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:

Then, we introduce the Predictive Path Elimination (PPE) algorithm, that learns a generalization of inverse dynamics and is provably sample and computationally efficient in EX-BMDPs when the endogenous state dynamics are near deterministic.

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.

However, it can also ignore controllable information. This may result in a failure to sufficiently explore the environment. 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. The over-abstraction problem was described at Misra et al. (2020), when the starting state is random. In Appendix A.3 we show inverse dynamics may over-abstract when the initial starting state is deterministic.

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 Footnote 1: Alg. 4 has time complexity of \(\mathcal{O}(S^{4}A^{4}H)\) compared to \(\mathcal{O}(S^{3}A^{3}H)\) for PPE. Furthermore, Alg. 4 requires an upper bound on \(S\), whereas PPE is adaptive to it. Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

```**Algorithm 1** PPE\((\delta,\eta)\): Predictive Path EliminationPPE performs iterations over the time steps \(h\in\{2,\cdots,H\}\). In the \(h^{th}\) iteration, it learns a policy cover \(\Psi_{h}\) for time step \(h\) containing open-loop policies. This is done by first augmenting the policy cover for previous time step by one step. Formally, we define \(\Upsilon_{h}=\Psi_{h-1}\circ\mathcal{A}=\{\pi\circ a\mid\pi\in\Psi_{h-1},a\in \mathcal{A}\}\) where \(\pi\circ a\) is an open-loop policy that follows \(\pi\) till time step \(h-1\) and then takes action \(a\). Since we assume the transition dynamics to be near-deterministic, therefore, we know that there exists a policy cover for time step \(h\) that is a subset of \(\Upsilon_{h}\) and whose size is equal to the number of reachable states at time step \(h\). Further, as the transitions are near-deterministic, we refer to an open-loop policy as a path, as we can view the policy as tracing a path in the latent transition model. PPE works by eliminating paths in \(\Upsilon_{h}\) so that we are left with just a single path for each reachable state. This is done by collecting a dataset \(\mathcal{D}\) of tuples \((x,v)\) where \(v\) is a uniformly sampled from \(\Upsilon_{h}\) and \(x\sim\mathbb{P}_{h}(x\mid v)\) (line 4). 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).

Intuitively, if \(\hat{f}_{h}(i\mid x)\) is sufficiently large, then we can hope that the path \(v_{i}\) visits the state \(\phi^{\star}(x)\). 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. For every pair of paths in \(\Upsilon_{h}\), we first compute a path prediction gap \(\widehat{\Delta}(\text{line 7})\). If the gap is too small, we show it implies that these paths reach the same endogenous state, hence we can eliminate a single redundant path from this pair (line 8). Finally, \(\Psi_{h}\) is defined as the set of all paths in \(\Upsilon_{h}\) which were not eliminated. PPE reduces RL to performing \(H\) standard classification problems. Further, the algorithm is very simple and in practice requires just a single hyperparameter (\(N\)). We believe these properties will make it well-suited for many problems.

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

However, for most problems, recovering an approximate deterministic transition dynamics suffices, which can be directly read from the path elimination data. We accomplish this by recovering a partition of paths in \(\Psi_{h-1}\times\mathcal{A}\) where two paths in the same partition set are said to be _merged_ with each other.

When we eliminate a path \(\upsilon_{j}\) on comparison with \(\upsilon_{i}\) in line 8, then all paths currently merged with \(\upsilon_{j}\) get merged with \(\upsilon_{i}\). We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

PPE runs in a reward-free setting.

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

_Realizability assumptions are common in theoretical analysis (e.g., Misra et al. (2020), Agarwal et al. (2020)). In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

**Theorem 1** (Sample Complexity).: _Fix \(\delta\in(0,1)\). Then, with probability greater than \(1-\delta\), PPE returns a policy cover \(\{\Psi_{h}\}_{h=1}^{H}\) such that for any \(h\in[H]\), \(\Psi_{h}\) is a \(\eta H\)-policy cover for time step \(h\) and \(|\Psi_{h}|\leq S\), which gives the total number of episodes used by PPE as \(\mathcal{O}\left(S^{2}A^{2}H\ln\frac{|\mathcal{F}|SAH}{\delta}\right)\).

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

We now show how a generalized inverse dynamics objective can eliminate such redundant paths.

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. 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.

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.

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 \(s_{1,a}\). For any state \(s_{h,g}\) we call \(g\) as its _type_ which can be \(a,b\) or \(c\). States with type \(a\) and \(b\) are considered _good_ states and those with type \(c\) are considered _bad_ states. 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 \(s_{h+1,c}\).

Lastly, PPE is able to solve the problem for all values of \(H\) and is significantly more sample efficient than baselines. Since the reward function of the combination-lock problem depends only on the endogenous state, we run PPE and then a value-iteration like algorithm (see Appendix D.2) to learn a near optimal policy.

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory.

Lastly, we observe that PPE is always able to learn a good decoder and is more sample efficient than baselines.

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 model \(\mathcal{F}\) using a two-layer convolutional neural network and train it using Adam optimization.

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.

We see that PPE is able to reduce the number of type 1 errors down to 0 using \(2\times 10^{5}\) episodes per time step. This is important since even a single type 1 error can cause exploration failures.

This is remarkable considering that PPE compares roughly \(2\times 10^{5}\) pairs of paths in the entire run. Hence, it makes only \(\leq 0.03\%\) type 2 errors. 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. In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems.

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:
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:
Looking at the claim and evidence, I'll evaluate its validity:

The claim states that PPE learns a generalization of inverse dynamics and is provably sample and computationally efficient in EX-BMDPs with near-deterministic endogenous state dynamics.

Summary of evaluation:
- The evidence strongly supports the claim through theoretical proofs (Theorem 1) showing sample complexity bounds
- Experimental results demonstrate PPE's effectiveness in combination lock and visual grid-world problems
- The algorithm shows better computational efficiency compared to previous work (O(S³A³H) vs O(S⁴A⁴H))
- The reviewer's comments acknowledge the theoretical contributions while noting some gaps in experimental details
- The evidence includes mathematical analysis, empirical validation, and comparison with baselines

The claim appears well-supported by the comprehensive theoretical and empirical evidence provided, though there are some minor gaps in experimental details as noted by the reviewer. The core assertion about efficiency and generalization is substantiated.

<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:

Then, we introduce the Predictive Path Elimination (PPE) algorithm, that learns a generalization of inverse dynamics and is provably sample and computationally efficient in EX-BMDPs when the endogenous state dynamics are near deterministic.

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.

However, it can also ignore controllable information. This may result in a failure to sufficiently explore the environment. 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. The over-abstraction problem was described at Misra et al. (2020), when the starting state is random. In Appendix A.3 we show inverse dynamics may over-abstract when the initial starting state is deterministic.

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 Footnote 1: Alg. 4 has time complexity of \(\mathcal{O}(S^{4}A^{4}H)\) compared to \(\mathcal{O}(S^{3}A^{3}H)\) for PPE. Furthermore, Alg. 4 requires an upper bound on \(S\), whereas PPE is adaptive to it. Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

