.\nBoosting of True Positive Feedback (Column 1 in Table 2). The feedback proba-\nbilities in Table 2 have been selected based on mathematical derivations (see Section 4). For\ncertain real-life data sets, however, it turns out that further boosting of Include actions can"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a7-934e-11ee-a4e1-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 10, "chroma:document": "be bene\ufb01cial. That is, pattern recognition accuracy can be enhanced by boosting rewarding of\nthese actions when they produce true positive outcomes. Penalizing of Exclude actions is then\nadjusted accordingly. In all brevity, we boost rewarding in this manner by replacings\u22121\nswith\n1.0 and1\nswith 0.0 in Column 1 of Table 2.\n3.3.4 Type II Feedback \u2013 Combating False Positive Output\nTable 3 covers Type II Feedback, that is, feedback that combats false positive output. Type II"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d21a8-934e-11ee-b5de-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 10, "chroma:document": "Feedback is given to the Tsetlin Automata of clauses C\u03c9\njwheny\u0338=\u03c9. The purpose of this\nfeedback is to introduce candidate literals that increase discrimination power. That is, a clause\nC\u03c9\njis supposed to output 0 when y\u0338=\u03c9, to discriminate between the outputs \u03c9\u2208{0,1}.\nHowever, if it erroneously evaluates to 1, we correct this by identifying the Tsetlin Automata\nthat have excluded a zero-valued literal from the clause. It is su\ufb03cient to merely include one"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4745-934e-11ee-97e3-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 10, "chroma:document": "of these literals to make the o\ufb00ending clause evaluate to 0 instead. This is because the clauses\nare conjunctive.\nBrief Analysis of Type II Feedback. Observe that Type II Feedback only penalizes the\nexclusion of literals of value 0 when a clause outputs 1 (false positive output). When a clause\noutputs 0 (the two leftmost columns in the table), only Inaction-feedback is given. This is to\navoid local minima by leaving it to Type I feedback to reinforce Include actions. Accordingly,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4746-934e-11ee-9926-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 10, "chroma:document": "together, Type I Feedback and Type II Feedback interact to minimize the expected output\nerror, moving towards a global optimum as explored in Section 4.\nTruth Value of Clause C\u03c9\nj 1 0\nTruth Value of Literal lk 1 0 1 0\nInclude Literal ( \u03b12\u2192lk\u2208L\u03c9\nj)P(Reward)s\u22121\nsNA 0 0\nP(Inaction)1\nsNAs\u22121\nss\u22121\ns\nP(Penalty) 0 NA1\ns1\ns\nExclude Literal ( \u03b11\u2192lk/\u2208L\u03c9\nj)P(Reward) 01\ns1\ns1\ns\nP(Inaction)1\nss\u22121\nss\u22121\nss\u22121\ns\nP(Penalty)s\u22121\ns0 0 0"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4747-934e-11ee-aaf7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 10, "chroma:document": "ss\u22121\nss\u22121\nss\u22121\ns\nP(Penalty)s\u22121\ns0 0 0\nTable 2: Type I Feedback \u2014 Feedback from the perspective of a single Tsetlin Automaton\ndeciding to either Include orExclude a given literal lkin the clause C\u03c9\nj. Type I Feedback is\ntriggered to increase the number of clauses that correctly evaluates to 1 for a given input X.\nTruth Value of Clause C\u03c9\nj 1 0\nTruth Value of Literal lk1 0 1 0\nInclude Literal ( \u03b12\u2192lk\u2208L\u03c9\nj)P(Reward) 0 NA 0 0\nP(Inaction) 1.0 NA 1.0 1.0\nP(Penalty) 0 NA 0 0"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4748-934e-11ee-b17e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 10, "chroma:document": "P(Inaction) 1.0 NA 1.0 1.0\nP(Penalty) 0 NA 0 0\nExclude Literal ( \u03b11\u2192lk/\u2208L\u03c9\nj)P(Reward) 0 0 0 0\nP(Inaction) 1.0 0 1.0 1.0\nP(Penalty) 0 1.0 0 0\nTable 3: Type II Feedback \u2014 Feedback from the perspective of a single Tsetlin Automaton\ndeciding to either Include orExclude a given literal lkin the clause C\u03c9\nj. Type II Feedback is\ntriggered to increase the discrimination power of the clauses.\n11"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4749-934e-11ee-8989-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "3.3.5 Allocating Clauses to Sub-Patterns\nFor the Tsetlin Machine output to be correct it is su\ufb03cient that the summation part of Eqn. 7,\u2211n/2\nj=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X), has the correct sign. That is, it is su\ufb03cient that a single clause,\nwith the appropriate polarity, outputs 1. In the case of noisy data, however, it can be bene\ufb01cial\nto introduce a margin T, referred to as the target of the summation. The intent is to make the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d474a-934e-11ee-b8b5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "available clauses distribute themselves across the sub-patterns in the data, so that Tclauses\nof the correct polarity evaluate to 1 for any particular input X. Additionally, the summation\ntargetTopens up for interplay between the clauses, including recti\ufb01cation of special cases.\nThe resource allocation mechanism that we introduce here is inspired by a \ufb01nite state\nautomaton-based solution to the fractional knapsack problem in unknown and stochastic en-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d474b-934e-11ee-a153-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "vironments [29]. The purpose of the mechanism is to ensure that the Tsetlin Machine only\nspends a few of the available clauses to represent each speci\ufb01c sub-pattern. This is achieved\nby randomly selecting clauses to receive feedback, reducing intensity when approaching the\nsummation target T.\nGenerating Type I Feedback. For Type I Feedback, we randomly select clauses C\u03c9\nj,\u03c9=\ny. Among these, the probability of receiving Type I Feedback is\nT\u2212clip(\u2211n/2\nj=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\n2T(8)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d474c-934e-11ee-8e54-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "j=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\n2T(8)\nwheny= 1, and\nT+clip(\u2211n/2\nj=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\n2T(9)\nwheny= 0. Here, the clip operation restricts the sum to lie between \u2212TandT.\nGenerating Type II Feedback. For Type II Feedback, we randomly select clauses\nC\u03c9\nj,\u03c9\u0338=y. The probability of receiving Type II Feedback is:\nT+clip(\u2211n/2\nj=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\n2T(10)\nwheny= 1, and\nT\u2212clip(\u2211n/2\nj=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\n2T(11)\nwheny= 0."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d474d-934e-11ee-bf0e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "j=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\n2T(11)\nwheny= 0.\nNotice how the feedback vanishes as the number of triggering clauses correctly approaches\nT/\u2212T. This is pertinent for e\ufb00ective use of the available pattern representation capacity. To\nexemplify, assume that the correct output is y= 1 for an input X. If the clause output sum\naccumulates to a total of Tor more, neither rewards nor penalties are provided to the involved"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d474e-934e-11ee-be4b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "Tsetlin Automata. This leaves the Tsetlin Automata free to learn other sub-patterns.\n3.3.6 The Tsetlin Machine Algorithm\nThe step-by-step procedure for learning conjunctive clauses can be found in Algorithm 1. The\nalgorithm takes a set of training examples ( X,y)\u2208Sas input. It then produces a collection of\nconjunctive clauses for predicting the output y. We will now take a closer look at the algorithm,\nline-by-line."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d474f-934e-11ee-8b9d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "line-by-line.\nLines 2-3. From the perspective of game theory, we form one team of Tsetlin Automata\nper clause, denoted A0\nj,A1\nj,j\u2208{1,...,n/ 2}. A team consists of 2 oautomata, each assigned\na user speci\ufb01ed number of states, \u03a6\u03c9\njk\u2208{\u03c61,\u03c62,...,\u03c6 2N},j\u2208{1,...,n/ 2},k\u2208{1,..., 2o},\n\u03c9\u2208{0,1}. The initial state is then set to \u03a6\u03c9\njk\u2190\u03c6N(the Exclude action).\nLine 5. The learning process is driven by a set of training examples S, sampled from the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4750-934e-11ee-b9a9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 11, "chroma:document": "input-output distribution P(X,y). Each training example ( X,y) is fed to the Tsetlin Machine,\none at a time, facilitating online learning.\n12"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4751-934e-11ee-b470-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 12, "chroma:document": "Algorithm 1 Tsetlin Machine\nInput Training data ( X,y)\u2208S\u223cP(X,y), Number of clauses n, Number of inputs o,\nPrecisions, TargetT\nOutput Trained conjunctive clauses C1\nj,C0\nj,j= 1,...,n/ 2\n1:function TrainTsetlinMachine (S,n,o,s,T )\n2:A1\n1,...,A1\nn/2\u2190CreateTATeams( n/2,2o,\u03c6N)\u22bfCreaten/2 teams of 2 oTsetlin\nAutomata (TA), one team A1\njper\nclauseC1\nj. Each TA is initialized to\nstate\u03c6N.\n3:A0\n1,...,A0\nn/2\u2190CreateTATeams( n/2,2o,\u03c6N)\u22bfCreate another n/2 teams of 2 oTA,\none teamA0\njper clause C0\nj. Each"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4752-934e-11ee-a9a2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 12, "chroma:document": "one teamA0\njper clause C0\nj. Each\nTA is initialized to state \u03c6N.\n4: repeat\n5:X,y\u2190GetTrainingExample( S) \u22bfMini-batches, random selection, etc.\n6:C1\n1,...,C1\nn/2\u2190ObtainClauses(A1\n1,...,A1\nn/2)\u22bfEach TA teamA1\njproduces a\nclauseC1\nj.\n7:C0\n1,...,C0\nn/2\u2190ObtainClauses(A0\n1,...,A0\nn/2)\u22bfEach TA teamA0\njproduces a\nclauseC0\nj.