Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

_We initiate a formal treatment of RL settings where the learner's observations are jointly generated by a latent endogenous state and an uncontrolled exogenous state, which is unaffected by the agent's actions and does not affect the agent's task.

The review comment to be evaluated is:

Strengths:

- EX-BMDPs seem to capture several popular simulated distractors that are currently being used in applied RL papers, such as random video backgrounds in MuJoCo tasks. I suspect the fact that the exogenous dynamics are relatively unconstrained will also make them a good model for practical tasks. The requirement for near-determinism in the endogenous dynamics is unfortunate, particularly given that it applies to the initial state distribution as well (often you want to perform well over a distribution of similar-but-not-identical tasks). However, I think this is not a major concern for exploration methods: so long as the possible initial states are reachable from some common starting state within $H$ steps, PPE will eventually explore them.
- The fact that it is possible to obtain a sample complexity bound completely independent of the complexity of $\Xi$ was surprising to me given the relatively complex form of exogenous noise in an EX-BMDP, and shows that the EX-BMDP class is at least tractable.
- The clarity of writing was appreciated, particularly when stating assumptions and comparing against past algorithms.

Weaknesses:

- I noticed that RND is missing from Section 3/Appendix A (shortcomings of existing methods), but does surprisingly well on the combination lock problem. Is it possible to give some intuition for where RND fails in general?
- The visual gridworld experiments are a little confusing. The takeaway seems to be that PPE drives accuracy to \~100% eventually on a less-toy problem. However, there are no baselines, so it is hard to determine how significant this achievement is. It would be useful to see how the baselines from Figure 2 fare in this setting, particularly in terms of sample efficiency.

The evidence to be evaluated is:We introduce a novel _Exogenous Block Markov Decision Process_ (EX-BMDP) setting to model systems with exogenous noise.

We start with describing the Block Markov Decision Process (BMDP) Du et al. (2019). This process consists of a finite set of observations \(\mathcal{X}\), a set of _latent_ states \(\mathcal{Z}\) with cardinality \(Z\), a finite set of actions \(\mathcal{A}\) with cardinality \(A\), a transition function \(T:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{Z})\), an emission function \(q:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{X})\), a reward function \(R:\mathcal{X}\times\mathcal{A}\rightarrow[0,1]\), a horizon \(H\in\mathbb{N}\), and a start state distribution \(\mu\in\Delta(\mathcal{Z})\). The agent interacts with the environment by repeatedly generating \(H\)-step trajectories \((z_{1},x_{1},a_{1},r_{1},\cdots,z_{H},x_{H},a_{H},r_{H})\) where \(z_{1}\sim\mu(\cdot)\) and for every \(h\in[H]\) we have \(x_{h}\sim q(\cdot\mid z_{h})\), \(r_{h}=R(x_{h},a_{h})\), and if \(h<H\), then \(z_{h+1}\sim T(\cdot\mid z_{h},a_{h})\). The agent does not observe the states \((z_{1},\cdots,z_{H})\), instead receiving only the observations \((x_{1},\cdots,x_{H})\) and rewards \((r_{1},\cdots,r_{H})\). We assume that the emission distributions of any two latent states are disjoint, usually referred as the _block assumption_: \(supp(q(\cdot\mid z_{1}))\cap supp(q(\cdot\mid z_{2}))=\emptyset\) when \(z_{1}\neq z_{2}\). The agent chooses actions using a policy \(\pi:\mathcal{X}\rightarrow\Delta(\mathcal{A})\). We also define the set of non-stationary policies \(\Pi_{\mathrm{NS}}=\Pi^{H}\) as a \(H\)-length tuple, with \((\pi_{1},\cdots,\pi_{H})\in\Pi_{\mathrm{NS}}\) denoting that the action at time step \(h\) is taken as \(a_{h}\sim\pi_{h}(\cdot\mid x_{h})\). The value \(V(\pi)\) of a policy \(\pi\) is the expected episodic sum of rewards \(V(\pi):=\mathbb{E}_{\pi}[\sum_{h=1}^{H}R(x_{h},a_{h})]\).

_An EX-BMDP is a BMDP such that the latent state can be decoupled into two parts \(z=(s,\xi)\) where \(s\in\mathcal{S}\) is the endogenous state and \(\xi\in\Xi\) is the exogenous state. For \(z\in\mathcal{Z}\) the initial distribution and transition functions are decoupled, that is: \(\mu(z)=\mu(s)\mu_{\xi}(\xi)\), and \(T(z^{\prime}\mid z,a)=T(s^{\prime}\mid s,a)T_{\xi}(\xi^{\prime}\mid\xi)\).

The endogenous state \(s\) captures the information that can be manipulated by the agent.

The exogenous state \(\xi\) captures all the other information that the agent cannot control and does not affect the information it can manipulate.

The decoupled dynamics assumption made in the EX-BMDP setting is a natural way to characterize exogenous noise; the type of noise that is not affected by our actions nor affects the endogenous state but may have non-trivial dynamic. This decoupling captures the movement of ducks, captured in the visual field of the agent in Figure 1, and many additional exogenous processes (e.g., movement of clouds in a navigation task).

The structure of EX-BMDPs allows to reduce the size of the policy cover significantly to \(|\mathcal{S}|\ll|\mathcal{Z}|=|\mathcal{S}|\,|\Xi|\) when the size of the exogenous state space is large. Specifically, we show that the set of policies that reach each _endogenous_ state, and _do not depend on the exogenous_ part of the state is also a policy cover (see Appendix B, Proposition 4).

Unfortunately, in the presence of exogenous noise, Homer distinguishes between two transitions that represent transition between the same latent endogenous states but different exogenous states. In our walk in the park example, even if the agent moves between same points in two transitions, the model maybe able to tell these transitions apart by looking at the position of ducks which may have different behaviour in the two transitions. This results in the Homer creating \(\mathcal{O}(|\mathcal{Z}|)\) many abstract states. We call this the _under-abstraction_ problem.

Such a representation can ignore all information that is not relevant for action prediction, which includes all exogenous/uncontrollable information.

Zhang et al. (2020) proposed learning a bisimulation metric to learn a representation which is invariant to exogenous noise.

In this section, we present an algorithm _Predictive Path Elimination_ (PPE) that we later show can provably solve any EX-BMDP with nearly deterministic dynamics and start state distribution of the endogenous state, while making no assumptions on the dynamics or start state distribution of the exogenous state (Algorithm 1). Before describing PPE, we highlight that PPE can be thought of as a computationally-efficient and simpler alternative to Algorithm 4 of Du et al. (2019) who studied rich-observation setting without exogenous noise.1

Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

```**Algorithm 1** PPE\((\delta,\eta)\): Predictive Path EliminationPPE performs iterations over the time steps \(h\in\{2,\cdots,H\}\). In the \(h^{th}\) iteration, it learns a policy cover \(\Psi_{h}\) for time step \(h\) containing open-loop policies. This is done by first augmenting the policy cover for previous time step by one step. Formally, we define \(\Upsilon_{h}=\Psi_{h-1}\circ\mathcal{A}=\{\pi\circ a\mid\pi\in\Psi_{h-1},a\in \mathcal{A}\}\) where \(\pi\circ a\) is an open-loop policy that follows \(\pi\) till time step \(h-1\) and then takes action \(a\). Since we assume the transition dynamics to be near-deterministic, therefore, we know that there exists a policy cover for time step \(h\) that is a subset of \(\Upsilon_{h}\) and whose size is equal to the number of reachable states at time step \(h\). Further, as the transitions are near-deterministic, we refer to an open-loop policy as a path, as we can view the policy as tracing a path in the latent transition model.

We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

PPE runs in a reward-free setting.

In each case, the sample complexity of learning a near-optimal policy, given the output of PPE, scales with the size of endogenous and not the exogenous state space. We provide the main sample complexity guarantee for PPE as well as additional intuition for why it works. We analyze the algorithm in near-deterministic MDPs defined as follows: Two transition functions \(T_{1}\) and \(T_{2}\) are \(\eta\)_-close_ if for all \(h\in[H],a\in\mathcal{A},s\in\mathcal{S}_{h}\) it holds that \(||T_{1}(\cdot\mid s,a)-T_{2}(\cdot\mid s,a)||_{1}\leq\eta\). Analogously, two starting distribution \(\mu_{1}\) and \(\mu_{2}\) are \(\eta\)-close if \(||\mu_{1}(\cdot)-\mu_{2}(\cdot)||_{1}\leq\eta\). We emphasize that near-deterministic dynamics are common in real-world applications like robotics. **Assumption 1** (Near deterministic endogenous dynamics).: _We assume the endogenous dynamics is \(\eta\)-close to a deterministic model \((\mu_{D,\eta},T_{D,\eta})\) where \(\eta\leq 1/(4SH)\).

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space. Further, since our analysis only uses standard uniform convergence arguments, it extends straightforwardly to infinitely large function classes by replacing \(\ln|\mathcal{F}|\) with other suitable complexity measures such as Rademacher complexity.

Let \(\mathbb{P}_{h}(\xi)\) denote the distribution over exogenous states at time step \(h\) which is independent of agent's policy. The Bayes optimal classifier (\(f^{*}_{h}:=f_{\Upsilon_{h},h}\)) of the prediction problem can be derived as:

\[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\]

The combination lock problem is defined for a given horizon \(H\) by an endogenous state space \(\mathcal{S}=\{s_{1,a}\}\cup\{s_{h,a},s_{h,b},s_{h,c}\}_{h=2}^{H}\), an exogenous state space \(\Xi=\{0,1\}^{H}\), an action space \(\mathcal{A}\) with 10 actions, and a deterministic endogenous start state of \(s_{1,a}\).

The exogenous state evolves as follows. We set \(\xi_{1}\in\{0,1\}^{H}\) where \(\xi_{1}(i)\sim\mathtt{Unf}\left(\{0,1\}\right)\) for each \(i\in[H]\). At time step \(h\), \(\xi_{h}\) is generated from \(\xi_{h-1}\) by uniformly flipping each bit in \(\xi_{h-1}\) independently with probability 0.1.

An observation \(x\) is generated stochastically from a latent state \(z=(s,\xi)\). We map \(s\) to a vector \(w\) encoding the identity of the state. We concatenate \((w,\xi)\), add Gaussian noise to each dimension, and multiply the result by a Hadamard matrix to generate \(x\). See Appendix F for full details.

Since the reward function of the combination-lock problem depends only on the endogenous state, we run PPE and then a value-iteration like algorithm (see Appendix D.2) to learn a near optimal policy.

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory. On inspection, we found that ID suffers from the under-abstraction issue highlighted earlier as it has difficulty separating observations from \(s_{3a}\) and \(s_{3b}\). On adding exogenous noise, the accuracy of Homer plummets significantly.

We test the ability of PPE to recover the latent endogenous transition dynamics in visual grid-world problem.2 The agent navigates in a \(N\times N\) grid world where each grid can contain a stationary object, the goal, or the agent. The agent's endogenous state is given by its position in the grid and its direction amongst four possible canonical directions. The agent can take five different actions for navigation. The world is visible to the agent as a \(8N\times 8N\) sized RGB image. We add exogenous noise as follows: at the beginning of each episode, we independently sample position, size and color of 5 ellipses. The position and size of these ellipses is perturbed after each time step independent of the action.

The exogenous state is given by the position, size and color of ellipses and is much larger than \(|\mathcal{S}|\leq 4N^{2}\).

In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems. We show that many existing RL algorithms fail in the presence of exogenous noise. We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores. We derive theoretical guarantees for PPE in near-deterministic setting and provide encouraging experimental evidence in support of our arguments. To our knowledge, this is the first such algorithm with guarantees for settings with exogenous noise.

Input: Figure 1: Left: An agent is walking next to a pond in a park and observes the world as an image. The world consists of a latent endogenous state, containing variable such as agent’s position, and a much larger latent exogenous state containing variables such as motion of ducks, ripples in the water, etc. Center: Graphical model of the EX-BMDP. Right: PPE learns a generalized form of inverse dynamics that recovers the endogenous state.

Input: Figure 2: Results on combination lock. Left: We show the latent transition dynamics of combination lock. Observations are not shown for brevity. Center: Shows minimal number of episodes needed to achieve a mean regret of at most V (π?)/2. Right: State decoding accuracy (in percent) of decoders learned by different methods. Solid lines implies no exogenous dimension while dashed lines imply an exogenous dimension of 100.

Input: Figure 3: Results on visual grid world. Left two: Shows sampled observations for the first two steps from the visual gridworld domain. The agent is depicted as a red-triangle, lava in orange, walls in grey, and the goal in green. Center Right: Shows errors of type 1 and type 2 made by the PPE in recovering the latent endogenous dynamics. Right: State decoding accuracy of PPE, Homer and a random uniform decoder. (see Section 6)

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim about initiating formal treatment of RL settings with jointly generated endogenous and exogenous states is well-supported. The paper thoroughly defines the EX-BMDP framework, provides mathematical formalization, demonstrates theoretical guarantees through PPE algorithm, and validates claims through experiments on combination lock and visual grid world problems. The reviewer's comments strengthen this assessment, noting the framework's ability to capture practical scenarios and achieve sample complexity bounds independent of exogenous state complexity. While there are some limitations (near-determinism requirement, lack of certain experimental comparisons), these don't detract significantly from the core claim's validity.

<score>1</score>

The claim is well-stated (score of 1) because:
1. It accurately reflects the paper's contributions without exaggeration
2. The mathematical framework and proofs provide rigorous formal treatment
3. Experimental results demonstrate practical applicability
4. The reviewer's comments validate the theoretical and practical significance
5. Limitations are appropriately acknowledged in the paper


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We initiate the formal study of latent state discovery in the presence of such _exogenous_ noise sources by proposing a new model, the Exogenous Block MDP (EX-BMDP), for rich observation RL.

The evidence to be evaluated is:We introduce a novel _Exogenous Block Markov Decision Process_ (EX-BMDP) setting to model systems with exogenous noise.

We start with describing the Block Markov Decision Process (BMDP) Du et al. (2019). This process consists of a finite set of observations \(\mathcal{X}\), a set of _latent_ states \(\mathcal{Z}\) with cardinality \(Z\), a finite set of actions \(\mathcal{A}\) with cardinality \(A\), a transition function \(T:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{Z})\), an emission function \(q:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{X})\), a reward function \(R:\mathcal{X}\times\mathcal{A}\rightarrow[0,1]\), a horizon \(H\in\mathbb{N}\), and a start state distribution \(\mu\in\Delta(\mathcal{Z})\). The agent interacts with the environment by repeatedly generating \(H\)-step trajectories \((z_{1},x_{1},a_{1},r_{1},\cdots,z_{H},x_{H},a_{H},r_{H})\) where \(z_{1}\sim\mu(\cdot)\) and for every \(h\in[H]\) we have \(x_{h}\sim q(\cdot\mid z_{h})\), \(r_{h}=R(x_{h},a_{h})\), and if \(h<H\), then \(z_{h+1}\sim T(\cdot\mid z_{h},a_{h})\). The agent does not observe the states \((z_{1},\cdots,z_{H})\), instead receiving only the observations \((x_{1},\cdots,x_{H})\) and rewards \((r_{1},\cdots,r_{H})\). We assume that the emission distributions of any two latent states are disjoint, usually referred as the _block assumption_: \(supp(q(\cdot\mid z_{1}))\cap supp(q(\cdot\mid z_{2}))=\emptyset\) when \(z_{1}\neq z_{2}\). The agent chooses actions using a policy \(\pi:\mathcal{X}\rightarrow\Delta(\mathcal{A})\). We also define the set of non-stationary policies \(\Pi_{\mathrm{NS}}=\Pi^{H}\) as a \(H\)-length tuple, with \((\pi_{1},\cdots,\pi_{H})\in\Pi_{\mathrm{NS}}\) denoting that the action at time step \(h\) is taken as \(a_{h}\sim\pi_{h}(\cdot\mid x_{h})\). The value \(V(\pi)\) of a policy \(\pi\) is the expected episodic sum of rewards \(V(\pi):=\mathbb{E}_{\pi}[\sum_{h=1}^{H}R(x_{h},a_{h})]\).

