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:

- 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 \(V(\pi):=\mathbb{E}_{\pi}[\sum_{h=1}^{H}R(x_{h},a_{h})]\).

**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 "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" appears well-stated. The evidence thoroughly supports this through:

1. Formal mathematical definition and analysis of EX-BMDP model
2. Demonstration of novel theoretical contributions (sample complexity bounds independent of exogenous state space)
3. Experimental validation on combination lock and visual gridworld problems
4. Clear differentiation from prior work through handling of exogenous noise
5. The reviewer's comments also support the novelty and significance

The claim accurately represents the contribution without exaggeration of its scope or impact.

<score>1</score>

Justification:
- The evidence provides comprehensive theoretical foundations through Definition 1 and subsequent analysis
- Experimental results demonstrate practical applications
- The reviewer affirms the novelty, noting "the fact that it is possible to obtain a sample complexity bound completely independent of the complexity of Ξ was surprising"
- The claim stays focused on introducing the model without overstating capabilities
- Limitations are acknowledged (e.g., "near-deterministic dynamics" requirement)
- Clear comparisons to prior work establish the novelty without diminishing previous contributions


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:

The algorithm, called Path Prediction and Elimination (PPE), learns a form of _multi-step inverse dynamics_ by predicting the identity of the path that generates an observation.

The evidence to be evaluated is:Another common approach in empirical works is based on modeling the inverse dynamics of the system, such as the ICM module of Pathak et al. (2017). In such approaches, we learn a representation by using consecutive observations to predict the action that was taken between them.

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).

1:Set \(\Psi_{1}=\{\bot\}\), stochasticity level \(\eta\leq\frac{1}{4SH}\)//\(\bot\) denotes an empty path 2:for\(h=2,\ldots,H\)do 3: Set \(N=16\left(\left|\Psi_{h-1}\circ A\right|\right)^{2}\log\left(\frac{\left| \mathcal{F}\right|\left|\Psi_{h-1}\right|\left|AH}{\delta}\right)\) 4: Collect a dataset \(\mathcal{D}\) of \(N\)_i.i.d._ tuples \((x,v)\) where \(v\sim\mathsf{Unf}(\Psi_{h-1}\circ\mathcal{A})\) and \(x\sim\mathbb{P}(x_{h}\mid v)\). 5: Solve multi-class classification problem: \(\hat{f}_{h}=\arg\max_{f\in\mathcal{F}}\sum_{(x,v)\in\mathcal{D}}\ln f(\mathrm{ idx}(v)\mid x)\).