\n8:v\u2190\u2211n/2\nj=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X) \u22bfCalculate sum of clause outputs.\n9: forj\u21901,...,n/ 2do \u22bfGive feedback to the TA teams.\n10: ify= 1then\n11: ifRandom()\u2264T\u2212clip(v,\u2212T,T)\n2Tthen"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4753-934e-11ee-8c0c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 12, "chroma:document": "11: ifRandom()\u2264T\u2212clip(v,\u2212T,T)\n2Tthen\n12: GenerateTypeIFeedback( X,C1\nj(X),A1\nj,o)\u22bfGive Type I Feedback to\nTA of clause C1\nj.\n13: end if\n14: ifRandom()\u2264T\u2212clip(v,\u2212T,T)\n2Tthen\n15: GenerateTypeIIFeedback( X,C0\nj(X),A0\nj,o)\u22bfGive Type II Feedback to\nTA of clause C0\nj.\n16: end if\n17: else ify= 0then\n18: ifRandom()\u2264T+clip(v,\u2212T,T)\n2Tthen\n19: GenerateTypeIIFeedback( X,C1\nj(X),A1\nj,o)\u22bfGive Type II Feedback to\nTA of clause C1\nj.\n20: end if\n21: ifRandom()\u2264T+clip(v,\u2212T,T)\n2Tthen"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4754-934e-11ee-a562-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 12, "chroma:document": "20: end if\n21: ifRandom()\u2264T+clip(v,\u2212T,T)\n2Tthen\n22: GenerateTypeIFeedback( X,C0\nj(X),A0\nj,o)\u22bfGive Type I Feedback to\nTA of clause C0\nj.\n23: end if\n24: end if\n25: end for\n26: until StopCriteria(S,C0\n1,...,C0\nn/2,C1\n1,...,C1\nn/2)\n27: return PruneAllExcludeClauses( C0\n1,...,C0\nn/2,C1\n1,...,C1\nn/2)\u22bfReturn completely\ntrained conjunctive clauses after pruning clauses where all literals have been excluded.\n28:end function\n13"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4755-934e-11ee-bb6d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "Algorithm 2 Type I Feedback\nInput Input vector Xof sizeo, Clause output c\u03c9\nj, Tsetlin Automata team A\u03c9\nj\n1:procedure GenerateTypeIFeedback (X,c\u03c9\nj,A\u03c9\nj,o)\n2: fork\u21901,..., 2odo \u22bfReward/Penalize all Tsetlin Automata in A\u03c9\nj.\n3:lk\u2190ObtainLiteral( X,k) \u22bfGet literallkfrom input X.\n4:\u03b1u\u2190G(\u03a6\u03c9\njk)\u22bfObtain corresponding automaton action according to Eqn. 1.\n5:\u03b2v\u2190SampleTypeIFeedback( \u03b1u,lk,c\u03c9\nj)\u22bfSample Type I Feedback from Table 2.\n6: \u03a6\u03c9\njk\u2190F(\u03a6\u03c9\njk,\u03b2v)\u22bfUpdate state of Tsetlin Automaton according to Eqn. 2."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4756-934e-11ee-afe6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "7: end for\n8:end procedure\nAlgorithm 3 Type II Feedback\nInput Input vector Xof sizeo, Clause output c\u03c9\nj, Tsetlin Automata team A\u03c9\nj\n1:procedure GenerateTypeIIFeedback (X,c\u03c9\nj,A\u03c9\nj,o)\n2: fork\u21901,..., 2odo \u22bfPenalize all Tsetlin Automata in A\u03c9\nj.\n3:lk\u2190ObtainLiteral( X,k) \u22bfGet literallkfrom input X.\n4:\u03b1u\u2190G(\u03a6\u03c9\njk)\u22bfObtain corresponding automaton action according to Eqn. 1.\n5:\u03b2v\u2190SampleTypeIIFeedback( \u03b1u,lk,c\u03c9\nj)\u22bfSample Type II Feedback from Table 3.\n6: \u03a6\u03c9\njk\u2190F(\u03a6\u03c9"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4757-934e-11ee-b245-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "6: \u03a6\u03c9\njk\u2190F(\u03a6\u03c9\njk,\u03b2v)\u22bfUpdate state of Tsetlin Automaton according to Eqn. 2.\n7: end for\n8:end procedure\nLine 6-7. In each iteration, the Tsetlin Automata decide whether to Include orExclude\ntheir assigned literals in their respective clauses. The result is a new set of clauses, C0\n1,...,C0\nn/2,\nC1\n1,...,C1\nn/2, for predicting y.\nLine 8. Based on the new clauses, the current example Xis evaluated, producing a clause"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4758-934e-11ee-87d4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "sumvaccording to Eqn. 7. The unit step function uis not used here because examples are\nnot classi\ufb01ed during training.\nLines 9-25. The next step is to randomly determine the clauses whose Tsetlin Automata\nare to receive Type I Feedback (Eqn. 8-9) and Type II Feedback (Eqn. 10-11), based on the\nclipped di\ufb00erence between vandT.\nAfter Type I or Type II Feedback have been triggered for a clause, the Tsetlin Automata"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4759-934e-11ee-9214-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "of the clause are rewarded/penalized according to Algorithm 2 and Algorithm 3. In all brevity,\nrewarding/penalizing is directly based on Table 2 and Table 3.\nLine 26. The above steps are iterated until a stopping criteria is ful\ufb01lled (for instance a\ncertain number of iterations over the dataset). The current clauses are then returned as the\noutput of the learning process, after the clauses without literals have been removed.\n3.4 Implementation Using Bitwise Operators"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d475a-934e-11ee-8a91-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "3.4 Implementation Using Bitwise Operators\nSmall memory footprint and speed of operation can be crucial in complex and large scale\npattern recognition. Being based on propositional formula, the Tsetlin Machine architecture\ncan naturally be represented with bits and manipulated upon using bitwise operators. However,\nit is not straightforward how to represent and update the Tsetlin Automata themselves. First"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d475b-934e-11ee-8b06-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "of all, the state index of each Tsetlin Automaton is an integer value. Further, the action of\nan automaton is decided upon using a smaller-than operator, while feedback is processed by\nmeans of increment and decrement operations.\nOne approach to bitwise operation is to jointly represent the state indexes of all of the\nTsetlin Automata of a clause C\u03c9\njwith multiple sequences of bits. Sequence 1 then contains the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d475c-934e-11ee-8b52-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 13, "chroma:document": "\ufb01rst bit of each state index, sequence 2 contains the second bit, and so on, as exempli\ufb01ed in\nFigure 5 for 24 Tsetlin Automata. The bene\ufb01t of this representation is that the action of each\nTsetlin Automaton is readily available from the most signi\ufb01cant bit (sequence 8 in the \ufb01gure).\n14"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d475d-934e-11ee-ac90-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 14, "chroma:document": "Thus, the output of the clause can be obtained from the input based on fast bitwise operators\n(NOT, AND, and CMP - comparison).\nIn the \ufb01gure, a setup for the Noisy XOR dataset from Section 5 is used for illustration\npurposes. First, the 12-bit input is extended to 24 bits by concatenating the original input\nwith the original input inverted. The resulting 24 bits are in turn connected with 24 Tsetlin\nAutomata. The \ufb01rst 12 control the non-negated input, while the second 12 control the negated"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d475e-934e-11ee-bcb5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 14, "chroma:document": "input.\nEmploying the latter bit-based representation reduces memory usage four times, compared\nto using a full 32-bit integer to represent the state of each and every Tsetlin Automaton.\nMore importantly, it is possible to increment/decrement the states of all of the automata in\nparallel with bitwise operations through customized increment/decrement procedures, signi\ufb01-\ncantly increasing learning speed. As an example, for the MNIST dataset (cf. Section 5), the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d475f-934e-11ee-b703-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 14, "chroma:document": "overall memory usage is approximately ten times smaller, learning speed 3.5 times faster, and\nclassi\ufb01cation speed 8 times faster with the bit-based representation.\nWhen deployed after training, only the state bit sequence containing the most signi\ufb01cant\nbit is required. The other bit sequences can be discarded because these bits are only used to\nkeep track of the learning. This provides a further reduction in memory usage."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4760-934e-11ee-9d60-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 14, "chroma:document": "State of 24 Tsetlin Automata 110101010011110101010011001010101100NOTAND100000000000 010000000000101000100000 010011010000111000010001 010010011011\u2026State Bits1-78CMPOutput\nInput0\nFigure 5: Bit-based representation of a clause for the Noisy XOR dataset (see Section 5).\nThe bit-based representation of the Tsetlin Automata states allows the actions of all of the\nautomata to be obtained directly from the most signi\ufb01cant bit (bit 8 in the \ufb01gure).\n4 Theoretical Analysis"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4761-934e-11ee-babb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 14, "chroma:document": "4 Theoretical Analysis\nIn this section, we analyse the Tsetlin Machine game formally. We base the analysis on par-\ntitioning the input space Xinto subsets that isolate the impact a single Tsetlin Automaton\nhas on classi\ufb01cation, mediated through its clause. We then study each of these subsets for the\ntwo output scenarios y= 0 andy= 1. From this investigation, we identify the Nash equi-\nlibria of Tsetlin Machine learning and compare the Nash equilibria clauses with the optimal"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4762-934e-11ee-a666-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 14, "chroma:document": "sub-patterns from De\ufb01nition 1.\n4.1 The Payo\ufb00 Matrix of the Tsetlin Machine Game\nBased on the learning dynamics described algorithmically in Section 3, we now introduce a\nmore succinct description of Tsetlin Machine learning by formally de\ufb01ning the payo\ufb00 matrix\nof the Tsetlin Machine game.