```**Algorithm 1** PPE\((\delta,\eta)\): Predictive Path EliminationPPE performs iterations over the time steps \(h\in\{2,\cdots,H\}\). In the \(h^{th}\) iteration, it learns a policy cover \(\Psi_{h}\) for time step \(h\) containing open-loop policies. This is done by first augmenting the policy cover for previous time step by one step. Formally, we define \(\Upsilon_{h}=\Psi_{h-1}\circ\mathcal{A}=\{\pi\circ a\mid\pi\in\Psi_{h-1},a\in \mathcal{A}\}\) where \(\pi\circ a\) is an open-loop policy that follows \(\pi\) till time step \(h-1\) and then takes action \(a\). Since we assume the transition dynamics to be near-deterministic, therefore, we know that there exists a policy cover for time step \(h\) that is a subset of \(\Upsilon_{h}\) and whose size is equal to the number of reachable states at time step \(h\). Further, as the transitions are near-deterministic, we refer to an open-loop policy as a path, as we can view the policy as tracing a path in the latent transition model. PPE works by eliminating paths in \(\Upsilon_{h}\) so that we are left with just a single path for each reachable state. This is done by collecting a dataset \(\mathcal{D}\) of tuples \((x,v)\) where \(v\) is a uniformly sampled from \(\Upsilon_{h}\) and \(x\sim\mathbb{P}_{h}(x\mid v)\) (line 4). 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).

Intuitively, if \(\hat{f}_{h}(i\mid x)\) is sufficiently large, then we can hope that the path \(v_{i}\) visits the state \(\phi^{\star}(x)\). 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. For every pair of paths in \(\Upsilon_{h}\), we first compute a path prediction gap \(\widehat{\Delta}(\text{line 7})\). If the gap is too small, we show it implies that these paths reach the same endogenous state, hence we can eliminate a single redundant path from this pair (line 8). Finally, \(\Psi_{h}\) is defined as the set of all paths in \(\Upsilon_{h}\) which were not eliminated. PPE reduces RL to performing \(H\) standard classification problems. Further, the algorithm is very simple and in practice requires just a single hyperparameter (\(N\)). We believe these properties will make it well-suited for many problems.