7: Calculate the path prediction gap: \(\widehat{\Delta}(i,j)=\frac{1}{N}\sum_{(x,v)\in\mathcal{D}}\left|\hat{f}_{h} (i|x)-\hat{f}_{h}(j|x)\right|\). 8: If \(\widehat{\Delta}(i,j)\leq\frac{5/8}{\left|\Psi_{h-1}\circ\mathcal{A}\right|}\), then eliminate path \(v\) with \(\mathrm{idx}(v)\!=\!j\). //\(v_{i}\) and \(v_{j}\) visit same state 9:\(\Psi_{h}\) is defined as the set of all paths in \(\Psi_{h-1}\circ\mathcal{A}\) that have not been eliminated in line 8.

```**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 train a classifier \(\hat{f}_{h}\) using \(\mathcal{D}\) by predicting the index \(\mathrm{idx}(v)\) of the path \(v\) from the observation \(x\) (line 5).

Further, we can view this prediction problem as learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action.

Recovering an endogenous state decoder. We can recover a endogenous state decoder \(\hat{\phi}_{h}\) for each time step \(h\in\{2,3,\cdots,H\}\) directly from \(\hat{f}_{h}\) as shown below: \[\hat{\phi}_{h}(x)=\min\left\{i\mid\hat{f}_{h}(i\mid x)\geq\max_{j}\hat{f}_{h}(j \mid x)-\mathcal{O}(\nicefrac{{1}}{{\Gamma_{h}}}),i\in[\left|\Upsilon_{h} \right|]\right\}.\] Intuitively, this assigns the observation to the path with smallest index that has the highest chance of visiting \(x\), and therefore, \(\phi^{\star}(x)\). We are implicitly using the decoder for exploring, since we rely on using \(\hat{f}_{h}\) for making planning decisions.

PPE can also be used to recover a latent endogenous transition dynamics.

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 \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

_We make a realizability assumption for the regression problem solved by PPE (line 5). We assume that \(\mathcal{F}\) is expressive enough to represent the Bayes optimal classifier of the regression problems created by PPE.

In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

Why does PPE work? We provide an asymptotic analysis to explain why PPE works. Consider a deterministic setting and the \(h^{th}\) iteration of PPE. Assume by induction that \(\Psi_{h-1}\) is an exact policy cover for time step \(h-1\).

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: \[f^{*}_{h}(\texttt{idx}(v)\mid x):=\mathbb{P}_{h}(v\mid x)=\frac{ \mathbb{P}_{h}(x\mid v)\mathbb{P}(v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{ P}_{h}(x\mid v^{\prime})\mathbb{P}(v^{\prime})}\stackrel{{(a)}}{{=}} \frac{\mathbb{P}_{h}(x\mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}( x\mid v^{\prime})}\stackrel{{(b)}}{{=}}\frac{\mathbb{P}_{h}(\phi^{*}(x)) \mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}(\phi^{*}(x))\mid v^{ \prime})},\] 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, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\] Let \(v_{1},v_{2}\in\Upsilon_{h}\) be two paths with indices \(i\) and \(j\) respectively. We define their exact path prediction gap as \(\Delta(i,j):=\mathbb{E}_{x_{h}}\left[\left|f^{*}_{h}(i\mid x_{h})-f^{*}_{h}(j \mid x_{h})\right|\right]\). Assume that \(v_{1}\) visits an endogenous state \(s\) at time step \(h\) and denote \(\omega(s)\) as the number of paths in \(\Upsilon_{h}\) that reaches \(s\). Then \(f^{*}_{h}(i\mid x_{h})=\nicefrac{{1}}{{\omega(s)}}\) if \(\phi^{*}(x_{h})=s\), and \(0\) otherwise. If \(v_{2}\) also visits \(s\) at time step \(h\), then \(f^{*}_{h}(i\mid x_{h})=f^{*}_{h}(j\mid x_{h})\) all \(x_{h}\). This implies \(\Delta(i,j)=0\) and \(\mathtt{PPE}\) will filter out the path with higher index since it detected both paths reach to the same endogenous state. Conversely, let \(v_{2}\) visit a different state at time step \(h\). If \(x\) is an observation that maps to \(s\), then \(f^{*}_{h}(i\mid x)=\nicefrac{{1}}{{\omega(s)}}\) and \(f^{*}_{h}(j\mid x)=0\). This gives \(\left|f^{*}_{h}(i\mid x)-f^{*}_{h}(j\mid x)\right|=\nicefrac{{1}}{{\omega(s)} }\geq\nicefrac{{1}}{{\left|\Upsilon_{h}\right|}}\) and, consequently, \(\Delta(i,j)>0\). In fact, we can show \(\Delta(i,j)\geq\mathcal{O}(\nicefrac{{1}}{{1}}\nicefrac{{1}}{{\left|\Upsilon_ {h}\right|}})\). Thus, \(\mathtt{PPE}\) will not eliminate these paths upon comparison.

We evaluate \(\mathtt{PPE}\) on two domains: a challenging exploration problem called _combination lock_ to test whether \(\mathtt{PPE}\) can learn an optimal policy and an accurate state decoder, and a visual-grid world with complex visual representations to test whether \(\mathtt{PPE}\) is able to recover the latent dynamics.

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).

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.

Since the problem has deterministic dynamics, we evaluate the accuracy of the learned transition model by measuring it in terms of accuracy of the elimination step (Algorithm 1, line 8), since this step induces our algorithm's mapping from observations to endogenous latent states.

Type 1 error computes whether PPE merged these paths, i.e., predicted them as mapping to the same abstract state, when they go to _different endogenous_ states.

Further, the agent is able to plan using the learned transition model and receive the optimal return.

As expected, PPE rapidly learns a highly accurate decoder while Homer performs only as well as a random uniform decoder.

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 appears well-stated. The evidence thoroughly demonstrates that PPE does learn multi-step inverse dynamics by predicting path identities from observations. This is supported by:

1. The algorithmic details showing path prediction and classification (Algorithm 1)
2. The mathematical formulation explaining how PPE learns to predict paths that generated observations
3. Explicit statements about learning "a generalized form of inverse dynamics"
4. Experimental validation showing PPE successfully learns state decoders and transition dynamics
5. Visual evidence in figures demonstrating the approach

The claim accurately reflects the algorithm's capabilities without exaggeration or overstatement of its abilities.

<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:

The algorithm, called Path Prediction and Elimination (PPE), learns a form of _multi-step inverse dynamics_ by predicting the identity of the path that generates an observation.

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:Another common approach in empirical works is based on modeling the inverse dynamics of the system, such as the ICM module of Pathak et al. (2017). In such approaches, we learn a representation by using consecutive observations to predict the action that was taken between them.

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).

1:Set \(\Psi_{1}=\{\bot\}\), stochasticity level \(\eta\leq\frac{1}{4SH}\)//\(\bot\) denotes an empty path 2:for\(h=2,\ldots,H\)do 3: Set \(N=16\left(\left|\Psi_{h-1}\circ A\right|\right)^{2}\log\left(\frac{\left| \mathcal{F}\right|\left|\Psi_{h-1}\right|\left|AH}{\delta}\right)\) 4: Collect a dataset \(\mathcal{D}\) of \(N\)_i.i.d._ tuples \((x,v)\) where \(v\sim\mathsf{Unf}(\Psi_{h-1}\circ\mathcal{A})\) and \(x\sim\mathbb{P}(x_{h}\mid v)\). 5: Solve multi-class classification problem: \(\hat{f}_{h}=\arg\max_{f\in\mathcal{F}}\sum_{(x,v)\in\mathcal{D}}\ln f(\mathrm{ idx}(v)\mid x)\).

7: Calculate the path prediction gap: \(\widehat{\Delta}(i,j)=\frac{1}{N}\sum_{(x,v)\in\mathcal{D}}\left|\hat{f}_{h} (i|x)-\hat{f}_{h}(j|x)\right|\). 8: If \(\widehat{\Delta}(i,j)\leq\frac{5/8}{\left|\Psi_{h-1}\circ\mathcal{A}\right|}\), then eliminate path \(v\) with \(\mathrm{idx}(v)\!=\!j\). //\(v_{i}\) and \(v_{j}\) visit same state 9:\(\Psi_{h}\) is defined as the set of all paths in \(\Psi_{h-1}\circ\mathcal{A}\) that have not been eliminated in line 8.