\nDe\ufb01nition 2. Consider a pattern recognition problem per De\ufb01nition 1 and a Tsetlin Ma-\nchine with nclausesC\u03c9\nj(X),j\u2208 {1,...,n/ 2},\u03c9\u2208 {0,1}. The resulting game consists of\n15"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4763-934e-11ee-96b4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 15, "chroma:document": "2noTsetlin Automata (players), one automaton per literal lk\u2208L={x1,...,x o,\u00afx1,..., \u00afxo},\nper clause C\u03c9\nj(X). LetA\u03c9\njkbe a Tsetlin Automaton with actions a\u03c9\njk\u2208 {\u03b11,\u03b12}, control-\nling the inclusion of literal lkin clauseC\u03c9\nj(\u03b11represents Exclude and \u03b12represents In-\nclude). Each cell in the payo\ufb00 matrix of the game then refers to a unique action con\ufb01guration\n\u03b1= (a0\n1,1,...,a0\nn/2,2o,a1\n1,1...,a1\nn/2,2o)\u2208{\u03b11,\u03b12}2no. Accordingly, the payo\ufb00 matrix consists of"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4764-934e-11ee-aa62-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 15, "chroma:document": "22nocells. For every cell, we de\ufb01ne the payo\ufb00 of each a\u03c9\njk-action in\u03b1using the four stochastic\nvariablesX,y,U\u03c9\nj, andR\u03c9\njk:\n\u2022Each game round, the distribution P(X,y)randomly produces a training example (X,y).\n\u2022Let the binary stochastic variable U\u03c9\njrefer to whether clause C\u03c9\njis updated ( U\u03c9\nj= 1)\nor ignored ( U\u03c9\nj= 0) in the given game round. The distribution of U\u03c9\njis decided as\nfollows. First, the clipped clause sum vc=clip(\u2211n/2\nj=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\nis"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4765-934e-11ee-b18b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 15, "chroma:document": "j=1C1\nj(X)\u2212\u2211n/2\nj=1C0\nj(X),\u2212T,T)\nis\ndetermined. This clipped clause sum is calculated from the clauses produced by the joint\naction con\ufb01guration \u03b1of the payo\ufb00 matrix cell, evaluated upon X. ClauseC\u03c9\njis then\nupdated with probability P(U\u03c9\nj= 1) =T\u2212vc\n2Tify= 1 and with probability P(U\u03c9\nj= 1) =\nT+vc\n2Tify= 0.\n\u2022Finally, we decide the payo\ufb00 R\u03c9\njk\u2208{\u2212 1,0,+1}(Penalty, Inaction, Reward) for each\nautomaton A\u03c9\njk. IfU\u03c9\nj= 0 the payo\ufb00 is 0. Otherwise, R\u03c9\njkis randomly set based on"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4766-934e-11ee-aed5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 15, "chroma:document": "jkis randomly set based on\nTable 2 ify=\u03c9(Type I Feedback) and Table 3 if y\u0338=\u03c9(Type II Feedback). We obtain\nthe distribution of R\u03c9\njkfrom the designated table by looking up: (1) the action selected\n(\u03b11or\u03b12), (2) the value of literal lkfromX, and (3) the value of the clause C\u03c9\nj(X).\nAccordingly, the payo\ufb00 of each action a\u03c9\njkin\u03b1becomesU\u03c9\nj\u00b7R\u03c9\njk.\nIn the following, we analyze the dynamics of the game using the above payo\ufb00 matrix.\nFigure 6: The subsets X0,X1,X\u03c9\nj, andX\u03c9"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4767-934e-11ee-a6e1-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 15, "chroma:document": "Figure 6: The subsets X0,X1,X\u03c9\nj, andX\u03c9\njk, pertinent for the theoretical analysis.\n4.2 Pertinent Input Space Subsets\nThe input space subsets ( X-subsets) of interest are listed in Table 4 and Table 5. Observe that\nall of theX-subsets in the tables are derived from the following four subsets: X0,X1,X1\nj, and\nX1\njk, illustrated in Figure 6.\nThe subsetX1={X|P(y= 0|X)\u2264P(y= 1|X)}is the subset of the input space where it\nis optimal to output y= 1. Conversely, the subset X0={X|P(y= 0|X)> P(y= 1|X)}is"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4768-934e-11ee-ac9e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 15, "chroma:document": "the compliment of X, where it is optimal to output y= 0. The subsetX1\njk={X|C\u03c9\njk(X) = 1},\nL\u03c9\njk\u2286L,lk\u2208Ljk, is the part of the input space in which the clause C\u03c9\njk(X) with literals\nL\u03c9\njk,lk\u2208L\u03c9\njk,evaluates to 1. Finally, the subset X1\nj={X|C\u03c9\nj(X) = 1},L\u03c9\nj=L\u03c9\njk\\lk, is a\nsuperset ofX1\njk, where the literal lkhas been removed from the conjunction.\n16"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4769-934e-11ee-85b2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "Recall that it is a single Tsetlin Automaton A\u03c9\njkthat decides whether to Include orExclude\nliterallkabove, which means that it in e\ufb00ect decides between clause C\u03c9\njk(X) (literal included)\nand clause C\u03c9\nj(X) (literal excluded).\nFurther note that we in this section ignore polarity to simplify notation. That is, we analyze\nhow the clauses with positive polarity in the Tsetlin Machine game capture the sub-patterns\nQ1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d476a-934e-11ee-bc4c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "Q1\nh,h\u2208{1,...,n/ 2},from De\ufb01nition 1. The analysis for negative polarity clauses follows the\nsame structure.\n4.3 Expected Payo\ufb00 for the Actions of a Single Tsetlin Automaton\nAs de\ufb01ned in the payo\ufb00 matrix of the Tsetlin Machine game, the stochasticity in Tsetlin\nMachine learning comes from: (i) the training samples ( X,y), which are drawn randomly\naccording to an unknown distribution P(X,y); (ii) the random selection of clauses for updating;"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d476b-934e-11ee-a643-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "and (iii) the random generation of the rewards and penalties from Type I feedback (Table 2).\nWe \ufb01rst focus on what happens with a speci\ufb01c Tsetlin Automaton each time its clause activates.\nThus, for the moment, we ignore stochasticity stemming from random clause selection, studying\na single clause.\nWe have collected the information needed to calculate the expected payo\ufb00 for the Exclude\naction in Table 4 and for the Include action in Table 5. The \ufb01rst column of the table enumerates"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d476c-934e-11ee-9b1f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "the possible y-values. The second column speci\ufb01es a partitioning of the input space Xfor each\ny-value. The third column contains the expected payo\ufb00 of the action covered by the table,\nfor the givenX-subset and output y. The expected payo\ufb00 has been obtained from Table 2\nand Table 3. The fourth column states the probability of obtaining a training example ( X,y)\nyX-Subset R Probability Noise 1\u2212\u03b3\n0X1\njk\u2229X10P(y= 0|X1\njk\u2229X1)P(X1\njk\u2229X1) (1\u2212\u03b3)P(X1\njk\u2229X1)\n0(X1\nj\\X1\njk)\u2229X1\u22121P(y= 0|[X1\nj\\X1\njk]\u2229X1)P([X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d476d-934e-11ee-881f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "0(X1\nj\\X1\njk)\u2229X1\u22121P(y= 0|[X1\nj\\X1\njk]\u2229X1)P([X1\nj\\X1\njk]\u2229X1)(1\u2212\u03b3)P([X1\nj\\X1\njk]\u2229X1)\n0X1\\X1\nj 0P(y= 0|X1\\X1\nj)P(X1\\X1\nj) (1\u2212\u03b3)P(X1\\X1\nj)\n0X1\njk\u2229X00P(y= 0|X1\njk\u2229X0)P(X1\njk\u2229X0) \u03b3P(X1\njk\u2229X0)\n0(X1\nj\\X1\njk)\u2229X0\u22121P(y= 0|[X1\nj\\X1\njk]\u2229X0)P([X1\nj\\X1\njk]\u2229X0)\u03b3P([X1\nj\\X1\njk]\u2229X0)\n0X0\\X1\nj 0P(y= 0|X0\\X1\nj)P(X0\\X1\nj) \u03b3P(X0\\X1\nj)\n1X1\njk\u2229X1\u2212s\u22121\nsP(y= 1|X1\njk\u2229X1)P(X1\njk\u2229X1) \u03b3P(X1\njk\u2229X1)\n1(X1\nj\\X1\njk)\u2229X1 1\nsP(y= 1|[X1\nj\\X1\njk]\u2229X1)P([X1\nj\\X1\njk]\u2229X1)\u03b3P([X1\nj\\X1\njk]\u2229X1)\n1X1\\X1\nj1\nsP(y= 1|X1\\X1\nj)P(X1\\X1\nj) \u03b3P(X1\\X1\nj)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d476e-934e-11ee-9a0f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "1X1\\X1\nj1\nsP(y= 1|X1\\X1\nj)P(X1\\X1\nj) \u03b3P(X1\\X1\nj)\n1X1\njk\u2229X0\u2212s\u22121\nsP(y= 1|X1\njk\u2229X0)P(X1\njk\u2229X0) (1\u2212\u03b3)P(X1\njk\u2229X0)\n1(X1\nj\\X1\njk)\u2229X0 1\nsP(y= 1|[X1\nj\\X1\njk]\u2229X0)P([X1\nj\\X1\njk]\u2229X0)(1\u2212\u03b3)P([X1\nj\\X1\njk]\u2229X0)\n1X0\\X1\nj1\nsP(y= 1|X0\\X1\nj)P(X0\\X1\nj) (1\u2212\u03b3)P(X0\\X1\nj)\nTable 4: Expected payo\ufb00 for the Exclude action.\nyX-Subset R Probability Noise 1\u2212\u03b3\n0X1\njk\u2229X10P(y= 0|X1\njk\u2229X1)P(X1\njk\u2229X1) (1\u2212\u03b3)P(X1\njk\u2229X1)\n0(X1\nj\\X1\njk)\u2229X10P(y= 0|[X1\nj\\X1\njk]\u2229X1)P([X1\nj\\X1\njk]\u2229X1)(1\u2212\u03b3)P([X1\nj\\X1\njk]\u2229X1)\n0X1\\X1\nj 0P(y= 0|X1\\X1\nj)P(X1\\X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d476f-934e-11ee-b576-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "j\\X1\njk]\u2229X1)\n0X1\\X1\nj 0P(y= 0|X1\\X1\nj)P(X1\\X1\nj) (1\u2212\u03b3)P(X1\\X1\nj)\n0X1\njk\u2229X00P(y= 0|X1\njk\u2229X0)P(X1\njk\u2229X0) \u03b3P(X1\njk\u2229X0)\n0(X1\nj\\X1\njk)\u2229X00P(y= 0|[X1\nj\\X1\njk]\u2229X0)P([X1\nj\\X1\njk]\u2229X0)\u03b3P([X1\nj\\X1\njk]\u2229X0)\n0X0\\X1\nj 0P(y= 0|X0\\X1\nj)P(X0\\X1\nj) \u03b3P(X0\\X1\nj)\n1X1\njk\u2229X1 s\u22121\nsP(y= 1|X1\njk\u2229X1)P(X1\njk\u2229X1) \u03b3P(X1\njk\u2229X1)\n1(X1\nj\\X1\njk)\u2229X1\u22121\nsP(y= 1|[X1\nj\\X1\njk]\u2229X1)P([X1\nj\\X1\njk]\u2229X1)\u03b3P([X1\nj\\X1\njk]\u2229X1)\n1X1\\X1\nj\u22121\nsP(y= 1|X1\\X1\nj)P(X1\\X1\nj) \u03b3P(X1\\X1\nj)\n1X1\njk\u2229X0 s\u22121\nsP(y= 1|X1\njk\u2229X0)P(X1\njk\u2229X0) (1\u2212\u03b3)P(X1\njk\u2229X0)\n1(X1\nj\\X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4770-934e-11ee-ac9c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 16, "chroma:document": "jk\u2229X0)P(X1\njk\u2229X0) (1\u2212\u03b3)P(X1\njk\u2229X0)\n1(X1\nj\\X1\njk)\u2229X0\u22121\nsP(y= 1|[X1\nj\\X1\njk]\u2229X0)P([X1\nj\\X1\njk]\u2229X0)(1\u2212\u03b3)P([X1\nj\\X1\njk]\u2229X0)\n1X0\\X1\nj\u22121\nsP(y= 1|X0\\X1\nj)P(X0\\X1\nj) (1\u2212\u03b3)P(X0\\X1\nj)\nTable 5: Expected payo\ufb00 for the Include action.