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

However, for most problems, recovering an approximate deterministic transition dynamics suffices, which can be directly read from the path elimination data. We accomplish this by recovering a partition of paths in \(\Psi_{h-1}\times\mathcal{A}\) where two paths in the same partition set are said to be _merged_ with each other.

When we eliminate a path \(\upsilon_{j}\) on comparison with \(\upsilon_{i}\) in line 8, then all paths currently merged with \(\upsilon_{j}\) get merged with \(\upsilon_{i}\). We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

PPE runs in a reward-free setting.

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

_Realizability assumptions are common in theoretical analysis (e.g., Misra et al. (2020), Agarwal et al. (2020)). In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

**Theorem 1** (Sample Complexity).: _Fix \(\delta\in(0,1)\). Then, with probability greater than \(1-\delta\), PPE returns a policy cover \(\{\Psi_{h}\}_{h=1}^{H}\) such that for any \(h\in[H]\), \(\Psi_{h}\) is a \(\eta H\)-policy cover for time step \(h\) and \(|\Psi_{h}|\leq S\), which gives the total number of episodes used by PPE as \(\mathcal{O}\left(S^{2}A^{2}H\ln\frac{|\mathcal{F}|SAH}{\delta}\right)\).

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

We now show how a generalized inverse dynamics objective can eliminate such redundant paths.

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. 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.

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.

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 \(s_{1,a}\). For any state \(s_{h,g}\) we call \(g\) as its _type_ which can be \(a,b\) or \(c\). States with type \(a\) and \(b\) are considered _good_ states and those with type \(c\) are considered _bad_ states. 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 \(s_{h+1,c}\).

Lastly, PPE is able to solve the problem for all values of \(H\) and is significantly more sample efficient than baselines. Since the reward function of the combination-lock problem depends only on the endogenous state, we run PPE and then a value-iteration like algorithm (see Appendix D.2) to learn a near optimal policy.

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory.

Lastly, we observe that PPE is always able to learn a good decoder and is more sample efficient than baselines.

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 model \(\mathcal{F}\) using a two-layer convolutional neural network and train it using Adam optimization.

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.

We see that PPE is able to reduce the number of type 1 errors down to 0 using \(2\times 10^{5}\) episodes per time step. This is important since even a single type 1 error can cause exploration failures.

This is remarkable considering that PPE compares roughly \(2\times 10^{5}\) pairs of paths in the entire run. Hence, it makes only \(\leq 0.03\%\) type 2 errors. 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. In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems.

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

Output:
After reviewing the claim and evidence, I find that it is moderately well-stated but has some elements of overstatement. The evidence shows that PPE does learn a generalized form of inverse dynamics and demonstrates sample efficiency in near-deterministic EX-BMDPs, supported by both theoretical guarantees (Theorem 1) and experimental results. However, the claim could be more precise about the conditions required - particularly that it requires near-deterministic dynamics (η≤1/(4SH)) and realizability assumptions. The experimental validation, while positive, is limited to relatively simple domains (combination lock and grid world).

<score>2</score>

Justification:
1. The evidence supports the core claim about learning generalized inverse dynamics and sample efficiency through:
- Theoretical proof in Theorem 1 showing sample complexity bounds
- Experimental results demonstrating effectiveness in combination lock and grid world domains
- Successful recovery of latent endogenous transition dynamics

2. Areas where the claim could be more precise:
- Should explicitly mention the near-deterministic requirement
- Could clarify the realizability assumptions needed
- Could acknowledge limitations in experimental validation scope

3. Reviewer comments suggest initial concerns about computational tractability and practical implementation were adequately addressed during discussion.

The score of 2 reflects that while there is strong supporting evidence for the core claims, some qualifying conditions could have been more explicitly stated in the claim itself.


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:

Then, we introduce the Predictive Path Elimination (PPE) algorithm, that learns a generalization of inverse dynamics and is provably sample and computationally efficient in EX-BMDPs when the endogenous state dynamics are near deterministic.

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.

However, it can also ignore controllable information. This may result in a failure to sufficiently explore the environment. 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. The over-abstraction problem was described at Misra et al. (2020), when the starting state is random. In Appendix A.3 we show inverse dynamics may over-abstract when the initial starting state is deterministic.