**Definition 1** (Exogenous Block Markov Decision Processes).: _An EX-BMDP is a BMDP such that the latent state can be decoupled into two parts \(z=(s,\xi)\) where \(s\in\mathcal{S}\) is the endogenous state and \(\xi\in\Xi\) is the exogenous state. For \(z\in\mathcal{Z}\) the initial distribution and transition functions are decoupled, that is: \(\mu(z)=\mu(s)\mu_{\xi}(\xi)\), and \(T(z^{\prime}\mid z,a)=T(s^{\prime}\mid s,a)T_{\xi}(\xi^{\prime}\mid\xi)\). _The observation space \(\mathcal{X}\) can be arbitrarily large to model which could be a high-dimensional real vector denoting an image, sound, or haptic data in an EX-BMDP.

The exogenous state \(\xi\) captures all the other information that the agent cannot control and does not affect the information it can manipulate. Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large. We note that the block assumption of the EX-BMDP implies the existence of two inverse mappings: \(\phi^{\star}:\mathcal{X}\rightarrow\mathcal{S}\) to map an observation to its endogenous state, and \(\phi^{\star}_{\xi}:\mathcal{X}\rightarrow\Xi\) to map it to its exogenous state.

The block assumption has been made by prior work (e.g., Du et al. (2019), Zhang et al. (2020)) to model many real-world settings where the observation is _rich_, i.e., it contains enough information to decode the latent state. The decoupled dynamics assumption made in the EX-BMDP setting is a natural way to characterize exogenous noise; the type of noise that is not affected by our actions nor affects the endogenous state but may have non-trivial dynamic. This decoupling captures the movement of ducks, captured in the visual field of the agent in Figure 1, and many additional exogenous processes (e.g., movement of clouds in a navigation task).

The structure of EX-BMDPs allows to reduce the size of the policy cover significantly to \(|\mathcal{S}|\ll|\mathcal{Z}|=|\mathcal{S}|\,|\Xi|\) when the size of the exogenous state space is large. Specifically, we show that the set of policies that reach each _endogenous_ state, and _do not depend on the exogenous_ part of the state is also a policy cover (see Appendix B, Proposition 4).

Unfortunately, in the presence of exogenous noise, Homer distinguishes between two transitions that represent transition between the same latent endogenous states but different exogenous states. In our walk in the park example, even if the agent moves between same points in two transitions, the model maybe able to tell these transitions apart by looking at the position of ducks which may have different behaviour in the two transitions. This results in the Homer creating \(\mathcal{O}(|\mathcal{Z}|)\) many abstract states. We call this the _under-abstraction_ problem.

Such a representation can ignore all information that is not relevant for action prediction, which includes all exogenous/uncontrollable information.

In this sense, inverse dynamics approaches result in an _over-abstraction_ problem where observations from different endogenous states can be mapped to the same abstract state.

Zhang et al. (2020) proposed learning a bisimulation metric to learn a representation which is invariant to exogenous noise.

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\). This shows that EX-BMDP is a highly non-trivial setting as we don't even have sample-efficient algorithms regardless of computationally-efficient.

In this section, we present an algorithm _Predictive Path Elimination_ (PPE) that we later show can provably solve any EX-BMDP with nearly deterministic dynamics and start state distribution of the endogenous state, while making no assumptions on the dynamics or start state distribution of the exogenous state (Algorithm 1). Before describing PPE, we highlight that PPE can be thought of as a computationally-efficient and simpler alternative to Algorithm 4 of Du et al. (2019) who studied rich-observation setting without exogenous noise.1

Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

If the reward function depends on the exogenous state then we can use the PSDP algorithm (Bagnell et al., 2004) to learn a near-optimal policy.

If the reward function only depends on the endogenous state, we can use a computationally cheaper value-iteration VI that uses the recovered transition dynamics.

In each case, the sample complexity of learning a near-optimal policy, given the output of PPE, scales with the size of endogenous and not the exogenous state space.

We analyze the algorithm in near-deterministic MDPs defined as follows: Two transition functions \(T_{1}\) and \(T_{2}\) are \(\eta\)_-close_ if for all \(h\in[H],a\in\mathcal{A},s\in\mathcal{S}_{h}\) it holds that \(||T_{1}(\cdot\mid s,a)-T_{2}(\cdot\mid s,a)||_{1}\leq\eta\). Analogously, two starting distribution \(\mu_{1}\) and \(\mu_{2}\) are \(\eta\)-close if \(||\mu_{1}(\cdot)-\mu_{2}(\cdot)||_{1}\leq\eta\). We emphasize that near-deterministic dynamics are common in real-world applications like robotics.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space. Further, since our analysis only uses standard uniform convergence arguments, it extends straightforwardly to infinitely large function classes by replacing \(\ln|\mathcal{F}|\) with other suitable complexity measures such as Rademacher complexity.

Let \(\mathbb{P}_{h}(\xi)\) denote the distribution over exogenous states at time step \(h\) which is independent of agent's policy. The Bayes optimal classifier (\(f^{*}_{h}:=f_{\Upsilon_{h},h}\)) of the prediction problem can be derived as:

where \((a)\) holds since all paths in \(\Upsilon_{h}\) are chosen uniformly, and \((b)\) critically uses the fact that for any open-loop policy \(v\) we have a factorization property, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\]

Our complete analysis in the Appendix generalizes the above reasoning to finite sample setting where we can only approximate \(f^{*}_{h}\) and \(\Delta\), as well as to EX-BMDPs with near-deterministic dynamics.

The combination lock problem is defined for a given horizon \(H\) by an endogenous state space \(\mathcal{S}=\{s_{1,a}\}\cup\{s_{h,a},s_{h,b},s_{h,c}\}_{h=2}^{H}\), an exogenous state space \(\Xi=\{0,1\}^{H}\), an action space \(\mathcal{A}\) with 10 actions, and a deterministic endogenous start state of \(s_{1,a}\).

Each instance of this problem is defined by two good action sequences \((a_{h})^{H}_{h=2},(a^{\prime}_{h})^{H}_{h=2}\) with \(a_{h}\neq a^{\prime}_{h}\), which are chosen uniformly randomly and kept fixed throughout. At \(h=1\), the agent is in \(s_{1,a}\) and action \(a_{1}\) leads to \(s_{2,a}\), \(a^{\prime}_{h}\) leads to \(s_{2,b}\), and all other actions lead to \(s_{2,c}\). For \(h>2\), taking action \(a_{h}\) in \(s_{h,a}\) leads to \(s_{h+1,a}\) and taking action \(a^{\prime}_{h}\) in \(s_{h,b}\) leads to \(s_{h+1,b}\). In all other cases involving taking an action in a state \(s_{h,g}\), we transition to the next bad state \(s_{h+1,c}\).

The exogenous state evolves as follows. We set \(\xi_{1}\in\{0,1\}^{H}\) where \(\xi_{1}(i)\sim\mathtt{Unf}\left(\{0,1\}\right)\) for each \(i\in[H]\). At time step \(h\), \(\xi_{h}\) is generated from \(\xi_{h-1}\) by uniformly flipping each bit in \(\xi_{h-1}\) independently with probability 0.1.

An observation \(x\) is generated stochastically from a latent state \(z=(s,\xi)\). We map \(s\) to a vector \(w\) encoding the identity of the state. We concatenate \((w,\xi)\), add Gaussian noise to each dimension, and multiply the result by a Hadamard matrix to generate \(x\). See Appendix F for full details. Our construction is inspired by prior work (Du et al., 2019; Misra et al., 2020).

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory. On inspection, we found that ID suffers from the under-abstraction issue highlighted earlier as it has difficulty separating observations from \(s_{3a}\) and \(s_{3b}\). On adding exogenous noise, the accuracy of Homer plummets significantly.

We test the ability of PPE to recover the latent endogenous transition dynamics in visual grid-world problem.2 The agent navigates in a \(N\times N\) grid world where each grid can contain a stationary object, the goal, or the agent. The agent's endogenous state is given by its position in the grid and its direction amongst four possible canonical directions. The agent can take five different actions for navigation. The world is visible to the agent as a \(8N\times 8N\) sized RGB image. We add exogenous noise as follows: at the beginning of each episode, we independently sample position, size and color of 5 ellipses. The position and size of these ellipses is perturbed after each time step independent of the action.

The exogenous state is given by the position, size and color of ellipses and is much larger than \(|\mathcal{S}|\leq 4N^{2}\).

In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems. We show that many existing RL algorithms fail in the presence of exogenous noise. We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores.

To our knowledge, this is the first such algorithm with guarantees for settings with exogenous noise.

Discovering and removing exogenous state variables and rewards for reinforcement learning.