```**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 train a classifier \(\hat{f}_{h}\) using \(\mathcal{D}\) by predicting the index \(\mathrm{idx}(v)\) of the path \(v\) from the observation \(x\) (line 5).

Further, we can view this prediction problem as learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action.

Recovering an endogenous state decoder. We can recover a endogenous state decoder \(\hat{\phi}_{h}\) for each time step \(h\in\{2,3,\cdots,H\}\) directly from \(\hat{f}_{h}\) as shown below: \[\hat{\phi}_{h}(x)=\min\left\{i\mid\hat{f}_{h}(i\mid x)\geq\max_{j}\hat{f}_{h}(j \mid x)-\mathcal{O}(\nicefrac{{1}}{{\Gamma_{h}}}),i\in[\left|\Upsilon_{h} \right|]\right\}.\] Intuitively, this assigns the observation to the path with smallest index that has the highest chance of visiting \(x\), and therefore, \(\phi^{\star}(x)\). We are implicitly using the decoder for exploring, since we rely on using \(\hat{f}_{h}\) for making planning decisions.

PPE can also be used to recover a latent endogenous transition dynamics.

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 \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

_We make a realizability assumption for the regression problem solved by PPE (line 5). We assume that \(\mathcal{F}\) is expressive enough to represent the Bayes optimal classifier of the regression problems created by PPE.

In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

Why does PPE work? We provide an asymptotic analysis to explain why PPE works. Consider a deterministic setting and the \(h^{th}\) iteration of PPE. Assume by induction that \(\Psi_{h-1}\) is an exact policy cover for time step \(h-1\).

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: \[f^{*}_{h}(\texttt{idx}(v)\mid x):=\mathbb{P}_{h}(v\mid x)=\frac{ \mathbb{P}_{h}(x\mid v)\mathbb{P}(v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{ P}_{h}(x\mid v^{\prime})\mathbb{P}(v^{\prime})}\stackrel{{(a)}}{{=}} \frac{\mathbb{P}_{h}(x\mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}( x\mid v^{\prime})}\stackrel{{(b)}}{{=}}\frac{\mathbb{P}_{h}(\phi^{*}(x)) \mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}(\phi^{*}(x))\mid v^{ \prime})},\] 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, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\] Let \(v_{1},v_{2}\in\Upsilon_{h}\) be two paths with indices \(i\) and \(j\) respectively. We define their exact path prediction gap as \(\Delta(i,j):=\mathbb{E}_{x_{h}}\left[\left|f^{*}_{h}(i\mid x_{h})-f^{*}_{h}(j \mid x_{h})\right|\right]\). Assume that \(v_{1}\) visits an endogenous state \(s\) at time step \(h\) and denote \(\omega(s)\) as the number of paths in \(\Upsilon_{h}\) that reaches \(s\). Then \(f^{*}_{h}(i\mid x_{h})=\nicefrac{{1}}{{\omega(s)}}\) if \(\phi^{*}(x_{h})=s\), and \(0\) otherwise. If \(v_{2}\) also visits \(s\) at time step \(h\), then \(f^{*}_{h}(i\mid x_{h})=f^{*}_{h}(j\mid x_{h})\) all \(x_{h}\). This implies \(\Delta(i,j)=0\) and \(\mathtt{PPE}\) will filter out the path with higher index since it detected both paths reach to the same endogenous state. Conversely, let \(v_{2}\) visit a different state at time step \(h\). If \(x\) is an observation that maps to \(s\), then \(f^{*}_{h}(i\mid x)=\nicefrac{{1}}{{\omega(s)}}\) and \(f^{*}_{h}(j\mid x)=0\). This gives \(\left|f^{*}_{h}(i\mid x)-f^{*}_{h}(j\mid x)\right|=\nicefrac{{1}}{{\omega(s)} }\geq\nicefrac{{1}}{{\left|\Upsilon_{h}\right|}}\) and, consequently, \(\Delta(i,j)>0\). In fact, we can show \(\Delta(i,j)\geq\mathcal{O}(\nicefrac{{1}}{{1}}\nicefrac{{1}}{{\left|\Upsilon_ {h}\right|}})\). Thus, \(\mathtt{PPE}\) will not eliminate these paths upon comparison.

We evaluate \(\mathtt{PPE}\) on two domains: a challenging exploration problem called _combination lock_ to test whether \(\mathtt{PPE}\) can learn an optimal policy and an accurate state decoder, and a visual-grid world with complex visual representations to test whether \(\mathtt{PPE}\) is able to recover the latent dynamics.

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).

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.