\n17"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4771-934e-11ee-900a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 17, "chroma:document": "for the corresponding X-subset and output y. The \ufb01fth column speci\ufb01es these probabilities\nassuming a static output noise probability P(y= 0|X1) =P(y= 1|X0) = 1\u2212\u03b3,\u03b3 > 0.5. If\nfor instance \u03b3= 0.75, the output is erroneously y= 0 forX\u2208X1andy= 1 forX\u2208X0,\non average, 25% of the time. The assumption of static noise is not a limitation of the Tsetlin\nMachine, however, the assumption simpli\ufb01es the theoretical analysis that follows."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4772-934e-11ee-b797-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 17, "chroma:document": "From Table 4 and Table 5, we can calculate the expected payo\ufb00 of each action by summing\nup the rewards (positive polarity) and penalties (negative polarity) in the R-column, each\nmultiplied with the probability of the X-subset and y-value of its row. The addend from the\n\ufb01rst row in Table 4 is for instance 0 \u00b7P(y= 0|X1\njk\u2229X1)P(X1\njk\u2229X1), which becomes zero\nbecause the reward is zero.\nLemma 1. Consider a literal lkand clause C\u03c9\nj. The expected payo\ufb00 of excluding lkfromC\u03c9\nj"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4773-934e-11ee-93ac-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 17, "chroma:document": "j. The expected payo\ufb00 of excluding lkfromC\u03c9\nj\nin the Tsetlin Machine game (De\ufb01nition 2) is:\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk) +1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\njk)\u2212s\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1)\u2212\ns\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1)\u2212\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X0).(12)\nProof. We prove Lemma 1 based on Table 4. The expected payo\ufb00 of the Exclude action can\nbe calculated as follows, assuming static noise probability 1 \u2212\u03b3:\n1\ns\u00b7\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X1) +\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\nj) +\n1\ns\u00b7(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X0) +\n1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\nj)\u2212\ns\u22121\ns\u00b7\u03b3\u00b7P(X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4774-934e-11ee-a11d-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 17, "chroma:document": "j\\X1\njk]\u2229X0) +\n1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\nj)\u2212\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1)\u2212\ns\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212\n(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1)\u2212\n\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X0). (13)\nSince (X1\nj\\X1\njk)\u2229(X1\\X1\nj) =\u2205and (X1\nj\\X1\njk)\u222a(X1\\X1\nj) =X1\\X1\njk, we can simplify to:\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk) +\n1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\njk)\u2212\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1)\u2212\ns\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212\n(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1)\u2212\n\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X0). (14)\nLemma 2. Consider a literal lkand clause C\u03c9\nj. The expected payo\ufb00 of including lkinC\u03c9\njin\nthe Tsetlin Machine game (De\ufb01nition 2) is:"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4775-934e-11ee-bb6e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 17, "chroma:document": "jin\nthe Tsetlin Machine game (De\ufb01nition 2) is:\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1) +s\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\njk). (15)\n18"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4776-934e-11ee-90df-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 18, "chroma:document": "Proof. The expected payo\ufb00 of the Include action can be derived from Table 5:\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1) +\ns\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212\n1\ns\u00b7\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X1)\u2212\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\nj)\u2212\n1\ns\u00b7(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X0)\u2212\n1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\nj), (16)\nwhich again simpli\ufb01es to:\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1) +\ns\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\u2212\n1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\njk). (17)\n4.4 Tsetlin Automaton Convergence Criteria\nMost bandit algorithms attempt to learn the action with the largest expected payo\ufb00. A Tsetlin"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4777-934e-11ee-a027-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 18, "chroma:document": "Automaton, however, can only learn this action if the expected payo\ufb00 of the action is posi-\ntive (i.e., the probability of receiving a reward is larger than the probability of receiving a\npenalty) [3]. If the latter criterion is not ful\ufb01lled, the Tsetlin Automaton instead moves to-\nwards its center states ( \u03c6Nand\u03c6N+1), that is, towards indecision. This property is crucial for\neliminating sub-optimal Nash equilibria in Tsetlin Machine learning, and is one of the reasons"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4778-934e-11ee-a6c5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 18, "chroma:document": "why it is not straightforward to replace the Tsetlin Automaton with another bandit algorithm,\nwithout loss in learning accuracy. So, in order to have a stable Nash equilibrium in Tsetlin\nMachine learning, each Tsetlin Automaton action in the equilibrium must not only have the\nlargest expected payo\ufb00, the expected payo\ufb00 must also be positive.\nLemma 3. By increasing the number of states, a two-action Tsetlin Automaton converges to"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4779-934e-11ee-965a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 18, "chroma:document": "performing an action \u03b1z,z\u2208{0,1},with probability arbitrarily close to unity i\ufb00:\n1. The action has positive expected payo\ufb00: E[R|\u03b1z]>0.\n2. The action has the largest expected payo\ufb00: E[R|\u03b1z]>E[R|\u03b11\u2212z].\nProof. The lemma follows trivially from the Tsetlin Automaton convergence proof found in [3].\n4.5 Nash Equilibria Without Noise\nLemma 4. Consider a pattern recognition problem per De\ufb01nition 1 without noise, P(y\u0338=\n\u03c9|X\u03c9) = 1\u2212\u03b3= 0, and a Tsetlin Machine game per De\ufb01nition 2 with a single clause C1(X)of"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d477a-934e-11ee-a7ec-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 18, "chroma:document": "positive polarity. The sub-patterns Q1\nh(X),h\u2208{1,...n/ 2},of the pattern recognition problem\nare then all Nash equilibria of the game. Additionally, the expected payo\ufb00 of each player action\nin each equilibrium is positive, hence not rejected by the Tsetlin Automata players.\n19"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d477b-934e-11ee-81ec-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 19, "chroma:document": "Proof. We \ufb01rst analyse the case when the Exclude action is optimal, before addressing the\nInclude action.\nExclude is Optimal. From Property 1 and Property 2 of De\ufb01nition 1, we have P(X1\njk|X1)<\n1\ns<P(X1\nj|X1)\u21d4P(X1\njk\u2229X1)\nP(X1)<1\ns<P(X1\nj\u2229X1)\nP(X1), which simpli\ufb01es toP(X1\njk)\nP(X1)<1\ns<P(X1\nj)\nP(X1)because\nof Property 4. From Property 4 we know in particular that [ X1\nj\\X1\njk]\u2229X0=\u2205. Under these\nconditions, we must verify that Exclude is also the preferred action according to Lemma 3."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d477c-934e-11ee-a0a4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 19, "chroma:document": "We \ufb01rst verify that the expected payo\ufb00 of Exclude (Eqn. 12) is positive to accommodate for\nCriterion 1 of Lemma 3. We here assume no noise, i.e., \u03b3= 1, and thus can remove the addends\ncontaining the factor (1 \u2212\u03b3) from Eqn. 12. We further know that P(X1\nj\\X1\njk]\u2229X0) = 0 because\n[X1\nj\\X1\njk]\u2229X0=\u2205. Hence, we can remove addends including that factor as well, ending up\nwith:\n1\ns\u00b7P(X1\\X1\njk)\u2212s\u22121\ns\u00b7P(X1\njk\u2229X1)>0. (18)\nBecauseX1\njis a subset ofX1(Property 4), we have:\n1\ns\u00b7P(X1\\X1\njk)\u2212s\u22121\ns\u00b7P(X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d477d-934e-11ee-9dca-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 19, "chroma:document": "1\ns\u00b7P(X1\\X1\njk)\u2212s\u22121\ns\u00b7P(X1\njk)>0. (19)\nDividing by P(X1) on each side we get:\n1\ns\u00b7P(X1\\X1\njk)\nP(X1)\u2212s\u22121\ns\u00b7P(X1\njk)\nP(X1)>0. (20)\nWithP(X1\njk)\nP(X1)<1\ns, we haveP(X1\\X1\njk)\nP(X1)>s\u22121\ns. This clearly entails that the left side is positive.\nWe next verify that the expected payo\ufb00 of Exclude is larger than that of Include to ensure\nCriterion 2 of Lemma 3. Performing the same simpli\ufb01cations as above, however, for Eqn. 15,\nwe end up with the following condition:\ns\u22121\ns\u00b7P(X1\njk)\nP(X1)\u22121\ns\u00b7P(X1\\X1\njk)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d477e-934e-11ee-a63b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 19, "chroma:document": "s\u22121\ns\u00b7P(X1\njk)\nP(X1)\u22121\ns\u00b7P(X1\\X1\njk)\nP(X1)<0. (21)\nFor the same reasons that the expected payo\ufb00 of the Exclude action was positive, the expected\npayo\ufb00 of Include is negative, which clearly is inferior.\nInclude is Optimal. Conversely, given that Include is optimal, we have:P(X1\njk)\nP(X1)>1\ns\n(Property 1) and X1\njk\u2229X0=\u2205(from Property 4). With no noise we can again simplify the\ncalculation of the expected payo\ufb00 of Include , reducing Eqn. 15 to:\ns\u22121\ns\u00b7P(X1\njk)\nP(X1)\u22121\ns\u00b7P(X1\\X1\njk)\nP(X1)>0. (22)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d477f-934e-11ee-86fc-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 19, "chroma:document": "s\u00b7P(X1\njk)\nP(X1)\u22121\ns\u00b7P(X1\\X1\njk)\nP(X1)>0. (22)\nWithP(X1\njk)\nP(X1)>1\ns(Property 1), we must haveP(X1\\X1\njk)\nP(X1)<s\u22121\ns. Accordingly, the expected\npayo\ufb00 of Include is positive.