Input: Figure 1: Left: An agent is walking next to a pond in a park and observes the world as an image. The world consists of a latent endogenous state, containing variable such as agent’s position, and a much larger latent exogenous state containing variables such as motion of ducks, ripples in the water, etc. Center: Graphical model of the EX-BMDP. Right: PPE learns a generalized form of inverse dynamics that recovers the endogenous state.

Input: Figure 2: Results on combination lock. Left: We show the latent transition dynamics of combination lock. Observations are not shown for brevity. Center: Shows minimal number of episodes needed to achieve a mean regret of at most V (π?)/2. Right: State decoding accuracy (in percent) of decoders learned by different methods. Solid lines implies no exogenous dimension while dashed lines imply an exogenous dimension of 100.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After careful review, the claim that this work "initiates the formal study of latent state discovery in the presence of exogenous noise sources" appears well-stated based on the extensive evidence provided. The paper:
- Introduces a novel formal model (EX-BMDP) with clear mathematical definitions
- Demonstrates theoretical analysis and proofs
- Shows limitations of existing approaches
- Provides experimental validation
- Acknowledges prior work while highlighting their novel contributions
- Clearly positions their work as addressing a gap in handling exogenous noise in rich observation RL

The evidence thoroughly supports the claim's scope and significance without overreaching.

<score>1</score>

Justification:
1. The formal mathematical framework (Definition 1) provides rigorous treatment
2. Experimental results on both synthetic (combination lock) and visual grid-world demonstrate practical relevance
3. Clear acknowledgment of building on prior work (e.g., Du et al. 2019) while highlighting novel contributions
4. Evidence shows limitations of existing approaches (Homer, ID) with exogenous noise
5. Claims are appropriately scoped to "initiating" formal study rather than claiming to completely solve the problem


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We initiate the formal study of latent state discovery in the presence of such _exogenous_ noise sources by proposing a new model, the Exogenous Block MDP (EX-BMDP), for rich observation RL.

The review comment to be evaluated is:

The paper is very well motivated. It has convinced me that being invariant to exogenous noise is important and that the EX-BMDP is useful abstraction for making this problem concrete. It is also very clearly presented. The argumentation is clear, and the flow of the paper is quite natural. Even the lengthy appendix is fairly easy to parse, with straightforward proofs and relevant details (e.g. specific claims about the unsuitability of alternative approaches).

One shortcoming is potential impact / applicability. I'm not convinced that environments with single starting states and near-deterministic dynamics is a very rich problem class. Or more specifically, I'm not convinced that restricting myself to this class would be preferable to working in the full space of MDPs with e.g. the possibility of aliasing a few states with a single-step inverse-dynamics approach. I am aware that impact is very hard to predict in advance, so I won't let this aspect unduly affect my scoring. And I'm convinced that even if PPE is never used in practice, this paper should still be accepted for its useful problem formulation and theoretical results. But a quick read would leave one with the impression that the alternatives (e.g. contrastive learning, inverse dynamics, bisimulation metrics) flaws outweigh their considerable advantages (e.g. applicability with stochastic dynamics) --  explicitly noting where alternative approaches are preferable would be appreciated.

I also noticed what I believe to be an omission in your related works: mutual information / empowerment based methods. "Variational Information Maximisation for Intrinsically Motivated Reinforcement Learning" seems particularly related in its search for predicable action sequences. And the follow-up work "Variational Intrinsic Control" appears to be a multi-step inverse dynamics analog that doesn't require open-loop policies. If you can show that these approaches are flawed in a way that PPE is not, then that would considerably raise my assessment of this work. And even if not, I believe this field (if not these specific works) should be acknowledged, as they similarly attempt to learn what in the environment is controllable.

While I agree that PPE is applicable in no-reward situations whereas bi-simulation metrics are not, your experiments all involve rewards, so I'm surprised a bi-simulation metric method was not included as a baseline.

While this isn't necessary for acceptance, it is worth noting that prior work has established much more challenging benchmarks for evaluating the representation of exogenous noise, and utilizing a pre-existing benchmark would make your empirical results considerably more impressive.

This is a small point, but it is not initially obvious why inverse dynamics fails on the combination lock problem. Does alliasing occur just when e.g. s_2a --> s_3a and s_2b-->s_3b have the same action?

The evidence to be evaluated is:We introduce a novel _Exogenous Block Markov Decision Process_ (EX-BMDP) setting to model systems with exogenous noise.

We start with describing the Block Markov Decision Process (BMDP) Du et al. (2019). This process consists of a finite set of observations \(\mathcal{X}\), a set of _latent_ states \(\mathcal{Z}\) with cardinality \(Z\), a finite set of actions \(\mathcal{A}\) with cardinality \(A\), a transition function \(T:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{Z})\), an emission function \(q:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{X})\), a reward function \(R:\mathcal{X}\times\mathcal{A}\rightarrow[0,1]\), a horizon \(H\in\mathbb{N}\), and a start state distribution \(\mu\in\Delta(\mathcal{Z})\). The agent interacts with the environment by repeatedly generating \(H\)-step trajectories \((z_{1},x_{1},a_{1},r_{1},\cdots,z_{H},x_{H},a_{H},r_{H})\) where \(z_{1}\sim\mu(\cdot)\) and for every \(h\in[H]\) we have \(x_{h}\sim q(\cdot\mid z_{h})\), \(r_{h}=R(x_{h},a_{h})\), and if \(h<H\), then \(z_{h+1}\sim T(\cdot\mid z_{h},a_{h})\). The agent does not observe the states \((z_{1},\cdots,z_{H})\), instead receiving only the observations \((x_{1},\cdots,x_{H})\) and rewards \((r_{1},\cdots,r_{H})\). We assume that the emission distributions of any two latent states are disjoint, usually referred as the _block assumption_: \(supp(q(\cdot\mid z_{1}))\cap supp(q(\cdot\mid z_{2}))=\emptyset\) when \(z_{1}\neq z_{2}\). The agent chooses actions using a policy \(\pi:\mathcal{X}\rightarrow\Delta(\mathcal{A})\). We also define the set of non-stationary policies \(\Pi_{\mathrm{NS}}=\Pi^{H}\) as a \(H\)-length tuple, with \((\pi_{1},\cdots,\pi_{H})\in\Pi_{\mathrm{NS}}\) denoting that the action at time step \(h\) is taken as \(a_{h}\sim\pi_{h}(\cdot\mid x_{h})\). The value \(V(\pi)\) of a policy \(\pi\) is the expected episodic sum of rewards \(V(\pi):=\mathbb{E}_{\pi}[\sum_{h=1}^{H}R(x_{h},a_{h})]\).

**Definition 1** (Exogenous Block Markov Decision Processes).: _An EX-BMDP is a BMDP such that the latent state can be decoupled into two parts \(z=(s,\xi)\) where \(s\in\mathcal{S}\) is the endogenous state and \(\xi\in\Xi\) is the exogenous state. For \(z\in\mathcal{Z}\) the initial distribution and transition functions are decoupled, that is: \(\mu(z)=\mu(s)\mu_{\xi}(\xi)\), and \(T(z^{\prime}\mid z,a)=T(s^{\prime}\mid s,a)T_{\xi}(\xi^{\prime}\mid\xi)\). _The observation space \(\mathcal{X}\) can be arbitrarily large to model which could be a high-dimensional real vector denoting an image, sound, or haptic data in an EX-BMDP.

The exogenous state \(\xi\) captures all the other information that the agent cannot control and does not affect the information it can manipulate. Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large. We note that the block assumption of the EX-BMDP implies the existence of two inverse mappings: \(\phi^{\star}:\mathcal{X}\rightarrow\mathcal{S}\) to map an observation to its endogenous state, and \(\phi^{\star}_{\xi}:\mathcal{X}\rightarrow\Xi\) to map it to its exogenous state.

The block assumption has been made by prior work (e.g., Du et al. (2019), Zhang et al. (2020)) to model many real-world settings where the observation is _rich_, i.e., it contains enough information to decode the latent state. The decoupled dynamics assumption made in the EX-BMDP setting is a natural way to characterize exogenous noise; the type of noise that is not affected by our actions nor affects the endogenous state but may have non-trivial dynamic. This decoupling captures the movement of ducks, captured in the visual field of the agent in Figure 1, and many additional exogenous processes (e.g., movement of clouds in a navigation task).

The structure of EX-BMDPs allows to reduce the size of the policy cover significantly to \(|\mathcal{S}|\ll|\mathcal{Z}|=|\mathcal{S}|\,|\Xi|\) when the size of the exogenous state space is large. Specifically, we show that the set of policies that reach each _endogenous_ state, and _do not depend on the exogenous_ part of the state is also a policy cover (see Appendix B, Proposition 4).

Unfortunately, in the presence of exogenous noise, Homer distinguishes between two transitions that represent transition between the same latent endogenous states but different exogenous states. In our walk in the park example, even if the agent moves between same points in two transitions, the model maybe able to tell these transitions apart by looking at the position of ducks which may have different behaviour in the two transitions. This results in the Homer creating \(\mathcal{O}(|\mathcal{Z}|)\) many abstract states. We call this the _under-abstraction_ problem.