Since the problem has deterministic dynamics, we evaluate the accuracy of the learned transition model by measuring it in terms of accuracy of the elimination step (Algorithm 1, line 8), since this step induces our algorithm's mapping from observations to endogenous latent states.

Type 1 error computes whether PPE merged these paths, i.e., predicted them as mapping to the same abstract state, when they go to _different endogenous_ states.

Further, the agent is able to plan using the learned transition model and receive the optimal return.

As expected, PPE rapidly learns a highly accurate decoder while Homer performs only as well as a random uniform decoder.

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, I find that the claim "The algorithm learns a form of multi-step inverse dynamics by predicting the identity of the path that generates an observation" is well-stated and accurately reflects the evidence without exaggeration.

The evidence clearly shows that PPE learns to predict which path generated an observation through its classifier f_h, which effectively learns the relationship between sequences of actions (paths) and resulting observations. This is demonstrated both theoretically in the mathematical formulation and empirically in the experimental results. The paper explicitly states this relationship to inverse dynamics and provides detailed explanation of how the algorithm accomplishes this through path prediction and elimination.

The review comments do not dispute this core claim about the algorithm's function, focusing instead on broader applicability concerns.

<score>1</score>

Justification:
1. Direct evidence from text: "we can view this prediction problem as learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action"

2. The algorithmic details in Algorithm 1 show exactly how this prediction mechanism works through the classifier f_h

3. The experimental results demonstrate the algorithm successfully learning these relationships in practice

4. No overstatement is present - the claim precisely describes what the algorithm does without exaggerating its capabilities or scope


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:

The algorithm, called Path Prediction and Elimination (PPE), learns a form of _multi-step inverse dynamics_ by predicting the identity of the path that generates an observation.

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:Another common approach in empirical works is based on modeling the inverse dynamics of the system, such as the ICM module of Pathak et al. (2017). In such approaches, we learn a representation by using consecutive observations to predict the action that was taken between them.

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).

1:Set \(\Psi_{1}=\{\bot\}\), stochasticity level \(\eta\leq\frac{1}{4SH}\)//\(\bot\) denotes an empty path 2:for\(h=2,\ldots,H\)do 3: Set \(N=16\left(\left|\Psi_{h-1}\circ A\right|\right)^{2}\log\left(\frac{\left| \mathcal{F}\right|\left|\Psi_{h-1}\right|\left|AH}{\delta}\right)\) 4: Collect a dataset \(\mathcal{D}\) of \(N\)_i.i.d._ tuples \((x,v)\) where \(v\sim\mathsf{Unf}(\Psi_{h-1}\circ\mathcal{A})\) and \(x\sim\mathbb{P}(x_{h}\mid v)\). 5: Solve multi-class classification problem: \(\hat{f}_{h}=\arg\max_{f\in\mathcal{F}}\sum_{(x,v)\in\mathcal{D}}\ln f(\mathrm{ idx}(v)\mid x)\).