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 Footnote 1: Alg. 4 has time complexity of \(\mathcal{O}(S^{4}A^{4}H)\) compared to \(\mathcal{O}(S^{3}A^{3}H)\) for PPE. Furthermore, Alg. 4 requires an upper bound on \(S\), whereas PPE is adaptive to it. Lastly, Du et al. (2019) assumed deterministic setting while we provide a generalization to near-determinism.

```**Algorithm 1** PPE\((\delta,\eta)\): Predictive Path EliminationPPE performs iterations over the time steps \(h\in\{2,\cdots,H\}\). In the \(h^{th}\) iteration, it learns a policy cover \(\Psi_{h}\) for time step \(h\) containing open-loop policies. This is done by first augmenting the policy cover for previous time step by one step. Formally, we define \(\Upsilon_{h}=\Psi_{h-1}\circ\mathcal{A}=\{\pi\circ a\mid\pi\in\Psi_{h-1},a\in \mathcal{A}\}\) where \(\pi\circ a\) is an open-loop policy that follows \(\pi\) till time step \(h-1\) and then takes action \(a\). Since we assume the transition dynamics to be near-deterministic, therefore, we know that there exists a policy cover for time step \(h\) that is a subset of \(\Upsilon_{h}\) and whose size is equal to the number of reachable states at time step \(h\). Further, as the transitions are near-deterministic, we refer to an open-loop policy as a path, as we can view the policy as tracing a path in the latent transition model. PPE works by eliminating paths in \(\Upsilon_{h}\) so that we are left with just a single path for each reachable state. This is done by collecting a dataset \(\mathcal{D}\) of tuples \((x,v)\) where \(v\) is a uniformly sampled from \(\Upsilon_{h}\) and \(x\sim\mathbb{P}_{h}(x\mid v)\) (line 4). 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).

Intuitively, if \(\hat{f}_{h}(i\mid x)\) is sufficiently large, then we can hope that the path \(v_{i}\) visits the state \(\phi^{\star}(x)\). 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. For every pair of paths in \(\Upsilon_{h}\), we first compute a path prediction gap \(\widehat{\Delta}(\text{line 7})\). If the gap is too small, we show it implies that these paths reach the same endogenous state, hence we can eliminate a single redundant path from this pair (line 8). Finally, \(\Psi_{h}\) is defined as the set of all paths in \(\Upsilon_{h}\) which were not eliminated. PPE reduces RL to performing \(H\) standard classification problems. Further, the algorithm is very simple and in practice requires just a single hyperparameter (\(N\)). We believe these properties will make it well-suited for many problems.

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

However, for most problems, recovering an approximate deterministic transition dynamics suffices, which can be directly read from the path elimination data. We accomplish this by recovering a partition of paths in \(\Psi_{h-1}\times\mathcal{A}\) where two paths in the same partition set are said to be _merged_ with each other.

When we eliminate a path \(\upsilon_{j}\) on comparison with \(\upsilon_{i}\) in line 8, then all paths currently merged with \(\upsilon_{j}\) get merged with \(\upsilon_{i}\). We then define an abstract state space \(\widehat{\mathcal{S}}_{h}\) for time step \(h\) that contains an abstract state \(j\) for each path \(\upsilon_{j}\in\Psi_{h}\). Further, we recover a latent deterministic transition dynamics for time step \(h-1\) as \(\hat{T}_{h-1}:\widehat{\mathcal{S}}_{h-1}\times\mathcal{A}\to\widehat{\mathcal{ S}}_{h}\) where we set \(\hat{T}_{h-1}(i,a)=j\) if the path \(\upsilon_{j}\in\Psi_{h}\) gets merged with path \(\upsilon_{i}^{\prime}\circ a\in\Psi_{h}\) where \(\upsilon_{i}^{\prime}\in\Psi_{h-1}\).

PPE runs in a reward-free setting.

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

_Realizability assumptions are common in theoretical analysis (e.g., Misra et al. (2020), Agarwal et al. (2020)). In practice, we use expressive neural networks to solve the regression problem, so we expect the realizability assumption to hold.

**Theorem 1** (Sample Complexity).: _Fix \(\delta\in(0,1)\). Then, with probability greater than \(1-\delta\), PPE returns a policy cover \(\{\Psi_{h}\}_{h=1}^{H}\) such that for any \(h\in[H]\), \(\Psi_{h}\) is a \(\eta H\)-policy cover for time step \(h\) and \(|\Psi_{h}|\leq S\), which gives the total number of episodes used by PPE as \(\mathcal{O}\left(S^{2}A^{2}H\ln\frac{|\mathcal{F}|SAH}{\delta}\right)\).

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

We now show how a generalized inverse dynamics objective can eliminate such redundant paths.

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. 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.

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.

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 \(s_{1,a}\). For any state \(s_{h,g}\) we call \(g\) as its _type_ which can be \(a,b\) or \(c\). States with type \(a\) and \(b\) are considered _good_ states and those with type \(c\) are considered _bad_ states. 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 \(s_{h+1,c}\).

Lastly, PPE is able to solve the problem for all values of \(H\) and is significantly more sample efficient than baselines. Since the reward function of the combination-lock problem depends only on the endogenous state, we run PPE and then a value-iteration like algorithm (see Appendix D.2) to learn a near optimal policy.

When there is no exogenous noise, Homer is able to learn a good state decoder with enough samples while ID fails to learn, in accordance with the theory.

Lastly, we observe that PPE is always able to learn a good decoder and is more sample efficient than baselines.

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 model \(\mathcal{F}\) using a two-layer convolutional neural network and train it using Adam optimization.

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.

We see that PPE is able to reduce the number of type 1 errors down to 0 using \(2\times 10^{5}\) episodes per time step. This is important since even a single type 1 error can cause exploration failures.