\nWe now turn to investigating whether the expected payo\ufb00 of Include is larger than the\nexpected payo\ufb00 of Exclude (Criterion 2 of Lemma 3), which we show by verifying that the\nexpected payo\ufb00 of Exclude is negative. We can again simplify Eqn. 12, however, this time\nwithout knowing that [ X1\nj\\X1\njk]\u2229X0=\u2205:\n1\ns\u00b7P(X1\\X1\njk)\u2212s\u22121"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4780-934e-11ee-b7e4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 19, "chroma:document": "j\\X1\njk]\u2229X0=\u2205:\n1\ns\u00b7P(X1\\X1\njk)\u2212s\u22121\ns\u00b7P(X1\njk)\u2212P([X1\nj\\X1\njk]\u2229X0)<0. (23)\nDividing by P(X1) on each side we get:\n1\ns\u00b7P(X1\\X1\njk)\nP(X1)\u2212s\u22121\ns\u00b7P(X1\njk)\nP(X1)\u2212P([X1\nj\\X1\njk]\u2229X0)\nP(X1)<0. (24)\n20"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4781-934e-11ee-9636-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 20, "chroma:document": "Again, withP(X1\njk)\nP(X1)>1\ns, we haveP(X1\\X1\njk)\nP(X1)<s\u22121\ns. From this result alone, the left side is\nnegative. Additionally, the addendP([X1\nj\\X1\njk]\u2229X0)\nP(X1)is negative as well, reducing the expected\npayo\ufb00 of Exclude even further. So, changing from Include toExclude gives a net expected\nloss.\nLemma 5. Consider a pattern recognition problem per De\ufb01nition 1 without noise, P(y\u0338=\n\u03c9|X\u03c9) = 1\u2212\u03b3= 0, and a Tsetlin Machine game per De\ufb01nition 2 with a single clause C1(X)of"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4782-934e-11ee-9547-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 20, "chroma:document": "positive polarity. If clause C1(X)deviates from all of the sub-patterns Q1\nh(X),h\u2208{1,...n/ 2},\nof the pattern recognition problem, it is either not a Nash equilibrium in the game or it is\nproduced by an action with negative expected payo\ufb00.\nProof. There are three kinds of sub-optimal con\ufb01gurations for a clause: (i) The clause is too\ninfrequent; (ii) The clause is too frequent; or (iii) The clause outputs 1 for input vectors from"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4783-934e-11ee-9178-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 20, "chroma:document": "X0. We must show that none of these ful\ufb01ll both criteria of Lemma 3.\nIn case (i) there exists at least one literal lkthat has been included in the clause, for which\nP(X1\njk\u2229X1)\nP(X1)<1\ns. That is, less than1\nsofX1is covered by the clause, in frequency. It is su\ufb03cient\nto show that the expected payo\ufb00 of Include (Eqn. 15) is negative for \u03b3= 1:\ns\u22121\ns\u00b7P(X1\njk\u2229X1)\u22121\ns\u00b7P(X1\\X1\njk)<0. (25)\nDividing by P(X1) on each side we get:\ns\u22121\ns\u00b7P(X1\njk\u2229X1)\nP(X1)\u22121\ns\u00b7P(X1\\X1\njk)\nP(X1)<0. (26)\nWithP(X1\njk\u2229X1)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4784-934e-11ee-9a83-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 20, "chroma:document": "s\u00b7P(X1\\X1\njk)\nP(X1)<0. (26)\nWithP(X1\njk\u2229X1)\nP(X1)<1\ns, we must haveP(X1\\X1\njk)\nP(X1)>s\u22121\ns. We can thus conclude that the expected\npayo\ufb00 of Include is negative.\nWe now turn to case (ii), that is, the case where the clause is too frequent. Then there\nmust exist an excluded literal lksuch thatP(X1\njk\u2229X1)\nP(X1)>1\ns. It is then su\ufb03cient to show that the\nexpected payo\ufb00 of Exclude is negative:\n1\ns\u00b7P(X1\\X1\njk)\nP(X1)\u2212s\u22121\ns\u00b7P(X1\njk\u2229X1)\nP(X1)\u2212P([X1\nj\\X1\njk]\u2229X0)\nP(X1)<0. (27)\nWithP(X1\njk\u2229X1)\nP(X1)>1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4785-934e-11ee-9207-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 20, "chroma:document": "jk]\u2229X0)\nP(X1)<0. (27)\nWithP(X1\njk\u2229X1)\nP(X1)>1\ns, we haveP(X1\\X1\njk)\nP(X1)<s\u22121\ns. Again, the left side is negative.\nApart from the sub-optimal con\ufb01gurations where too many or too few literals are included\nin the clause, we must also consider case (iii), that is, scenarios where P([X1\nj\\X1\njk]\u2229X0)>0.\nThis means that the corresponding clause C1\nj(X) outputs 1 for input vectors X\u2208[X1\nj\\X1\njk]\u2229X0.\nAccordingly, it outputs 1 when P(y= 1|X)<P(y= 0|X), which is sub-optimal and violates"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4786-934e-11ee-ae99-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 20, "chroma:document": "Property 4 of De\ufb01nition 1. It must then clearly exist a literal lkthat has been excluded\nfrom the clause and that is zero for input vectors X\u2208[X1\nj\\X1\njk]\u2229X0. If not, an optimal\ncon\ufb01guration would not exist, which it does by problem de\ufb01nition. Including lkin the clause\nwould make the clause output 0 for input vectors X\u2208[X1\nj\\X1\njk]\u2229X0. This will thus eliminate\nX\u2208[X1\nj\\X1\njk]\u2229X0from the input space captured by the clause. Accordingly, we must ensure"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4787-934e-11ee-9149-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 20, "chroma:document": "that excluding lk= 0 gives a negative expected payo\ufb00:\n1\ns\u00b7P(X1\\X1\njk)\u2212s\u22121\ns\u00b7P(X1\njk\u2229X1)\u2212P([X1\nj\\X1\njk]\u2229X0)<0. (28)\nWe can assume that1\ns\u00b7P(X1\\X1\njk) ands\u22121\ns\u00b7P(X1\njk\u2229X1) are roughly equal. If they are not,\nwe have already determined in the previous two paragraphs that the resulting clause is not a\nNash equilibrium. Thus, what remains is to show that P([X1\nj\\X1\njk]\u2229X0)>0, which is stated\nby case (iii) itself.\n21"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4788-934e-11ee-bac4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 21, "chroma:document": "Theorem 1. Consider a pattern recognition problem per De\ufb01nition 1 without noise, P(y\u0338=\n\u03c9|X\u03c9) = 1\u2212\u03b3= 0, and a Tsetlin Machine game per De\ufb01nition 2 with a single clause C1(X)of\npositive polarity. The sub-patterns Q1\nh(X),h\u2208{1,...n/ 2},are then the only Nash equilibria\nof the game, where the expected payo\ufb00 of each player action is positive, hence not rejected by\nthe Tsetlin Automata players.\nProof. This theorem follows from Lemma 3, Lemma 4 and Lemma 5.\n4.6 Nash Equilibria with Static Noise"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4789-934e-11ee-9419-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 21, "chroma:document": "4.6 Nash Equilibria with Static Noise\nLemma 6. Consider a pattern recognition problem per De\ufb01nition 1 with static noise, P(y\u0338=\n\u03c9|X\u03c9) = 1\u2212\u03b3,0.5< \u03b3 < 1.0, and output classes of equal size, P(X0) =P(X1) = 0.5.\nConsider further a Tsetlin Machine game per De\ufb01nition 2 with a single clause C1(X)of positive\npolarity. Each sub-pattern Q1\nh(X),h\u2208{1,...n/ 2},of the pattern recognition problem is a Nash\nequilibrium in the game when sis scaled by a factor t,\nb1\n\u03b3<t<b2\nb2\nb3(1\u2212\u03b3) + 2(\u03b3\u22120.5), (29)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d478a-934e-11ee-9c7a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 21, "chroma:document": "b1\n\u03b3<t<b2\nb2\nb3(1\u2212\u03b3) + 2(\u03b3\u22120.5), (29)\nwhere,b2= min j,kP(X1)\nP(X1\njk)\u00b7s,b2>1,andb3= max j,kP(X1)\nP(X1\nj)\u00b7s,0< b 3<1,de\ufb01ne a worst-case\nupper bound for twhen Exclude is optimal. Further, b1= max j,kP(X1)\nP(X1\njk)\u00b7s,0< b1<1,de\ufb01nes\na worst-case lower-bound when Include is optimal. Additionally, the expected payo\ufb00 of each\nplayer action in each equilibrium is positive, hence not rejected by the Tsetlin Automata players."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d478b-934e-11ee-8fea-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 21, "chroma:document": "As seen, increasing noise (reduced \u03b3) tightens the lower bound, but loosens the upper.\nFurther, an increasing b2loosens the upper bound, while a decreasing b1loosens the lower\nbound.\nProof. We \ufb01rst analyse the case when Exclude is optimal, before addressing Include .\nExclude is Optimal. Since Exclude is optimal, we have (follows from De\ufb01nition 1):\nP(X1\njk\u2229X0) = 0,P([X1\nj\\X1\njk]\u2229X0) = 0,P(X1\njk\u2229X1) =P(X1\njk),P([X1\nj\\X1\njk]\u2229X1) =\nP(X1\nj\\X1\njk), andP(X0\\X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d478c-934e-11ee-a290-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 21, "chroma:document": "j\\X1\njk]\u2229X1) =\nP(X1\nj\\X1\njk), andP(X0\\X1\njk) =P(X0). We consider Criterion 1 of Lemma 3 \ufb01rst, i.e., that\nthe expected payo\ufb00 from Eqn. 12 is positive, which simpli\ufb01es to:\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk) +1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0)\u2212\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk)\u2212(1\u2212\u03b3)\u00b7P(X1\nj\\X1\njk)>0. (30)\nDividing by P(X1) on each side, we get:\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\nP(X1)+1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0)\nP(X1)\u2212\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk)\nP(X1)\u2212(1\u2212\u03b3)\u00b7P(X1\nj\\X1\njk)\nP(X1)>0. (31)\nClearly, to maintain positive expected payo\ufb00 after the introduction of static noise, we need to"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d478d-934e-11ee-9f00-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 21, "chroma:document": "compensate for the addends introduced by the noise, i.e., the (1 \u2212\u03b3)-addends. To this end, we\narti\ufb01cially scale up the speci\ufb01city parameter sby a factor t. Simplifying further, we replace\nP(X0)\nP(X1)with 1:\n1\nts\u00b7\u03b3\u00b7P(X1\\X1\njk)\nP(X1)+1\nts\u00b7(1\u2212\u03b3)\u2212\nts\u22121\nts\u00b7\u03b3\u00b7P(X1\njk)\nP(X1)\u2212(1\u2212\u03b3)\u00b7P(X1\nj\\X1\njk)\nP(X1)>0. (32)\n22"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d478e-934e-11ee-8aae-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 22, "chroma:document": "Since we consider the case where excluding literal lkis optimal, we have b2>1 such that:\nP(X1\njk)\nP(X1)=1\nb2s\u21d2b2=P(X1)\nP(X1\njk)\u00b7s. That is, b2is an unknown factor that measures exactly how\nmuch including lkconstrains the clause beyond1\ns, according to Property 1 and Property 2 of\nDe\ufb01nition 1. This in turn, means thatP(X1\\X1\njk)\nP(X1)= 1\u2212P(X1\njk)\nP(X1)=b2s\u22121\nb2s. We \ufb01nally introduce\nb3,0<b3<1. The role of b3is to exactly determine how much excluding literal lkloosens up\nthe clause beyond1\ns:P(X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d478f-934e-11ee-b7c8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 22, "chroma:document": "the clause beyond1\ns:P(X1\nj)\nP(X1)=1\nb3s\u21d2b3=P(X1)\nP(X1\nj)\u00b7s. This leads toP(X1\nj\\X1\njk)\nP(X1)=P(X1\nj)\nP(X1)\u2212P(X1\njk)\nP(X1)=\n1\nb3s\u22121\nb2s. With these amendments, we revisit the condition that the expected payo\ufb00 of Exclude\nmust be positive:\n1\nts\u00b7\u03b3\u00b7b2s\u22121\nb2s+1\nts\u00b7(1\u2212\u03b3)\u2212ts\u22121\nts\u00b7\u03b3\u00b71\nb2s\u2212(1\u2212\u03b3)\u00b7(1\nb3s\u22121\nb2s)\n>0.