7: Calculate the path prediction gap: \(\widehat{\Delta}(i,j)=\frac{1}{N}\sum_{(x,v)\in\mathcal{D}}\left|\hat{f}_{h} (i|x)-\hat{f}_{h}(j|x)\right|\). 8: If \(\widehat{\Delta}(i,j)\leq\frac{5/8}{\left|\Psi_{h-1}\circ\mathcal{A}\right|}\), then eliminate path \(v\) with \(\mathrm{idx}(v)\!=\!j\). //\(v_{i}\) and \(v_{j}\) visit same state 9:\(\Psi_{h}\) is defined as the set of all paths in \(\Psi_{h-1}\circ\mathcal{A}\) that have not been eliminated in line 8.

```**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 train a classifier \(\hat{f}_{h}\) using \(\mathcal{D}\) by predicting the index \(\mathrm{idx}(v)\) of the path \(v\) from the observation \(x\) (line 5).

Further, we can view this prediction problem as learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action.

Recovering an endogenous state decoder. We can recover a endogenous state decoder \(\hat{\phi}_{h}\) for each time step \(h\in\{2,3,\cdots,H\}\) directly from \(\hat{f}_{h}\) as shown below: \[\hat{\phi}_{h}(x)=\min\left\{i\mid\hat{f}_{h}(i\mid x)\geq\max_{j}\hat{f}_{h}(j \mid x)-\mathcal{O}(\nicefrac{{1}}{{\Gamma_{h}}}),i\in[\left|\Upsilon_{h} \right|]\right\}.\] Intuitively, this assigns the observation to the path with smallest index that has the highest chance of visiting \(x\), and therefore, \(\phi^{\star}(x)\). We are implicitly using the decoder for exploring, since we rely on using \(\hat{f}_{h}\) for making planning decisions.

PPE can also be used to recover a latent endogenous transition dynamics.

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 \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

_We make a realizability assumption for the regression problem solved by PPE (line 5). We assume that \(\mathcal{F}\) is expressive enough to represent the Bayes optimal classifier of the regression problems created by PPE.

In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

Why does PPE work? We provide an asymptotic analysis to explain why PPE works. Consider a deterministic setting and the \(h^{th}\) iteration of PPE. Assume by induction that \(\Psi_{h-1}\) is an exact policy cover for time step \(h-1\).

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: \[f^{*}_{h}(\texttt{idx}(v)\mid x):=\mathbb{P}_{h}(v\mid x)=\frac{ \mathbb{P}_{h}(x\mid v)\mathbb{P}(v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{ P}_{h}(x\mid v^{\prime})\mathbb{P}(v^{\prime})}\stackrel{{(a)}}{{=}} \frac{\mathbb{P}_{h}(x\mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}( x\mid v^{\prime})}\stackrel{{(b)}}{{=}}\frac{\mathbb{P}_{h}(\phi^{*}(x)) \mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}(\phi^{*}(x))\mid v^{ \prime})},\] 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, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\] Let \(v_{1},v_{2}\in\Upsilon_{h}\) be two paths with indices \(i\) and \(j\) respectively. We define their exact path prediction gap as \(\Delta(i,j):=\mathbb{E}_{x_{h}}\left[\left|f^{*}_{h}(i\mid x_{h})-f^{*}_{h}(j \mid x_{h})\right|\right]\). Assume that \(v_{1}\) visits an endogenous state \(s\) at time step \(h\) and denote \(\omega(s)\) as the number of paths in \(\Upsilon_{h}\) that reaches \(s\). Then \(f^{*}_{h}(i\mid x_{h})=\nicefrac{{1}}{{\omega(s)}}\) if \(\phi^{*}(x_{h})=s\), and \(0\) otherwise. If \(v_{2}\) also visits \(s\) at time step \(h\), then \(f^{*}_{h}(i\mid x_{h})=f^{*}_{h}(j\mid x_{h})\) all \(x_{h}\). This implies \(\Delta(i,j)=0\) and \(\mathtt{PPE}\) will filter out the path with higher index since it detected both paths reach to the same endogenous state. Conversely, let \(v_{2}\) visit a different state at time step \(h\). If \(x\) is an observation that maps to \(s\), then \(f^{*}_{h}(i\mid x)=\nicefrac{{1}}{{\omega(s)}}\) and \(f^{*}_{h}(j\mid x)=0\). This gives \(\left|f^{*}_{h}(i\mid x)-f^{*}_{h}(j\mid x)\right|=\nicefrac{{1}}{{\omega(s)} }\geq\nicefrac{{1}}{{\left|\Upsilon_{h}\right|}}\) and, consequently, \(\Delta(i,j)>0\). In fact, we can show \(\Delta(i,j)\geq\mathcal{O}(\nicefrac{{1}}{{1}}\nicefrac{{1}}{{\left|\Upsilon_ {h}\right|}})\). Thus, \(\mathtt{PPE}\) will not eliminate these paths upon comparison.

We evaluate \(\mathtt{PPE}\) on two domains: a challenging exploration problem called _combination lock_ to test whether \(\mathtt{PPE}\) can learn an optimal policy and an accurate state decoder, and a visual-grid world with complex visual representations to test whether \(\mathtt{PPE}\) is able to recover the latent dynamics.

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).

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.

Since the problem has deterministic dynamics, we evaluate the accuracy of the learned transition model by measuring it in terms of accuracy of the elimination step (Algorithm 1, line 8), since this step induces our algorithm's mapping from observations to endogenous latent states.

Type 1 error computes whether PPE merged these paths, i.e., predicted them as mapping to the same abstract state, when they go to _different endogenous_ states.

Further, the agent is able to plan using the learned transition model and receive the optimal return.

As expected, PPE rapidly learns a highly accurate decoder while Homer performs only as well as a random uniform decoder.

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:
Based on my review of the evidence and reviewer comments, I'll evaluate the claim: "The algorithm, called Path Prediction and Elimination (PPE), learns a form of multi-step inverse dynamics by predicting the identity of the path that generates an observation."

The evidence strongly supports this claim through multiple elements:
1. The algorithm explicitly uses path prediction to learn relationships between actions and observations across multiple timesteps
2. The mathematical formulation shows how PPE predicts path indices from observations, effectively learning inverse dynamics
3. The evidence directly states "we can view this prediction problem as learning a multistep inverse dynamics model"
4. The experimental results demonstrate the practical application of this approach

The claim is precise and accurately reflects what the algorithm does without overstating its capabilities. The reviewer comments also acknowledge the theoretical basis for using inverse dynamics.

<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:

The algorithm, called Path Prediction and Elimination (PPE), learns a form of _multi-step inverse dynamics_ by predicting the identity of the path that generates an observation.

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:Another common approach in empirical works is based on modeling the inverse dynamics of the system, such as the ICM module of Pathak et al. (2017). In such approaches, we learn a representation by using consecutive observations to predict the action that was taken between them.

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).

1:Set \(\Psi_{1}=\{\bot\}\), stochasticity level \(\eta\leq\frac{1}{4SH}\)//\(\bot\) denotes an empty path 2:for\(h=2,\ldots,H\)do 3: Set \(N=16\left(\left|\Psi_{h-1}\circ A\right|\right)^{2}\log\left(\frac{\left| \mathcal{F}\right|\left|\Psi_{h-1}\right|\left|AH}{\delta}\right)\) 4: Collect a dataset \(\mathcal{D}\) of \(N\)_i.i.d._ tuples \((x,v)\) where \(v\sim\mathsf{Unf}(\Psi_{h-1}\circ\mathcal{A})\) and \(x\sim\mathbb{P}(x_{h}\mid v)\). 5: Solve multi-class classification problem: \(\hat{f}_{h}=\arg\max_{f\in\mathcal{F}}\sum_{(x,v)\in\mathcal{D}}\ln f(\mathrm{ idx}(v)\mid x)\).

7: Calculate the path prediction gap: \(\widehat{\Delta}(i,j)=\frac{1}{N}\sum_{(x,v)\in\mathcal{D}}\left|\hat{f}_{h} (i|x)-\hat{f}_{h}(j|x)\right|\). 8: If \(\widehat{\Delta}(i,j)\leq\frac{5/8}{\left|\Psi_{h-1}\circ\mathcal{A}\right|}\), then eliminate path \(v\) with \(\mathrm{idx}(v)\!=\!j\). //\(v_{i}\) and \(v_{j}\) visit same state 9:\(\Psi_{h}\) is defined as the set of all paths in \(\Psi_{h-1}\circ\mathcal{A}\) that have not been eliminated in line 8.