Such a representation can ignore all information that is not relevant for action prediction, which includes all exogenous/uncontrollable information.

In this sense, inverse dynamics approaches result in an _over-abstraction_ problem where observations from different endogenous states can be mapped to the same abstract state.

Zhang et al. (2020) proposed learning a bisimulation metric to learn a representation which is invariant to exogenous noise.

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\). This shows that EX-BMDP is a highly non-trivial setting as we don't even have sample-efficient algorithms regardless of computationally-efficient.

In this section, we present an algorithm _Predictive Path Elimination_ (PPE) that we later show can provably solve any EX-BMDP with nearly deterministic dynamics and start state distribution of the endogenous state, while making no assumptions on the dynamics or start state distribution of the exogenous state (Algorithm 1). Before describing PPE, we highlight that PPE can be thought of as a computationally-efficient and simpler alternative to Algorithm 4 of Du et al. (2019) who studied rich-observation setting without exogenous noise.1

Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

If the reward function depends on the exogenous state then we can use the PSDP algorithm (Bagnell et al., 2004) to learn a near-optimal policy.

If the reward function only depends on the endogenous state, we can use a computationally cheaper value-iteration VI that uses the recovered transition dynamics.

In each case, the sample complexity of learning a near-optimal policy, given the output of PPE, scales with the size of endogenous and not the exogenous state space.

We analyze the algorithm in near-deterministic MDPs defined as follows: Two transition functions \(T_{1}\) and \(T_{2}\) are \(\eta\)_-close_ if for all \(h\in[H],a\in\mathcal{A},s\in\mathcal{S}_{h}\) it holds that \(||T_{1}(\cdot\mid s,a)-T_{2}(\cdot\mid s,a)||_{1}\leq\eta\). Analogously, two starting distribution \(\mu_{1}\) and \(\mu_{2}\) are \(\eta\)-close if \(||\mu_{1}(\cdot)-\mu_{2}(\cdot)||_{1}\leq\eta\). We emphasize that near-deterministic dynamics are common in real-world applications like robotics.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space. Further, since our analysis only uses standard uniform convergence arguments, it extends straightforwardly to infinitely large function classes by replacing \(\ln|\mathcal{F}|\) with other suitable complexity measures such as Rademacher complexity.

Let \(\mathbb{P}_{h}(\xi)\) denote the distribution over exogenous states at time step \(h\) which is independent of agent's policy. The Bayes optimal classifier (\(f^{*}_{h}:=f_{\Upsilon_{h},h}\)) of the prediction problem can be derived as:

where \((a)\) holds since all paths in \(\Upsilon_{h}\) are chosen uniformly, and \((b)\) critically uses the fact that for any open-loop policy \(v\) we have a factorization property, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\]

Our complete analysis in the Appendix generalizes the above reasoning to finite sample setting where we can only approximate \(f^{*}_{h}\) and \(\Delta\), as well as to EX-BMDPs with near-deterministic dynamics.

The combination lock problem is defined for a given horizon \(H\) by an endogenous state space \(\mathcal{S}=\{s_{1,a}\}\cup\{s_{h,a},s_{h,b},s_{h,c}\}_{h=2}^{H}\), an exogenous state space \(\Xi=\{0,1\}^{H}\), an action space \(\mathcal{A}\) with 10 actions, and a deterministic endogenous start state of \(s_{1,a}\).

Each instance of this problem is defined by two good action sequences \((a_{h})^{H}_{h=2},(a^{\prime}_{h})^{H}_{h=2}\) with \(a_{h}\neq a^{\prime}_{h}\), which are chosen uniformly randomly and kept fixed throughout. At \(h=1\), the agent is in \(s_{1,a}\) and action \(a_{1}\) leads to \(s_{2,a}\), \(a^{\prime}_{h}\) leads to \(s_{2,b}\), and all other actions lead to \(s_{2,c}\). For \(h>2\), taking action \(a_{h}\) in \(s_{h,a}\) leads to \(s_{h+1,a}\) and taking action \(a^{\prime}_{h}\) in \(s_{h,b}\) leads to \(s_{h+1,b}\). In all other cases involving taking an action in a state \(s_{h,g}\), we transition to the next bad state \(s_{h+1,c}\).

The exogenous state evolves as follows. We set \(\xi_{1}\in\{0,1\}^{H}\) where \(\xi_{1}(i)\sim\mathtt{Unf}\left(\{0,1\}\right)\) for each \(i\in[H]\). At time step \(h\), \(\xi_{h}\) is generated from \(\xi_{h-1}\) by uniformly flipping each bit in \(\xi_{h-1}\) independently with probability 0.1.

An observation \(x\) is generated stochastically from a latent state \(z=(s,\xi)\). We map \(s\) to a vector \(w\) encoding the identity of the state. We concatenate \((w,\xi)\), add Gaussian noise to each dimension, and multiply the result by a Hadamard matrix to generate \(x\). See Appendix F for full details. Our construction is inspired by prior work (Du et al., 2019; Misra et al., 2020).

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory. On inspection, we found that ID suffers from the under-abstraction issue highlighted earlier as it has difficulty separating observations from \(s_{3a}\) and \(s_{3b}\). On adding exogenous noise, the accuracy of Homer plummets significantly.

We test the ability of PPE to recover the latent endogenous transition dynamics in visual grid-world problem.2 The agent navigates in a \(N\times N\) grid world where each grid can contain a stationary object, the goal, or the agent. The agent's endogenous state is given by its position in the grid and its direction amongst four possible canonical directions. The agent can take five different actions for navigation. The world is visible to the agent as a \(8N\times 8N\) sized RGB image. We add exogenous noise as follows: at the beginning of each episode, we independently sample position, size and color of 5 ellipses. The position and size of these ellipses is perturbed after each time step independent of the action.

The exogenous state is given by the position, size and color of ellipses and is much larger than \(|\mathcal{S}|\leq 4N^{2}\).

In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems. We show that many existing RL algorithms fail in the presence of exogenous noise. We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores.

To our knowledge, this is the first such algorithm with guarantees for settings with exogenous noise.

Discovering and removing exogenous state variables and rewards for reinforcement learning.

Input: Figure 1: Left: An agent is walking next to a pond in a park and observes the world as an image. The world consists of a latent endogenous state, containing variable such as agent’s position, and a much larger latent exogenous state containing variables such as motion of ducks, ripples in the water, etc. Center: Graphical model of the EX-BMDP. Right: PPE learns a generalized form of inverse dynamics that recovers the endogenous state.

Input: Figure 2: Results on combination lock. Left: We show the latent transition dynamics of combination lock. Observations are not shown for brevity. Center: Shows minimal number of episodes needed to achieve a mean regret of at most V (π?)/2. Right: State decoding accuracy (in percent) of decoders learned by different methods. Solid lines implies no exogenous dimension while dashed lines imply an exogenous dimension of 100.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim that this work "initiates the formal study of latent state discovery in the presence of exogenous noise sources" appears somewhat overstated. While the paper does introduce a novel model (EX-BMDP) and makes important contributions, the review points out omitted related work, particularly mutual information/empowerment based methods that similarly attempted to learn controllable aspects of environments. The evidence shows thorough theoretical development and experimental validation, but claiming to "initiate" the formal study overlooks prior contributions in this space.

<score>4</score>

Justification:
1. The paper makes significant novel contributions by formalizing the EX-BMDP model and providing theoretical guarantees, supported by detailed mathematical definitions and proofs.

2. However, the reviewer identifies relevant prior work that was omitted, particularly:
- "Variational Information Maximisation for Intrinsically Motivated Reinforcement Learning"
- "Variational Intrinsic Control"
These papers apparently tackled similar problems of identifying controllable aspects of environments.

3. While the paper advances the field significantly, claiming to "initiate" the formal study overstates its position as the first work in this space, when other approaches had previously explored related concepts.

4. The thorough theoretical framework and experimental results support the paper's contributions, but more modest framing acknowledging building upon prior work would have been more appropriate.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We initiate the formal study of latent state discovery in the presence of such _exogenous_ noise sources by proposing a new model, the Exogenous Block MDP (EX-BMDP), for rich observation RL.

The review comment to be evaluated is:

Strengths:

- The problem setting is important and seems to be relevant to many real-world problems.
- While the idea of using inverse dynamics and even multi-step inverse dynamics [1] to learn representations in these types of environments to learn controllable representations is not novel, this paper provides a theoretical basis for this choice, which I think is valuable.
- I found sections 1 through 5 to be very well written.

Weaknesses:

- In the experiment section, it seems as though you are testing the planning/policy learning abilities with PPE in both of the environments. However, as far as I can tell, there is no indication of how you perform this planning in PPE in the main text. The paper simply switches from talking about learning policy covers to showing regret for environments with only a small mention of the fact that you could plan using the data collected with PPE. The way the experiments are run and the choice to test the planning performance / how the state representations are obtained in experiments should be explained in more detail in my opinion.
- The paper discusses prior works with inverse dynamics seemingly only in the context of their applications in representation learning in two prior works on exploration. I think the application of multi-step inverse dynamics in goal-conditioned environments described in [1] is worth mentioning, since for a fixed goal this algorithm seems like it is doing something similar to PPL. Related is the lack of a "related works" section in the paper, which would be helpful for framing PPL in the context of prior works.
- There is no conclusion section, which I feel would improve the paper.

My major questions are:

1. How is the policy learning done in the experiment section? How are representations found in the decoding accuracy plot?
2. How does this paper relate to prior works on inverse dynamics?

[1] Paster, Keiran, et al. Planning from Pixels Using Inverse Dynamics Models. 2020. openreview.net, https://openreview.net/forum?id=V6BjBgku7Ro.

The evidence to be evaluated is:We introduce a novel _Exogenous Block Markov Decision Process_ (EX-BMDP) setting to model systems with exogenous noise.

We start with describing the Block Markov Decision Process (BMDP) Du et al. (2019). This process consists of a finite set of observations \(\mathcal{X}\), a set of _latent_ states \(\mathcal{Z}\) with cardinality \(Z\), a finite set of actions \(\mathcal{A}\) with cardinality \(A\), a transition function \(T:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{Z})\), an emission function \(q:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{X})\), a reward function \(R:\mathcal{X}\times\mathcal{A}\rightarrow[0,1]\), a horizon \(H\in\mathbb{N}\), and a start state distribution \(\mu\in\Delta(\mathcal{Z})\). The agent interacts with the environment by repeatedly generating \(H\)-step trajectories \((z_{1},x_{1},a_{1},r_{1},\cdots,z_{H},x_{H},a_{H},r_{H})\) where \(z_{1}\sim\mu(\cdot)\) and for every \(h\in[H]\) we have \(x_{h}\sim q(\cdot\mid z_{h})\), \(r_{h}=R(x_{h},a_{h})\), and if \(h<H\), then \(z_{h+1}\sim T(\cdot\mid z_{h},a_{h})\). The agent does not observe the states \((z_{1},\cdots,z_{H})\), instead receiving only the observations \((x_{1},\cdots,x_{H})\) and rewards \((r_{1},\cdots,r_{H})\). We assume that the emission distributions of any two latent states are disjoint, usually referred as the _block assumption_: \(supp(q(\cdot\mid z_{1}))\cap supp(q(\cdot\mid z_{2}))=\emptyset\) when \(z_{1}\neq z_{2}\). The agent chooses actions using a policy \(\pi:\mathcal{X}\rightarrow\Delta(\mathcal{A})\). We also define the set of non-stationary policies \(\Pi_{\mathrm{NS}}=\Pi^{H}\) as a \(H\)-length tuple, with \((\pi_{1},\cdots,\pi_{H})\in\Pi_{\mathrm{NS}}\) denoting that the action at time step \(h\) is taken as \(a_{h}\sim\pi_{h}(\cdot\mid x_{h})\). The value \(V(\pi)\) of a policy \(\pi\) is the expected episodic sum of rewards \(V(\pi):=\mathbb{E}_{\pi}[\sum_{h=1}^{H}R(x_{h},a_{h})]\).

**Definition 1** (Exogenous Block Markov Decision Processes).: _An EX-BMDP is a BMDP such that the latent state can be decoupled into two parts \(z=(s,\xi)\) where \(s\in\mathcal{S}\) is the endogenous state and \(\xi\in\Xi\) is the exogenous state. For \(z\in\mathcal{Z}\) the initial distribution and transition functions are decoupled, that is: \(\mu(z)=\mu(s)\mu_{\xi}(\xi)\), and \(T(z^{\prime}\mid z,a)=T(s^{\prime}\mid s,a)T_{\xi}(\xi^{\prime}\mid\xi)\). _The observation space \(\mathcal{X}\) can be arbitrarily large to model which could be a high-dimensional real vector denoting an image, sound, or haptic data in an EX-BMDP.

The exogenous state \(\xi\) captures all the other information that the agent cannot control and does not affect the information it can manipulate. Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large. We note that the block assumption of the EX-BMDP implies the existence of two inverse mappings: \(\phi^{\star}:\mathcal{X}\rightarrow\mathcal{S}\) to map an observation to its endogenous state, and \(\phi^{\star}_{\xi}:\mathcal{X}\rightarrow\Xi\) to map it to its exogenous state.

The block assumption has been made by prior work (e.g., Du et al. (2019), Zhang et al. (2020)) to model many real-world settings where the observation is _rich_, i.e., it contains enough information to decode the latent state. The decoupled dynamics assumption made in the EX-BMDP setting is a natural way to characterize exogenous noise; the type of noise that is not affected by our actions nor affects the endogenous state but may have non-trivial dynamic. This decoupling captures the movement of ducks, captured in the visual field of the agent in Figure 1, and many additional exogenous processes (e.g., movement of clouds in a navigation task).

The structure of EX-BMDPs allows to reduce the size of the policy cover significantly to \(|\mathcal{S}|\ll|\mathcal{Z}|=|\mathcal{S}|\,|\Xi|\) when the size of the exogenous state space is large. Specifically, we show that the set of policies that reach each _endogenous_ state, and _do not depend on the exogenous_ part of the state is also a policy cover (see Appendix B, Proposition 4).

Unfortunately, in the presence of exogenous noise, Homer distinguishes between two transitions that represent transition between the same latent endogenous states but different exogenous states. In our walk in the park example, even if the agent moves between same points in two transitions, the model maybe able to tell these transitions apart by looking at the position of ducks which may have different behaviour in the two transitions. This results in the Homer creating \(\mathcal{O}(|\mathcal{Z}|)\) many abstract states. We call this the _under-abstraction_ problem.

Such a representation can ignore all information that is not relevant for action prediction, which includes all exogenous/uncontrollable information.

In this sense, inverse dynamics approaches result in an _over-abstraction_ problem where observations from different endogenous states can be mapped to the same abstract state.

Zhang et al. (2020) proposed learning a bisimulation metric to learn a representation which is invariant to exogenous noise.

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\). This shows that EX-BMDP is a highly non-trivial setting as we don't even have sample-efficient algorithms regardless of computationally-efficient.

In this section, we present an algorithm _Predictive Path Elimination_ (PPE) that we later show can provably solve any EX-BMDP with nearly deterministic dynamics and start state distribution of the endogenous state, while making no assumptions on the dynamics or start state distribution of the exogenous state (Algorithm 1). Before describing PPE, we highlight that PPE can be thought of as a computationally-efficient and simpler alternative to Algorithm 4 of Du et al. (2019) who studied rich-observation setting without exogenous noise.1

Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

If the reward function depends on the exogenous state then we can use the PSDP algorithm (Bagnell et al., 2004) to learn a near-optimal policy.

If the reward function only depends on the endogenous state, we can use a computationally cheaper value-iteration VI that uses the recovered transition dynamics.

In each case, the sample complexity of learning a near-optimal policy, given the output of PPE, scales with the size of endogenous and not the exogenous state space.

We analyze the algorithm in near-deterministic MDPs defined as follows: Two transition functions \(T_{1}\) and \(T_{2}\) are \(\eta\)_-close_ if for all \(h\in[H],a\in\mathcal{A},s\in\mathcal{S}_{h}\) it holds that \(||T_{1}(\cdot\mid s,a)-T_{2}(\cdot\mid s,a)||_{1}\leq\eta\). Analogously, two starting distribution \(\mu_{1}\) and \(\mu_{2}\) are \(\eta\)-close if \(||\mu_{1}(\cdot)-\mu_{2}(\cdot)||_{1}\leq\eta\). We emphasize that near-deterministic dynamics are common in real-world applications like robotics.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space. Further, since our analysis only uses standard uniform convergence arguments, it extends straightforwardly to infinitely large function classes by replacing \(\ln|\mathcal{F}|\) with other suitable complexity measures such as Rademacher complexity.

Let \(\mathbb{P}_{h}(\xi)\) denote the distribution over exogenous states at time step \(h\) which is independent of agent's policy. The Bayes optimal classifier (\(f^{*}_{h}:=f_{\Upsilon_{h},h}\)) of the prediction problem can be derived as:

where \((a)\) holds since all paths in \(\Upsilon_{h}\) are chosen uniformly, and \((b)\) critically uses the fact that for any open-loop policy \(v\) we have a factorization property, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\]

Our complete analysis in the Appendix generalizes the above reasoning to finite sample setting where we can only approximate \(f^{*}_{h}\) and \(\Delta\), as well as to EX-BMDPs with near-deterministic dynamics.

The combination lock problem is defined for a given horizon \(H\) by an endogenous state space \(\mathcal{S}=\{s_{1,a}\}\cup\{s_{h,a},s_{h,b},s_{h,c}\}_{h=2}^{H}\), an exogenous state space \(\Xi=\{0,1\}^{H}\), an action space \(\mathcal{A}\) with 10 actions, and a deterministic endogenous start state of \(s_{1,a}\).