(33)\nThrough a sequence of standard algebraic operations, the above condition can be simpli\ufb01ed as\nfollows:\n\u03b3\u00b7b2s\u22121\nb2s+ (1\u2212\u03b3)\u2212(ts\u22121)\u00b7\u03b3\u00b71\nb2s\u2212ts\u00b7(1\u2212\u03b3)\u00b7(1\nb3s\u22121\nb2s)\n>0 (34)\n\u03b3\u00b7b2s\u22121"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4790-934e-11ee-b566-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 22, "chroma:document": "b2s\u2212ts\u00b7(1\u2212\u03b3)\u00b7(1\nb3s\u22121\nb2s)\n>0 (34)\n\u03b3\u00b7b2s\u22121\nb2s+ (1\u2212\u03b3)\u2212(ts\u22121)\u00b7\u03b3\u00b71\nb2s\u2212ts\u00b7(1\u2212\u03b3)\u00b7b2\u2212b3\nb2b3s>0 (35)\n\u03b3\u00b7(b2s\u22121) +b2s\u00b7(1\u2212\u03b3)\u2212(ts\u22121)\u00b7\u03b3\u2212ts\u00b7(1\u2212\u03b3)\u00b7b2\u2212b3\nb3>0 (36)\nb2s\u03b3\u2212\u03b3+b2s\u2212b2s\u03b3\u2212ts\u03b3+\u03b3\u2212ts\u00b7(1\u2212\u03b3)\u00b7b2\u2212b3\nb3>0 (37)\nb2\u03b3+b2\u2212b2\u03b3\u2212t\u03b3\u2212t\u00b7(1\u2212\u03b3)\u00b7b2\u2212b3\nb3>0 (38)\nb2\u03b3+b2\u2212b2\u03b3\u2212t\u03b3\u2212t\u00b7(1\u2212\u03b3)\u00b7(b2\nb3\u22121)\n>0 (39)\nb2\u03b3+b2\u2212b2\u03b3\u2212t\u03b3\u2212(\nt\u00b7(1\u2212\u03b3)\u00b7b2\nb3\u2212t\u00b7(1\u2212\u03b3))\n>0 (40)\nb2\u03b3+b2\u2212b2\u03b3\u2212t\u03b3\u2212t\u00b7(1\u2212\u03b3)\u00b7b2\nb3+t\u00b7(1\u2212\u03b3)>0 (41)\nb2\u2212t\u03b3\u2212t\u00b7(1\u2212\u03b3)\u00b7b2\nb3+t\u00b7(1\u2212\u03b3)>0 (42)\nb2\u2212t\u03b3\u2212b2\nb3t+b2\nb3t\u03b3+t\u2212t\u03b3 > 0 (43)\nb2\u2212t(\n\u03b3+b2\nb3\u2212b2\nb3\u03b3\u22121 +\u03b3)\n>0 (44)\nb2\n\u03b3+b2\nb3\u2212b2"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4791-934e-11ee-862b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 22, "chroma:document": "b2\u2212t(\n\u03b3+b2\nb3\u2212b2\nb3\u03b3\u22121 +\u03b3)\n>0 (44)\nb2\n\u03b3+b2\nb3\u2212b2\nb3\u03b3\u22121 +\u03b3> t (45)\nb2\nb2\nb3(1\u2212\u03b3) + 2(\u03b3\u22121\n2)> t (46)\nThis means that it is possible to increase tup to a certain boundary, weakening reinforcement\nofExclude , without making the expected payo\ufb00 of Exclude turn negative.\nWe now show that the expected payo\ufb00 of Include is negative, ful\ufb01lling Criterion 2 of\nLemma 3:\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1) +s\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\njk)<0. (47)"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4792-934e-11ee-acf0-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 22, "chroma:document": "1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\njk)<0. (47)\nSince Exclude is optimal, we can simplify as follows (per De\ufb01nition 1):\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk)\u22121\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X0)<0. (48)\n23"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4793-934e-11ee-9396-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 23, "chroma:document": "Introducing the scaling factor t>1 forsand dividing by P(X1) we get:\nts\u22121\nts\u00b7\u03b3\u00b7P(X1\njk)\nP(X1)\u22121\nts\u00b7\u03b3\u00b7P(X1\\X1\njk)\nP(X1)\u22121\nts\u00b7(1\u2212\u03b3)\u00b7P(X0)\nP(X1)<0. (49)\nBecause we have established that Eqn. 32 is positive, it follows that the above equation is\nnegative. This is due to the fact Eqn. 32 is still positive after removing the last addend, and\nthe above equation is simply the negated version of the latter.\nInclude is Optimal. To ful\ufb01ll Criterion 1 of Lemma 3 we must show that the expected"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4794-934e-11ee-a5d2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 23, "chroma:document": "payo\ufb00 of Include is positive:\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk\u2229X1) +s\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X1\njk\u2229X0)\u2212\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X0\\X1\njk)>0. (50)\nSince Include is optimal, we have: P(X1\njk\u2229X0) = 0,P(X1\njk\u2229X1) =P(X1\njk) andP(X0\\X1\njk) =\nP(X0), again per Property 4 of De\ufb01nition 1. Accordingly, we can simplify the above expression\nto:\ns\u22121\ns\u00b7\u03b3\u00b7P(X1\njk)\u22121\ns\u00b7\u03b3\u00b7P(X1\\X1\njk)\u22121\ns\u00b7(1\u2212\u03b3)\u00b7P(X0)>0. (51)\nTo compensate for the addends introduced by noise, we again increase the s-parameter by"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4795-934e-11ee-8afb-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 23, "chroma:document": "arti\ufb01cially multiplying it with a factor t>1. Dividing by P(X1) we get:\nts\u22121\nts\u00b7\u03b3\u00b7P(X1\njk)\nP(X1)\u22121\nts\u00b7\u03b3\u00b7P(X1\\X1\njk)\nP(X1)\u22121\nts\u00b7(1\u2212\u03b3)\u00b7P(X0)\nP(X1)>0. (52)\nNow we introduce 0 < b1<1, which tells exactly how much largerP(X1\njk)\nP(X1)is compared to1\ns,\ni.e.,P(X1\njk)\nP(X1)=1\nb1s\u21d2b1=P(X1)\nP(X1\njk)\u00b7sas determined by Property 1 and Property 2 of De\ufb01nition 1,\nwe can simplify the condition as follows, also replacingP(X0)\nP(X1)with 1:\nts\u22121\nts\u00b7\u03b3\u00b71\nb1s\u22121\nts\u00b7\u03b3\u00b7b1s\u22121\nb1s\u22121\nts\u00b7(1\u2212\u03b3)>0 (53)\nts\u22121\nt\u00b7\u03b3\u00b71\nb1\u22121"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4796-934e-11ee-b3fe-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 23, "chroma:document": "ts\u00b7\u03b3\u00b7b1s\u22121\nb1s\u22121\nts\u00b7(1\u2212\u03b3)>0 (53)\nts\u22121\nt\u00b7\u03b3\u00b71\nb1\u22121\nt\u00b7\u03b3\u00b7b1s\u22121\nb1\u2212s\nt\u00b7(1\u2212\u03b3)>0 (54)\n(ts\u22121)\u00b7\u03b3\u2212\u03b3\u00b7(b1s\u22121)\u2212b1s\u00b7(1\u2212\u03b3)>0 (55)\nts\u03b3\u2212\u03b3\u2212b1s\u03b3+\u03b3\u2212b1s+b1s\u03b3 > 0 (56)\nts\u03b3\u2212b1s > 0 (57)\nb1\n\u03b3< t (58)\nThis leads to the following range for t, recovering a Nash equilibrium corresponding to the\nsub-patterns of the pattern recognition problem:\nb1\n\u03b3< t <b2\nb2\nb3(1\u2212\u03b3) + 2(\u03b3\u22121\n2)(59)\nThat is, scaling sby a factor tallows us to recover from some degree of static noise, as governed\nbyb1,b2,andb3."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4797-934e-11ee-8f99-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 23, "chroma:document": "byb1,b2,andb3.\nWe \ufb01nally need to verify that the expected payo\ufb00 of Exclude is negative, in order to ful\ufb01ll\nCriterion 2 of Lemma 3:\n1\ns\u00b7\u03b3\u00b7P(X1\\X1\njk) +1\ns\u00b7(1\u2212\u03b3)\u00b7P(X0)\u2212s\u22121\ns\u00b7\u03b3\u00b7P(X1\njk)\u2212\n(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1)\u2212\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X0)<0. (60)\n24"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4798-934e-11ee-942b-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 24, "chroma:document": "Again, introducing the scaling factor t>1 forsand dividing by P(X1) we get:\n1\nts\u00b7\u03b3\u00b7P(X1\\X1\njk)\nP(X1)+1\nts\u00b7(1\u2212\u03b3)\u00b7P(X0)\nP(X1)\u2212ts\u22121\nts\u00b7\u03b3\u00b7P(X1\njk)\nP(X1)\u2212\n(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1)\nP(X1)\u2212\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X0)\nP(X1)<0. (61)\nSince we have established that Eqn. 52 is positive, it follows that the above equation is negative.\nThis is due to the above equation simply being the negated version of the latter, subtracting\ntwo extra addends, clearly making it negative."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d4799-934e-11ee-b4e2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 24, "chroma:document": "two extra addends, clearly making it negative.\nLemma 7. Consider a pattern recognition problem per De\ufb01nition 1 with static noise, P(y\u0338=\n\u03c9|X\u03c9) = 1\u2212\u03b3,0.5< \u03b3 < 1.0, and output classes of equal size, P(y= 0) =P(y= 1) = 0.5.\nConsider further a Tsetlin Machine game per De\ufb01nition 2 with only a single clause C1(X)of\npositive polarity. If clause C1(X)deviates from all of the sub-patterns Q1\nh(X),h\u2208{1,...n/ 2},\nof the pattern recognition problem, it is either not a Nash equilibrium in the game or it is"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d479a-934e-11ee-a5b7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 24, "chroma:document": "produced by an action with negative payo\ufb00.\nProof. For too infrequent or too frequent clauses, the proof follows the structure of the corre-\nsponding proof for the noise-free cases (i) and (ii). However, we need to introduce the scaling\nfactort, as demonstrated above.