```**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 train a classifier \(\hat{f}_{h}\) using \(\mathcal{D}\) by predicting the index \(\mathrm{idx}(v)\) of the path \(v\) from the observation \(x\) (line 5).

Further, we can view this prediction problem as learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action.

Recovering an endogenous state decoder. We can recover a endogenous state decoder \(\hat{\phi}_{h}\) for each time step \(h\in\{2,3,\cdots,H\}\) directly from \(\hat{f}_{h}\) as shown below: \[\hat{\phi}_{h}(x)=\min\left\{i\mid\hat{f}_{h}(i\mid x)\geq\max_{j}\hat{f}_{h}(j \mid x)-\mathcal{O}(\nicefrac{{1}}{{\Gamma_{h}}}),i\in[\left|\Upsilon_{h} \right|]\right\}.\] Intuitively, this assigns the observation to the path with smallest index that has the highest chance of visiting \(x\), and therefore, \(\phi^{\star}(x)\). We are implicitly using the decoder for exploring, since we rely on using \(\hat{f}_{h}\) for making planning decisions.

PPE can also be used to recover a latent endogenous transition dynamics.

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 \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

_We make a realizability assumption for the regression problem solved by PPE (line 5). We assume that \(\mathcal{F}\) is expressive enough to represent the Bayes optimal classifier of the regression problems created by PPE.

In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

Why does PPE work? We provide an asymptotic analysis to explain why PPE works. Consider a deterministic setting and the \(h^{th}\) iteration of PPE. Assume by induction that \(\Psi_{h-1}\) is an exact policy cover for time step \(h-1\).

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: \[f^{*}_{h}(\texttt{idx}(v)\mid x):=\mathbb{P}_{h}(v\mid x)=\frac{ \mathbb{P}_{h}(x\mid v)\mathbb{P}(v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{ P}_{h}(x\mid v^{\prime})\mathbb{P}(v^{\prime})}\stackrel{{(a)}}{{=}} \frac{\mathbb{P}_{h}(x\mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}( x\mid v^{\prime})}\stackrel{{(b)}}{{=}}\frac{\mathbb{P}_{h}(\phi^{*}(x)) \mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}(\phi^{*}(x))\mid v^{ \prime})},\] 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, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\] Let \(v_{1},v_{2}\in\Upsilon_{h}\) be two paths with indices \(i\) and \(j\) respectively. We define their exact path prediction gap as \(\Delta(i,j):=\mathbb{E}_{x_{h}}\left[\left|f^{*}_{h}(i\mid x_{h})-f^{*}_{h}(j \mid x_{h})\right|\right]\). Assume that \(v_{1}\) visits an endogenous state \(s\) at time step \(h\) and denote \(\omega(s)\) as the number of paths in \(\Upsilon_{h}\) that reaches \(s\). Then \(f^{*}_{h}(i\mid x_{h})=\nicefrac{{1}}{{\omega(s)}}\) if \(\phi^{*}(x_{h})=s\), and \(0\) otherwise. If \(v_{2}\) also visits \(s\) at time step \(h\), then \(f^{*}_{h}(i\mid x_{h})=f^{*}_{h}(j\mid x_{h})\) all \(x_{h}\). This implies \(\Delta(i,j)=0\) and \(\mathtt{PPE}\) will filter out the path with higher index since it detected both paths reach to the same endogenous state. Conversely, let \(v_{2}\) visit a different state at time step \(h\). If \(x\) is an observation that maps to \(s\), then \(f^{*}_{h}(i\mid x)=\nicefrac{{1}}{{\omega(s)}}\) and \(f^{*}_{h}(j\mid x)=0\). This gives \(\left|f^{*}_{h}(i\mid x)-f^{*}_{h}(j\mid x)\right|=\nicefrac{{1}}{{\omega(s)} }\geq\nicefrac{{1}}{{\left|\Upsilon_{h}\right|}}\) and, consequently, \(\Delta(i,j)>0\). In fact, we can show \(\Delta(i,j)\geq\mathcal{O}(\nicefrac{{1}}{{1}}\nicefrac{{1}}{{\left|\Upsilon_ {h}\right|}})\). Thus, \(\mathtt{PPE}\) will not eliminate these paths upon comparison.