Each instance of this problem is defined by two good action sequences \((a_{h})^{H}_{h=2},(a^{\prime}_{h})^{H}_{h=2}\) with \(a_{h}\neq a^{\prime}_{h}\), which are chosen uniformly randomly and kept fixed throughout. At \(h=1\), the agent is in \(s_{1,a}\) and action \(a_{1}\) leads to \(s_{2,a}\), \(a^{\prime}_{h}\) leads to \(s_{2,b}\), and all other actions lead to \(s_{2,c}\). For \(h>2\), taking action \(a_{h}\) in \(s_{h,a}\) leads to \(s_{h+1,a}\) and taking action \(a^{\prime}_{h}\) in \(s_{h,b}\) leads to \(s_{h+1,b}\). In all other cases involving taking an action in a state \(s_{h,g}\), we transition to the next bad state \(s_{h+1,c}\).

The exogenous state evolves as follows. We set \(\xi_{1}\in\{0,1\}^{H}\) where \(\xi_{1}(i)\sim\mathtt{Unf}\left(\{0,1\}\right)\) for each \(i\in[H]\). At time step \(h\), \(\xi_{h}\) is generated from \(\xi_{h-1}\) by uniformly flipping each bit in \(\xi_{h-1}\) independently with probability 0.1.

An observation \(x\) is generated stochastically from a latent state \(z=(s,\xi)\). We map \(s\) to a vector \(w\) encoding the identity of the state. We concatenate \((w,\xi)\), add Gaussian noise to each dimension, and multiply the result by a Hadamard matrix to generate \(x\). See Appendix F for full details. Our construction is inspired by prior work (Du et al., 2019; Misra et al., 2020).

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory. On inspection, we found that ID suffers from the under-abstraction issue highlighted earlier as it has difficulty separating observations from \(s_{3a}\) and \(s_{3b}\). On adding exogenous noise, the accuracy of Homer plummets significantly.

We test the ability of PPE to recover the latent endogenous transition dynamics in visual grid-world problem.2 The agent navigates in a \(N\times N\) grid world where each grid can contain a stationary object, the goal, or the agent. The agent's endogenous state is given by its position in the grid and its direction amongst four possible canonical directions. The agent can take five different actions for navigation. The world is visible to the agent as a \(8N\times 8N\) sized RGB image. We add exogenous noise as follows: at the beginning of each episode, we independently sample position, size and color of 5 ellipses. The position and size of these ellipses is perturbed after each time step independent of the action.

The exogenous state is given by the position, size and color of ellipses and is much larger than \(|\mathcal{S}|\leq 4N^{2}\).

In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems. We show that many existing RL algorithms fail in the presence of exogenous noise. We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores.

To our knowledge, this is the first such algorithm with guarantees for settings with exogenous noise.

Discovering and removing exogenous state variables and rewards for reinforcement learning.

Input: Figure 1: Left: An agent is walking next to a pond in a park and observes the world as an image. The world consists of a latent endogenous state, containing variable such as agent’s position, and a much larger latent exogenous state containing variables such as motion of ducks, ripples in the water, etc. Center: Graphical model of the EX-BMDP. Right: PPE learns a generalized form of inverse dynamics that recovers the endogenous state.

Input: Figure 2: Results on combination lock. Left: We show the latent transition dynamics of combination lock. Observations are not shown for brevity. Center: Shows minimal number of episodes needed to achieve a mean regret of at most V (π?)/2. Right: State decoding accuracy (in percent) of decoders learned by different methods. Solid lines implies no exogenous dimension while dashed lines imply an exogenous dimension of 100.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find this claim to be well-stated for several reasons:

1. The paper thoroughly establishes and formalizes the EX-BMDP model with detailed mathematical definitions and properties.

2. The evidence demonstrates that this is indeed an initial formal study, as shown by:
- Novel mathematical formalization of exogenous noise in RL settings
- Clear differentiation from prior work (Du et al. 2019, Zhang et al. 2020)
- Introduction of new theoretical results regarding Bellman rank
- Development and analysis of a new algorithm (PPE)

3. The reviewer comments support this being a valuable theoretical contribution while noting some limitations in experimental details, which doesn't affect the validity of this being an initiating formal study.

The claim avoids overstatement by limiting itself to "initiating the formal study" rather than claiming to solve all aspects of the problem.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We initiate the formal study of latent state discovery in the presence of such _exogenous_ noise sources by proposing a new model, the Exogenous Block MDP (EX-BMDP), for rich observation RL.

The review comment to be evaluated is:

STRENGTHS

(S1) The problem of efficiently learning in high-dimensional observation spaces is a good one, and the authors provided an excellent discussion of the technical components of this problem in the early parts of the paper. I thought the definition and presentation of the EX-BMDP was particularly good.

(S2) I appreciate the authors providing experimental results to show that PPE performs well in some regards.

WEAKNESSES

(W1) I'm afraid I found the discussion of PPE to be a bit difficult to decode. Even looking at Algorithm 1, I find myself with a number of important questions unanswered, chief among them being "what about $\phi^*_e$"? It strikes me as odd that nowhere in this algorithm are the inverse mappings from observation to exogenous/endogenous state represented or used. Why is this?

(W2) I'm a little concerned with the computational tractability of the algorithm. My understanding from the paper is that one seeks some sort of "covering" of the state space--in what sense is it practical to obtain and/or store this covering? If the problem setting consists of high-dimensional observations this seems especially challenging.

(W3) From my naive perspective, it seems that the ultimate goal here is to use PPE and then actually perform reinforcement learning. However, I didn't understand from the brief discussion at the end of Section 4 (or, frankly, from reading the Appendix) how exactly that would be accomplished. I feel this is important enough that it should appear in the main paper.

POST-DISCUSSION COMMENTS

During the discussion, I feel as though the authors adequately addressed each of the issues I raised above, and so I'm happy to raise my score to accept.

The evidence to be evaluated is:We introduce a novel _Exogenous Block Markov Decision Process_ (EX-BMDP) setting to model systems with exogenous noise.

We start with describing the Block Markov Decision Process (BMDP) Du et al. (2019). This process consists of a finite set of observations \(\mathcal{X}\), a set of _latent_ states \(\mathcal{Z}\) with cardinality \(Z\), a finite set of actions \(\mathcal{A}\) with cardinality \(A\), a transition function \(T:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{Z})\), an emission function \(q:\mathcal{Z}\times\mathcal{A}\rightarrow\Delta(\mathcal{X})\), a reward function \(R:\mathcal{X}\times\mathcal{A}\rightarrow[0,1]\), a horizon \(H\in\mathbb{N}\), and a start state distribution \(\mu\in\Delta(\mathcal{Z})\). The agent interacts with the environment by repeatedly generating \(H\)-step trajectories \((z_{1},x_{1},a_{1},r_{1},\cdots,z_{H},x_{H},a_{H},r_{H})\) where \(z_{1}\sim\mu(\cdot)\) and for every \(h\in[H]\) we have \(x_{h}\sim q(\cdot\mid z_{h})\), \(r_{h}=R(x_{h},a_{h})\), and if \(h<H\), then \(z_{h+1}\sim T(\cdot\mid z_{h},a_{h})\). The agent does not observe the states \((z_{1},\cdots,z_{H})\), instead receiving only the observations \((x_{1},\cdots,x_{H})\) and rewards \((r_{1},\cdots,r_{H})\). We assume that the emission distributions of any two latent states are disjoint, usually referred as the _block assumption_: \(supp(q(\cdot\mid z_{1}))\cap supp(q(\cdot\mid z_{2}))=\emptyset\) when \(z_{1}\neq z_{2}\). The agent chooses actions using a policy \(\pi:\mathcal{X}\rightarrow\Delta(\mathcal{A})\). We also define the set of non-stationary policies \(\Pi_{\mathrm{NS}}=\Pi^{H}\) as a \(H\)-length tuple, with \((\pi_{1},\cdots,\pi_{H})\in\Pi_{\mathrm{NS}}\) denoting that the action at time step \(h\) is taken as \(a_{h}\sim\pi_{h}(\cdot\mid x_{h})\). The value \(V(\pi)\) of a policy \(\pi\) is the expected episodic sum of rewards \(V(\pi):=\mathbb{E}_{\pi}[\sum_{h=1}^{H}R(x_{h},a_{h})]\).

**Definition 1** (Exogenous Block Markov Decision Processes).: _An EX-BMDP is a BMDP such that the latent state can be decoupled into two parts \(z=(s,\xi)\) where \(s\in\mathcal{S}\) is the endogenous state and \(\xi\in\Xi\) is the exogenous state. For \(z\in\mathcal{Z}\) the initial distribution and transition functions are decoupled, that is: \(\mu(z)=\mu(s)\mu_{\xi}(\xi)\), and \(T(z^{\prime}\mid z,a)=T(s^{\prime}\mid s,a)T_{\xi}(\xi^{\prime}\mid\xi)\). _The observation space \(\mathcal{X}\) can be arbitrarily large to model which could be a high-dimensional real vector denoting an image, sound, or haptic data in an EX-BMDP.

The exogenous state \(\xi\) captures all the other information that the agent cannot control and does not affect the information it can manipulate. Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large. We note that the block assumption of the EX-BMDP implies the existence of two inverse mappings: \(\phi^{\star}:\mathcal{X}\rightarrow\mathcal{S}\) to map an observation to its endogenous state, and \(\phi^{\star}_{\xi}:\mathcal{X}\rightarrow\Xi\) to map it to its exogenous state.