\nCase (iii) requires some more scrutiny. That is, we need to show that when P([X1\nj\\X1\njk]\u2229\nX0)>0, the expected payo\ufb00 of Exclude is negative:\n1\nts\u00b7\u03b3\u00b7P(X1\\X1\njk) +\n1\nts\u00b7(1\u2212\u03b3)\u00b7P(X0)\u2212\nts\u22121\nts\u00b7\u03b3\u00b7P(X1\njk)\u2212\n(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1)\u2212"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d479b-934e-11ee-ad0f-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 24, "chroma:document": "ts\u22121\nts\u00b7\u03b3\u00b7P(X1\njk)\u2212\n(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1)\u2212\n\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X0)<0. (62)\nAgain we can assume thatts\u22121\nts\u00b7\u03b3\u00b7P(X1\njk)\u22121\nts\u00b7\u03b3\u00b7P(X1\\X1\njk)\u22121\nts\u00b7(1\u2212\u03b3)\u00b7P(X0) is close\nto zero. Otherwise, we did not have a Nash equilibrium in the \ufb01rst place, due to case (i) and\ncase (ii). Thus, we only need to verify that:\n(1\u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1) +\u03b3\u00b7P([X1\nj\\X1\njk]\u2229X0)>0.\nThe above condition is clearly true because we have P([X1\nj\\X1\njk]\u2229X0)>0 by case (iii) itself.\nAdditionally, because we have noise, (1 \u2212\u03b3)\u00b7P([X1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d479c-934e-11ee-8656-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 24, "chroma:document": "Additionally, because we have noise, (1 \u2212\u03b3)\u00b7P([X1\nj\\X1\njk]\u2229X1) is strictly positive as well.\nTheorem 2. Consider a pattern recognition problem per De\ufb01nition 1 with static noise, P(y\u0338=\n\u03c9|X\u03c9) = 1\u2212\u03b3,0.5< \u03b3 < 1.0, and output classes of equal size, P(y= 0) =P(y= 1) = 0.5.\nConsider further a Tsetlin Machine game per De\ufb01nition 2 with only a single clause C1(X)of\npositive polarity. The sub-patterns Q1\nh(X),h\u2208{1,...n/ 2},of the pattern recognition problem"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d479d-934e-11ee-8cb9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 24, "chroma:document": "are then the only Nash equilibria of the game, where the expected payo\ufb00 of each player action\nis positive, hence not rejected by the Tsetlin Automata players.\nProof. This theorem follows from Lemma Lemma 3, Lemma 6, and Lemma 7.\n4.7 Coordination of Multiple Tsetlin Automata Teams\nTheorem 3. Consider a pattern recognition problem per De\ufb01nition 1 without noise, P(y\u0338=\n\u03c9|X\u03c9) = 1\u2212\u03b3= 0, and a Tsetlin Machine game per De\ufb01nition 2 with T\u00b7nclauses. Then the"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d479e-934e-11ee-a67e-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 24, "chroma:document": "only Nash equilibrium of the Tsetlin Machine game is a con\ufb01guration where TTsetlin Machine\nclausesC\u03c9\nj,\u03c9\u2208{0,1},j\u2208{1,...,T\u00b7n/2}, duplicate each sub-pattern Q\u03c9\nh,\u03c9\u2208{0,1},h\u2208\n{1,...,n/ 2}.\n25"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d479f-934e-11ee-b443-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "Proof. We already known from Theorem 1 that if a clause C\u03c9\njis not duplicating any of the sub-\npatternsQ\u03c9\nh, the Tsetlin Automata of the clause are not in a Nash Equilibrium. We therefore\nonly consider the case where all clauses C\u03c9\njare duplicating some sub-pattern Q\u03c9\nh. Let us \ufb01rst\nconsider positive polarity clauses ( \u03c9= 1). Assume that more than Tof these clauses duplicate\na particular sub-pattern Q1\nh. Then, clearly, there are fewer than Tclauses duplicating some\nother sub-pattern Q1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a0-934e-11ee-8891-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "other sub-pattern Q1\nh\u2032,h\u2032\u0338=h. The Tsetlin Automata that take part in duplicating Q1\nhwill then\nonly receive Inaction-feedback when facing sub-pattern Q1\nh, as seen from Eqn. 8. However,\nwhen they observe sub-pattern Q1\nh\u2032they will receive Type I Feedback with probability larger\nthan 0. Type I Feedback pulls the Tsetlin Automata away from Q1\nhand towards Q1\nh\u2032. Hence,\nwe do not have a Nash equilibrium. For the same reasons, we do not have a Nash equilibrium"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a1-934e-11ee-8ead-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "when more than Tnegative polarity clauses capture a sub-pattern Q0\nhdue to Eqn. 9. When\nthere areTclauses per sub-pattern Q\u03c9\nh, on the other hand, the whole system is in equilibrium\ndue to Theorem 1 and Eqns. 8-11.\nThe proof for static noise follows the above structure, however, relies on Theorem 2 instead.\nThe proof is left out here for the sake of brevity.\nRemark 1. Note that a Tsetlin Machine in practice also learns robustly when the total"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a2-934e-11ee-95a4-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "number of clauses available is less than T\u00b7n. This is because of the gradually increasing\nprobability of receiving feedback the farther away one is from T/\u2212T, which leads to a load\nbalancing e\ufb00ect among the clauses. This e\ufb00ect dynamically distributes the clauses among the\navailable sub-patterns.\nRemark 2. A traditional Learning Automata approach to pattern recognition would use\nclassi\ufb01cation accuracy as payo\ufb00 (utility) function, to give feedback to the Learning Automata."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a3-934e-11ee-a5b9-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "In general, this leads to the Vanishing Signal to Noise Ratio Problem described in Section 1.3.\nThis is due to the relatively small e\ufb00ect each single Learning Automaton has on overall classi-\n\ufb01cation accuracy, and due to the noise introduced by the random behaviour of the team as a\nwhole. In the Tsetlin Machine game, on the other hand, each single automaton gets feedback\ndirectly from the value of its literal and the value of its clause, as laid out in Table 2 and Ta-"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a4-934e-11ee-9dff-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "ble 3. We will now show that this local feedback scheme can be a generalized ordinal potential\ngame [30], with classi\ufb01cation accuracy being the global payo\ufb00 (utility) function.\nTheorem 4. LetMrefer to the payo\ufb00 matrix of the Tsetlin Machine game from De\ufb01nition 2.\nFurther, let a\u2032\nianda\u2032\u2032\nirefer to the two actions available to a single automaton, while a\u2212irefers\nto the remaining actions in the action con\ufb01guration \u03b1. Finally, let the summation target Tbe1,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a5-934e-11ee-9b28-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "only considering positive clauses. The resulting Tsetlin Machine game is a generalized ordinal\npotential game with regards to classi\ufb01cation accuracy, under the constraints of De\ufb01nition 1.\nThat is, we have: P(y= \u02c6y;a\u2032\ni,a\u2212i)\u2212P(y= \u02c6y;a\u2032\u2032\ni,a\u2212i)>0\u21d2M(a\u2032\ni,a\u2212i)\u2212M(a\u2032\u2032\ni,a\u2212i)>0.\nProof. Classi\ufb01cation accuracy is clearly maximised only when C1\nj,j\u2208{1,...,n/ 2},are du-\nplicatingQ1\nh,h\u2208{1,...,n/ 2}. Any mismatch will either produce false positive output if the\nTsetlin Machine clauses C1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a6-934e-11ee-af74-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "Tsetlin Machine clauses C1\njare missing literals, or false negative output if not all of the sub-\npatternsQ1\nhare completely covered. Thus, during learning, there are two ways classi\ufb01cation\naccuracy can increase by switching action from a\u2032\u2032\nitoa\u2032\ni: either by a decrease in the rate of\nfalse negatives, P(y= 1,\u02c6y= 0), or by a decrease in the rate of false positives, P(y= 0,\u02c6y= 1).\nLet us \ufb01rst consider decreasing P(y= 1,\u02c6y= 0). This can only be achieved by turning"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a7-934e-11ee-87b1-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "false negative output into true positive output. Clearly, for a particular input X, we have false\nnegative output if there exists an hsuch thatQ1\nh(X) = 1, with C1\nj(X) = 0 for all j. LetL1\nj\nbe the literals of clause C1\njandK1\nhbe the literals of Q1\nh. We can correct the false negative\nerror if and only if we have ajandkso thatL1\nj\\{lk}\u2286K1\nh,lk/\u2208K1\nh, implyinglk= 0. The\nfalse negative error can then be corrected by excluding lkfrom clause C1\nj. Examining Feedback\nType I forC1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a8-934e-11ee-b4f1-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 25, "chroma:document": "j. Examining Feedback\nType I forC1\nj(X) = 0,lk= 0,in Table 2, which applies for y= 1, we observe that the expected\npayo\ufb00 of excluding lkis larger that the expected payo\ufb00 of including it.\n26"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47a9-934e-11ee-bfc6-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "We consider decreasing P(y= 0,\u02c6y= 1) next. This can be achieved by turning false\npositive output into true negative output. Clearly, for a particular input X, we have false\npositive output if and only if there exists a jsuch thatC1\nj(X) = 1, with Q1\nh(X) = 0 for all\nh. LetL1\njbe the literals of clause C1\njandK1\nhbe the literals of Q1\nh. We can correct the false\npositive output if and only if there exists only onejsuch thatC1\nj(X) = 1, and if this particular\nC1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47aa-934e-11ee-8ca7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "j(X) = 1, and if this particular\nC1\nj(X) has left out a literal lkof value 0, lk= 0,lk/\u2208L1\nj. Then the false positive output can\nbe corrected by including lk= 0 inC1\nj. Examining Feedback Type II for C1\nj(X) = 1,lk= 0,in\nTable 3, which applies for y= 0, we observe that the expected payo\ufb00 of including lkis larger\nthat the expected payo\ufb00 of excluding it.