We evaluate \(\mathtt{PPE}\) on two domains: a challenging exploration problem called _combination lock_ to test whether \(\mathtt{PPE}\) can learn an optimal policy and an accurate state decoder, and a visual-grid world with complex visual representations to test whether \(\mathtt{PPE}\) is able to recover the latent dynamics.

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).

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.

Since the problem has deterministic dynamics, we evaluate the accuracy of the learned transition model by measuring it in terms of accuracy of the elimination step (Algorithm 1, line 8), since this step induces our algorithm's mapping from observations to endogenous latent states.

Type 1 error computes whether PPE merged these paths, i.e., predicted them as mapping to the same abstract state, when they go to _different endogenous_ states.

Further, the agent is able to plan using the learned transition model and receive the optimal return.

As expected, PPE rapidly learns a highly accurate decoder while Homer performs only as well as a random uniform decoder.

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 that "PPE learns a form of multi-step inverse dynamics by predicting the identity of the path that generates an observation" is well-stated and supported by multiple pieces of evidence:

1. The algorithm explicitly learns to predict path indices from observations (Algorithm 1, line 5)
2. This is described as "learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action"
3. The theoretical analysis shows how PPE uses this prediction to recover endogenous state information
4. Experimental results demonstrate PPE successfully learning accurate state decoders and transition dynamics

The review comments do not challenge this specific claim. The evidence directly supports the technical description without exaggeration.

<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:

The algorithm, called Path Prediction and Elimination (PPE), learns a form of _multi-step inverse dynamics_ by predicting the identity of the path that generates an observation.

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:Another common approach in empirical works is based on modeling the inverse dynamics of the system, such as the ICM module of Pathak et al. (2017). In such approaches, we learn a representation by using consecutive observations to predict the action that was taken between them.

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).

1:Set \(\Psi_{1}=\{\bot\}\), stochasticity level \(\eta\leq\frac{1}{4SH}\)//\(\bot\) denotes an empty path 2:for\(h=2,\ldots,H\)do 3: Set \(N=16\left(\left|\Psi_{h-1}\circ A\right|\right)^{2}\log\left(\frac{\left| \mathcal{F}\right|\left|\Psi_{h-1}\right|\left|AH}{\delta}\right)\) 4: Collect a dataset \(\mathcal{D}\) of \(N\)_i.i.d._ tuples \((x,v)\) where \(v\sim\mathsf{Unf}(\Psi_{h-1}\circ\mathcal{A})\) and \(x\sim\mathbb{P}(x_{h}\mid v)\). 5: Solve multi-class classification problem: \(\hat{f}_{h}=\arg\max_{f\in\mathcal{F}}\sum_{(x,v)\in\mathcal{D}}\ln f(\mathrm{ idx}(v)\mid x)\).