This is remarkable considering that PPE compares roughly \(2\times 10^{5}\) pairs of paths in the entire run. Hence, it makes only \(\leq 0.03\%\) type 2 errors. 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. In this work, we introduce the EX-BMDP setting, an RL setting that models exogenous noise, ubiquitous in many real-world systems.

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:
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:
Let me assess the claim based on the provided evidence and review comments:

Summary: The claim states that PPE learns a generalized inverse dynamics and is provably sample/computationally efficient in EX-BMDPs with near-deterministic endogenous dynamics. The evidence strongly supports this through:
1. Theoretical guarantees (Theorem 1) showing sample complexity bounds
2. Experimental validation on combination lock and visual gridworld tasks
3. Empirical results showing PPE successfully learns accurate state decoders and transition models
4. Review comments acknowledge the significance of sample complexity bounds being independent of exogenous state complexity

While the claim has some limitations (e.g., near-deterministic requirement noted in the review), it carefully qualifies its scope and is well-supported by both theoretical and empirical evidence.

<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 sample complexity of PPE depends polynomially on the size of the latent endogenous state space while not directly depending on the size of the observation space, nor the exogenous state space.

The evidence to be evaluated is:_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 endogenous state \(s\) captures the information that can be manipulated by the agent.

We assume that the set of all endogenous states \(\mathcal{S}\) is finite with cardinality \(S\).

Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large.

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).

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\).

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).

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.

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.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

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

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

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

Output:
After reviewing the claim and evidence, I find the claim to be well-supported by both theoretical and empirical evidence. The text explicitly states "Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space" and demonstrates this through:

1. Theoretical analysis showing policy cover reduction to |S| rather than |S||Ξ|
2. Experimental results in Figure 2 showing similar performance between scenarios with no exogenous dimension (solid lines) and large exogenous dimension of 100 (dashed lines)
3. Clear mathematical formulation showing dependence on endogenous state space size

The evidence directly supports the polynomial dependence on latent endogenous state space while showing independence from observation and exogenous space sizes.

<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 sample complexity of PPE depends polynomially on the size of the latent endogenous state space while not directly depending on the size of the observation space, nor the exogenous state space.

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

We assume that the set of all endogenous states \(\mathcal{S}\) is finite with cardinality \(S\).

Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large.

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).

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\).

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).

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.

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.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

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

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 appears to be well-supported but slightly overstated. While the evidence clearly shows that PPE's sample complexity depends polynomially on the endogenous state space size and works with arbitrarily large observation/exogenous spaces, the claim's definitive tone about "not directly depending" on these spaces could be more nuanced. The evidence shows this independence in specific contexts (near-deterministic dynamics, single start state) rather than universally. The empirical results in Figure 2 support the claim's core message about PPE's scalability with exogenous dimensions.

<score>2</score>

Justification:
1. The evidence explicitly states "Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space" and demonstrates this with examples where the exogenous state space is "arbitrarily large."

2. However, there are important caveats that the claim doesn't acknowledge:
- The paper assumes "near-deterministic dynamics and start state distribution"
- The Bellman rank can scale as O(|Ξ|) (mentioned in Appendix A)
- The review points out limitations regarding "single starting states and near-deterministic dynamics"

3. While the claim's core message is supported by both theoretical and empirical evidence (Figure 2 shows PPE maintaining performance with large exogenous dimensions), a more precise statement would acknowledge these constraints.

The score of 2 reflects that while the claim is fundamentally accurate, it would be better stated with some qualifying language about the specific conditions under which it holds.


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 sample complexity of PPE depends polynomially on the size of the latent endogenous state space while not directly depending on the size of the observation space, nor the exogenous state space.

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

We assume that the set of all endogenous states \(\mathcal{S}\) is finite with cardinality \(S\).

Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large.

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).