\nRemark 4. The ordinal potential game is generalized because the Tsetlin Machine game"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47ab-934e-11ee-bac7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "matrixMrewards action switches that have no impact on classi\ufb01cation accuracy. That is,\nwe do not have: M(a\u2032\ni,a\u2212i)\u2212M(a\u2032\u2032\ni,a\u2212i)>0\u21d2P(y= \u02c6y;a\u2032\ni,a\u2212i)\u2212P(y= \u02c6y;a\u2032\u2032\ni,a\u2212i)>0.\nIn particular, consider false negative output that cannot be corrected by excluding a single\nliteral from a single clause. Yet, Mstill rewards excluding literals of value 0 ,lk= 0,when\nC1\nj(X) = 0, eventually correcting the false negative output. Similarly, when multiple clauses"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47ac-934e-11ee-8651-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "output 1, producing a false positive output, no single Include action can alone correct the\nfalse positive output. Yet, including literals of value 0 is still rewarded in the game matrix\nM, again, eventually \ufb01xing the false positive output. Thus, in this sense, the Tsetlin Machine\ngame matrix has a longer planning horizon than just greedily optimizing classi\ufb01cation accuracy\nin single steps. Note that these latter dynamics do not produce any additional Nash equilibria,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47ad-934e-11ee-a656-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "as established by Theorem 3. This means that there is a one-to-one correspondence between\nthe equilibria in the potential game and in optimizing classi\ufb01cation accuracy.\nRemark 5. As summarized in [3], the behaviour of \ufb01nite state Learning Automata collectives\nhave been studied by several researchers, showing that rational behaviour can be obtained in\ncertain cases, when memory is in\ufb01nite and the game\u2019s payo\ufb00 matrix is constrained [31, 15, 32,"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47ae-934e-11ee-b62a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "33, 34]. Proofs for variable structure Learning Automata are more general, on the other hand.\nThat is, for identical payo\ufb00 games, the Learning Automata collective converges to a Nash\nequilibrium with probability arbitrarily close to unity [3]. Further, for zero-sum games with\na saddle point, there exists Learning Automata collectives that can obtain the Von Neumann\nvalue [35, 36]. In the next section, we study the behaviour of Tsetlin Machine games empirically"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47af-934e-11ee-873a-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "on various datasets, including the arti\ufb01cial Noisy XOR dataset where the optimal classi\ufb01cation\naccuracy and the corresponding Nash equilibria are known.\n5 Empirical Results\nIn this section, we evaluate the Tsetlin Machine empirically using \ufb01ve datasets:\n\u2022Binary Iris Dataset. This is the classical Iris Dataset, however, with features in binary\nform.\n\u2022Binary Digits Dataset. This is the classical digits dataset, again with features in\nbinary form."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b0-934e-11ee-b6e5-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "binary form.\n\u2022Axis & Allies Board Game Dataset. This new dataset involves optimal move pre-\ndiction in a minimalistic, yet intricate, mini-game from the Axis & Allies board game.\n\u2022Noisy XOR Dataset with Non-informative Features. This arti\ufb01cial dataset is de-\nsigned to reveal particular \u201dblind zones\u201d of pattern recognition algorithms. The dataset\ncaptures the renowned XOR-relation. Furthermore, the dataset contains a large number"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b1-934e-11ee-ada1-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 26, "chroma:document": "of random non-informative features to measure susceptibility towards the curse of dimen-\nsionality [37]. To examine robustness towards noise we have further randomly inverted\n40% of the outputs.\n27"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b2-934e-11ee-82b2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 27, "chroma:document": "\u2022MNIST Dataset. The MNIST dataset is a larger scale dataset used extensively to\nbenchmark machine learning algorithms. We have included this dataset to investigate the\nscalability of the Tsetlin Machine, as well as the behaviour of longer learning processes.\nFor these datasets, we form ensembles of 50 to 1000 independent replications with di\ufb00erent\nrandom number streams. We do this to minimize the variance of the reported results and to"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b3-934e-11ee-94e8-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 27, "chroma:document": "provide the foundation for a statistical analysis of the merits of the di\ufb00erent schemes evaluated.\nTogether with the Tsetlin Machine, we also evaluate several classical machine learning tech-\nniques using the same random number streams. This includes Multilayer Perceptron Networks,\nthe Naive Bayes Classi\ufb01er, Support Vector Machines, and Logistic Regression. Where appro-\npriate, the di\ufb00erent schemes are optimized by means of relatively light hyper-parameter grid"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b4-934e-11ee-b681-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 27, "chroma:document": "searches. As an example, Figure 7 captures the impact the sparameter of the Tsetlin Ma-\nchine has on mean accuracy, for the Noisy XOR Dataset. Each point in the plot measures the\nmean accuracy of 100 di\ufb00erent replications of the XOR-experiment for a particular value of s.\nClearly, accuracy increases with sup to a certain point, before it degrades gradually. Based\non the plot, for the Noisy XOR-experiment, we decided to use an svalue of 3.9.\n 0.5 0.6 0.7 0.8 0.9 1"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b5-934e-11ee-b18c-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 27, "chroma:document": "0.5 0.6 0.7 0.8 0.9 1\n 1 1.5  2 2.5  3 3.5  4 4.5  5 5.5  6Accuracy\nsTsetlin Machine\nFigure 7: The mean accuracy of the Tsetlin Machine (y-axis) on the Noisy XOR Dataset for\ndi\ufb00erent values of the parameter s(x-axis).\n5.1 The Binary Iris Dataset\nWe \ufb01rst evaluate the Tsetlin Machine on the classical Iris dataset2. This dataset consists of\n150 examples with four inputs (Sepal Length, Sepal Width, Petal Length and Petal Width),\nand three possible outputs (Setosa, Versicolour, and Virginica)."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b6-934e-11ee-a181-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 27, "chroma:document": "We increase the challenge by transforming the four input values into one consecutive se-\nquence of 16 bits, four bits per \ufb02oat. It is thus necessary to also learn how to segment the 16\n2UCI Machine Learning Repository [https://archive.ics.uci.edu/ml/datasets/iris].\n28"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b7-934e-11ee-a2e7-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 28, "chroma:document": "bits into four partitions, and extract the numeric information. We refer to the new dataset as\nthe Binary Iris Dataset.\nWe partition this dataset into a training set and a test set, with 80 percent of the data\nbeing used for training. We here randomly produce 1000 training and test data partitions. For\neach ensemble, we also randomly reinitialize the competing algorithms, to gain information on\nstability and robustness. The results are reported in Table 6."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b8-934e-11ee-8a88-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 28, "chroma:document": "Technique/Accuracy ( %) Mean 5 %ile 95 %ileMin. Max.\nTsetlin Machine 95.0\u00b10.2 86.7 100.0 80.0100.0\nNaive Bayes 91.6\u00b10.3 83.3 96.7 70.0100.0\nLogistic Regression 92.6\u00b10.2 86.7 100.0 76.7100.0\nMultilayer Perceptron Networks 93.8\u00b10.2 86.7 100.0 80.0100.0\nSVM 93.6\u00b10.3 86.7 100.0 76.7100.0\nTable 6: The Binary Iris Dataset \u2013 accuracy on test data.\nThe Tsetlin Machine3used here employs 300 clauses, and uses an s-value of 3.0 and a"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47b9-934e-11ee-8e11-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 28, "chroma:document": "summation target Tof 10. Furthermore, the individual Tsetlin Automata each has 100 states.\nThis Tsetlin Machine is run for 500 epochs, and it is the accuracy after the \ufb01nal epoch that is\nreported. Propositional formulas with higher test accuracy are often found in preceding epochs\nbecause of the random exploration of the Tsetlin Machine. However, to avoid over\ufb01tting\nto the test set by handpicking the best con\ufb01guration found, we instead simply use the last\ncon\ufb01guration produced."}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47ba-934e-11ee-83ae-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 28, "chroma:document": "con\ufb01guration produced.\nIn Table 6, we list mean accuracy with 95% con\ufb01dence intervals, 5 and 95 percentiles, as\nwell as the minimum and maximum accuracy obtained, across the 1000 experiment runs we\nexecuted. As seen, the Tsetlin Machine provides the highest mean accuracy. For the 95 %ile\nscores, however, most of the schemes obtain 100% accuracy. This can be explained by the\nsmall size of the test set, which merely contains 30 examples. Thus it is easier to stumble"}


created_at: 2023-12-05 09:10:18
topic: persistent://default/default/a0c119a8-dc6b-46d3-8158-5f41a08fd57f
id: 152d47bb-934e-11ee-b9b2-ac198e74a6f7
encoding: FLOAT32
metadata:
{"source": "source_documents\\1804.01508.pdf", "page": 28, "chroma:document": "upon a random con\ufb01guration that happens to provide fault-free classi\ufb01cation. Since the test\nset is merely a sample of the corresponding real-world problem, it is reasonable to assume that\nhigher mean accuracy translates to more robust performance overall.\nThe training set, on the other hand, reveals subtler di\ufb00erences between the schemes. The\nresults obtained on the training set are shown in Table 7. As seen, the SVM here provides the"}