7: Calculate the path prediction gap: \(\widehat{\Delta}(i,j)=\frac{1}{N}\sum_{(x,v)\in\mathcal{D}}\left|\hat{f}_{h} (i|x)-\hat{f}_{h}(j|x)\right|\). 8: If \(\widehat{\Delta}(i,j)\leq\frac{5/8}{\left|\Psi_{h-1}\circ\mathcal{A}\right|}\), then eliminate path \(v\) with \(\mathrm{idx}(v)\!=\!j\). //\(v_{i}\) and \(v_{j}\) visit same state 9:\(\Psi_{h}\) is defined as the set of all paths in \(\Psi_{h-1}\circ\mathcal{A}\) that have not been eliminated in line 8.

```**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 train a classifier \(\hat{f}_{h}\) using \(\mathcal{D}\) by predicting the index \(\mathrm{idx}(v)\) of the path \(v\) from the observation \(x\) (line 5).

Further, we can view this prediction problem as learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action.

Recovering an endogenous state decoder. We can recover a endogenous state decoder \(\hat{\phi}_{h}\) for each time step \(h\in\{2,3,\cdots,H\}\) directly from \(\hat{f}_{h}\) as shown below: \[\hat{\phi}_{h}(x)=\min\left\{i\mid\hat{f}_{h}(i\mid x)\geq\max_{j}\hat{f}_{h}(j \mid x)-\mathcal{O}(\nicefrac{{1}}{{\Gamma_{h}}}),i\in[\left|\Upsilon_{h} \right|]\right\}.\] Intuitively, this assigns the observation to the path with smallest index that has the highest chance of visiting \(x\), and therefore, \(\phi^{\star}(x)\). We are implicitly using the decoder for exploring, since we rely on using \(\hat{f}_{h}\) for making planning decisions.

PPE can also be used to recover a latent endogenous transition dynamics.

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 \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

_We make a realizability assumption for the regression problem solved by PPE (line 5). We assume that \(\mathcal{F}\) is expressive enough to represent the Bayes optimal classifier of the regression problems created by PPE.

In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

Why does PPE work? We provide an asymptotic analysis to explain why PPE works. Consider a deterministic setting and the \(h^{th}\) iteration of PPE. Assume by induction that \(\Psi_{h-1}\) is an exact policy cover for time step \(h-1\).

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: \[f^{*}_{h}(\texttt{idx}(v)\mid x):=\mathbb{P}_{h}(v\mid x)=\frac{ \mathbb{P}_{h}(x\mid v)\mathbb{P}(v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{ P}_{h}(x\mid v^{\prime})\mathbb{P}(v^{\prime})}\stackrel{{(a)}}{{=}} \frac{\mathbb{P}_{h}(x\mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}( x\mid v^{\prime})}\stackrel{{(b)}}{{=}}\frac{\mathbb{P}_{h}(\phi^{*}(x)) \mid v)}{\sum_{v^{\prime}\in\Upsilon_{h}}\mathbb{P}_{h}(\phi^{*}(x))\mid v^{ \prime})},\] 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, \[\mathbb{P}_{h}(x\mid v)=q\left(x\mid\phi^{*}(x),\phi^{*}_{\xi}(x)\right) \mathbb{P}_{h}(\phi^{*}(x)\mid v)\mathbb{P}_{h}(\phi^{*}_{\xi}(x)).\] Let \(v_{1},v_{2}\in\Upsilon_{h}\) be two paths with indices \(i\) and \(j\) respectively. We define their exact path prediction gap as \(\Delta(i,j):=\mathbb{E}_{x_{h}}\left[\left|f^{*}_{h}(i\mid x_{h})-f^{*}_{h}(j \mid x_{h})\right|\right]\). Assume that \(v_{1}\) visits an endogenous state \(s\) at time step \(h\) and denote \(\omega(s)\) as the number of paths in \(\Upsilon_{h}\) that reaches \(s\). Then \(f^{*}_{h}(i\mid x_{h})=\nicefrac{{1}}{{\omega(s)}}\) if \(\phi^{*}(x_{h})=s\), and \(0\) otherwise. If \(v_{2}\) also visits \(s\) at time step \(h\), then \(f^{*}_{h}(i\mid x_{h})=f^{*}_{h}(j\mid x_{h})\) all \(x_{h}\). This implies \(\Delta(i,j)=0\) and \(\mathtt{PPE}\) will filter out the path with higher index since it detected both paths reach to the same endogenous state. Conversely, let \(v_{2}\) visit a different state at time step \(h\). If \(x\) is an observation that maps to \(s\), then \(f^{*}_{h}(i\mid x)=\nicefrac{{1}}{{\omega(s)}}\) and \(f^{*}_{h}(j\mid x)=0\). This gives \(\left|f^{*}_{h}(i\mid x)-f^{*}_{h}(j\mid x)\right|=\nicefrac{{1}}{{\omega(s)} }\geq\nicefrac{{1}}{{\left|\Upsilon_{h}\right|}}\) and, consequently, \(\Delta(i,j)>0\). In fact, we can show \(\Delta(i,j)\geq\mathcal{O}(\nicefrac{{1}}{{1}}\nicefrac{{1}}{{\left|\Upsilon_ {h}\right|}})\). Thus, \(\mathtt{PPE}\) will not eliminate these paths upon comparison.

We evaluate \(\mathtt{PPE}\) on two domains: a challenging exploration problem called _combination lock_ to test whether \(\mathtt{PPE}\) can learn an optimal policy and an accurate state decoder, and a visual-grid world with complex visual representations to test whether \(\mathtt{PPE}\) is able to recover the latent dynamics.

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).

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.

Since the problem has deterministic dynamics, we evaluate the accuracy of the learned transition model by measuring it in terms of accuracy of the elimination step (Algorithm 1, line 8), since this step induces our algorithm's mapping from observations to endogenous latent states.

Type 1 error computes whether PPE merged these paths, i.e., predicted them as mapping to the same abstract state, when they go to _different endogenous_ states.

Further, the agent is able to plan using the learned transition model and receive the optimal return.

As expected, PPE rapidly learns a highly accurate decoder while Homer performs only as well as a random uniform decoder.

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 that "PPE learns a form of multi-step inverse dynamics by predicting the identity of the path that generates an observation" appears to be well-stated. The evidence strongly supports this through:

1. The algorithmic details showing PPE explicitly learns to predict path identities from observations
2. The mathematical formulation demonstrating how PPE uses multi-step path prediction rather than just single-step inverse dynamics
3. Experimental validation in both combination lock and visual grid world domains
4. Clear theoretical analysis explaining why this approach works
5. The reviewer comments do not dispute this specific claim

The claim accurately reflects what the algorithm does without exaggeration or overstatement of capabilities.

<score>1</score>

Justification:
- The explanation of how PPE works directly shows it learns to predict path identities: "We train a classifier f̂h using D by predicting the index idx(v) of the path v from the observation x"
- The text explicitly states "Further, we can view this prediction problem as learning a multistep inverse dynamics model since the open-loop policy contains information about all previous actions and not just the last action"
- The experimental results validate that this approach works in practice
- The claim makes no overreaching statements about capabilities beyond what is demonstrated
- The reviewer comments focus on other aspects and do not challenge this specific characterization of PPE