The block assumption has been made by prior work (e.g., Du et al. (2019), Zhang et al. (2020)) to model many real-world settings where the observation is _rich_, i.e., it contains enough information to decode the latent state. The decoupled dynamics assumption made in the EX-BMDP setting is a natural way to characterize exogenous noise; the type of noise that is not affected by our actions nor affects the endogenous state but may have non-trivial dynamic. This decoupling captures the movement of ducks, captured in the visual field of the agent in Figure 1, and many additional exogenous processes (e.g., movement of clouds in a navigation task).

The structure of EX-BMDPs allows to reduce the size of the policy cover significantly to \(|\mathcal{S}|\ll|\mathcal{Z}|=|\mathcal{S}|\,|\Xi|\) when the size of the exogenous state space is large. Specifically, we show that the set of policies that reach each _endogenous_ state, and _do not depend on the exogenous_ part of the state is also a policy cover (see Appendix B, Proposition 4).

Unfortunately, in the presence of exogenous noise, Homer distinguishes between two transitions that represent transition between the same latent endogenous states but different exogenous states. In our walk in the park example, even if the agent moves between same points in two transitions, the model maybe able to tell these transitions apart by looking at the position of ducks which may have different behaviour in the two transitions. This results in the Homer creating \(\mathcal{O}(|\mathcal{Z}|)\) many abstract states. We call this the _under-abstraction_ problem.

Such a representation can ignore all information that is not relevant for action prediction, which includes all exogenous/uncontrollable information.

In this sense, inverse dynamics approaches result in an _over-abstraction_ problem where observations from different endogenous states can be mapped to the same abstract state.

Zhang et al. (2020) proposed learning a bisimulation metric to learn a representation which is invariant to exogenous noise.

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\). This shows that EX-BMDP is a highly non-trivial setting as we don't even have sample-efficient algorithms regardless of computationally-efficient.

In this section, we present an algorithm _Predictive Path Elimination_ (PPE) that we later show can provably solve any EX-BMDP with nearly deterministic dynamics and start state distribution of the endogenous state, while making no assumptions on the dynamics or start state distribution of the exogenous state (Algorithm 1). Before describing PPE, we highlight that PPE can be thought of as a computationally-efficient and simpler alternative to Algorithm 4 of Du et al. (2019) who studied rich-observation setting without exogenous noise.1

Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

If the reward function depends on the exogenous state then we can use the PSDP algorithm (Bagnell et al., 2004) to learn a near-optimal policy.

If the reward function only depends on the endogenous state, we can use a computationally cheaper value-iteration VI that uses the recovered transition dynamics.

In each case, the sample complexity of learning a near-optimal policy, given the output of PPE, scales with the size of endogenous and not the exogenous state space.

We analyze the algorithm in near-deterministic MDPs defined as follows: Two transition functions \(T_{1}\) and \(T_{2}\) are \(\eta\)_-close_ if for all \(h\in[H],a\in\mathcal{A},s\in\mathcal{S}_{h}\) it holds that \(||T_{1}(\cdot\mid s,a)-T_{2}(\cdot\mid s,a)||_{1}\leq\eta\). Analogously, two starting distribution \(\mu_{1}\) and \(\mu_{2}\) are \(\eta\)-close if \(||\mu_{1}(\cdot)-\mu_{2}(\cdot)||_{1}\leq\eta\). We emphasize that near-deterministic dynamics are common in real-world applications like robotics.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space. Further, since our analysis only uses standard uniform convergence arguments, it extends straightforwardly to infinitely large function classes by replacing \(\ln|\mathcal{F}|\) with other suitable complexity measures such as Rademacher complexity.

Let \(\mathbb{P}_{h}(\xi)\) denote the distribution over exogenous states at time step \(h\) which is independent of agent's policy. The Bayes optimal classifier (\(f^{*}_{h}:=f_{\Upsilon_{h},h}\)) of the prediction problem can be derived as:

where \((a)\) holds since all paths in \(\Upsilon_{h}\) are chosen uniformly, and \((b)\) critically uses the fact that for any open-loop policy \(v\) we have a factorization property, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\]

Our complete analysis in the Appendix generalizes the above reasoning to finite sample setting where we can only approximate \(f^{*}_{h}\) and \(\Delta\), as well as to EX-BMDPs with near-deterministic dynamics.

The combination lock problem is defined for a given horizon \(H\) by an endogenous state space \(\mathcal{S}=\{s_{1,a}\}\cup\{s_{h,a},s_{h,b},s_{h,c}\}_{h=2}^{H}\), an exogenous state space \(\Xi=\{0,1\}^{H}\), an action space \(\mathcal{A}\) with 10 actions, and a deterministic endogenous start state of \(s_{1,a}\).

Each instance of this problem is defined by two good action sequences \((a_{h})^{H}_{h=2},(a^{\prime}_{h})^{H}_{h=2}\) with \(a_{h}\neq a^{\prime}_{h}\), which are chosen uniformly randomly and kept fixed throughout. At \(h=1\), the agent is in \(s_{1,a}\) and action \(a_{1}\) leads to \(s_{2,a}\), \(a^{\prime}_{h}\) leads to \(s_{2,b}\), and all other actions lead to \(s_{2,c}\). For \(h>2\), taking action \(a_{h}\) in \(s_{h,a}\) leads to \(s_{h+1,a}\) and taking action \(a^{\prime}_{h}\) in \(s_{h,b}\) leads to \(s_{h+1,b}\). In all other cases involving taking an action in a state \(s_{h,g}\), we transition to the next bad state \(s_{h+1,c}\).

The exogenous state evolves as follows. We set \(\xi_{1}\in\{0,1\}^{H}\) where \(\xi_{1}(i)\sim\mathtt{Unf}\left(\{0,1\}\right)\) for each \(i\in[H]\). At time step \(h\), \(\xi_{h}\) is generated from \(\xi_{h-1}\) by uniformly flipping each bit in \(\xi_{h-1}\) independently with probability 0.1.

An observation \(x\) is generated stochastically from a latent state \(z=(s,\xi)\). We map \(s\) to a vector \(w\) encoding the identity of the state. We concatenate \((w,\xi)\), add Gaussian noise to each dimension, and multiply the result by a Hadamard matrix to generate \(x\). See Appendix F for full details. Our construction is inspired by prior work (Du et al., 2019; Misra et al., 2020).

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory. On inspection, we found that ID suffers from the under-abstraction issue highlighted earlier as it has difficulty separating observations from \(s_{3a}\) and \(s_{3b}\). On adding exogenous noise, the accuracy of Homer plummets significantly.

We test the ability of PPE to recover the latent endogenous transition dynamics in visual grid-world problem.2 The agent navigates in a \(N\times N\) grid world where each grid can contain a stationary object, the goal, or the agent. The agent's endogenous state is given by its position in the grid and its direction amongst four possible canonical directions. The agent can take five different actions for navigation. The world is visible to the agent as a \(8N\times 8N\) sized RGB image. We add exogenous noise as follows: at the beginning of each episode, we independently sample position, size and color of 5 ellipses. The position and size of these ellipses is perturbed after each time step independent of the action.

The exogenous state is given by the position, size and color of ellipses and is much larger than \(|\mathcal{S}|\leq 4N^{2}\).

In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems. We show that many existing RL algorithms fail in the presence of exogenous noise. We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores.

To our knowledge, this is the first such algorithm with guarantees for settings with exogenous noise.

Discovering and removing exogenous state variables and rewards for reinforcement learning.

Input: Figure 1: Left: An agent is walking next to a pond in a park and observes the world as an image. The world consists of a latent endogenous state, containing variable such as agent’s position, and a much larger latent exogenous state containing variables such as motion of ducks, ripples in the water, etc. Center: Graphical model of the EX-BMDP. Right: PPE learns a generalized form of inverse dynamics that recovers the endogenous state.

Input: Figure 2: Results on combination lock. Left: We show the latent transition dynamics of combination lock. Observations are not shown for brevity. Center: Shows minimal number of episodes needed to achieve a mean regret of at most V (π?)/2. Right: State decoding accuracy (in percent) of decoders learned by different methods. Solid lines implies no exogenous dimension while dashed lines imply an exogenous dimension of 100.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, I find that the claim about initiating a formal study of latent state discovery with exogenous noise through the EX-BMDP model is well-stated. The evidence strongly supports this through:

1. Detailed mathematical formalization of the EX-BMDP model
2. Clear differentiation from existing approaches
3. Thorough theoretical analysis including limitations (Bellman rank scaling)
4. Empirical validation through experiments
5. Reviewer comments acknowledge the excellent definition and presentation of EX-BMDP

The claim avoids overstatement by:
- Not claiming to solve all cases (specifies "near-deterministic dynamics")
- Acknowledging computational challenges
- Presenting limitations transparently

<score>1</score>