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\).

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).

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.

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.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

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

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

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

Output:
After reviewing the claim and evidence, the claim appears largely accurate but slightly overstated. While the evidence demonstrates that PPE's sample complexity primarily depends on endogenous state space size and shows independence from exogenous state space (demonstrated in Figure 2's results and theoretical analysis), the claim makes a strong assertion about "polynomial" dependence that isn't explicitly proven in the provided evidence. The experiments show favorable scaling, but the precise polynomial relationship isn't clearly established. Additionally, while the claim about independence from observation space size is supported by statements in the evidence, the absolute independence isn't comprehensively demonstrated.

<score>2</score>

Justification:
1. Supporting evidence:
- "Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space"
- Figure 2 shows consistent performance even with large exogenous dimension (100)
- The algorithm works with arbitrarily large observation spaces ("can be arbitrarily large to model")

2. Limitations/Overstatements:
- The specific "polynomial" dependence isn't explicitly shown in the evidence
- The evidence shows independence from exogenous state space size practically, but theoretical bounds still show some scaling ("Bellman rank for EX-BMDP can scale as O(|Ξ|)")
- The reviewer points out some gaps in experimental details and methodology

3. Reviewer comments support this assessment, noting the theoretical basis is valuable but questioning some experimental details and relationships to prior work.


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 sample complexity of PPE depends polynomially on the size of the latent endogenous state space while not directly depending on the size of the observation space, nor the exogenous state space.

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

We assume that the set of all endogenous states \(\mathcal{S}\) is finite with cardinality \(S\).

Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large.

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).

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\).

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).

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.

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.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

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

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

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

Output:
After reviewing the claim and evidence, I find that the claim is generally well-supported but slightly overstated. While the evidence clearly shows that PPE's sample complexity depends on the endogenous state space size, the claim about "not directly depending" on observation and exogenous state spaces needs more nuance. The evidence shows that direct dependence is avoided, but there are implicit connections - for example, the evidence mentions Bellman rank scaling with O(|Ξ|). The experimental results in Figure 2 support the practical effectiveness but also show some performance differences with larger exogenous dimensions.

<score>2</score>

Justification:
1. Supporting evidence:
- The paper explicitly states "Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space"
- The structure allows reduction to |S| << |Z| = |S||Ξ|
- Experimental results show PPE working effectively even with large exogenous dimensions

2. Qualifying factors:
- Bellman rank scales with O(|Ξ|), suggesting some indirect dependence
- Reviewer note W2 raises valid concerns about computational tractability with high-dimensional observations
- Figure 2(c) shows some performance degradation with exogenous dimension of 100 (dashed lines vs solid)

The claim would be better stated if it acknowledged these indirect dependencies and computational considerations while maintaining its core assertion about polynomial dependence on endogenous state space size.


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 sample complexity of PPE depends polynomially on the size of the latent endogenous state space while not directly depending on the size of the observation space, nor the exogenous state space.

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

We assume that the set of all endogenous states \(\mathcal{S}\) is finite with cardinality \(S\).

Again, we make no assumptions on the exogenous dynamics nor on its cardinality \(|\Xi|\) which may be arbitrarily large.

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).

However, we show in Appendix A that Bellman rank for EX-BMDP can scale as \(\mathcal{O}(|\Xi|)\).

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).

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.

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.

Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space.

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

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 appears to be well-stated. The evidence shows that:
1. The algorithm's sample complexity depends on the endogenous state space size (|S|)
2. The evidence explicitly states "Our sample complexity guarantees do not depend directly on the size of observation space or the exogenous space"
3. Figure 2 demonstrates consistent performance even with large exogenous dimension (100)
4. The theoretical results and empirical validation align well with the claim
5. No significant overstatement or exaggeration is found in comparing the claim to the evidence

The claim accurately reflects what is demonstrated in both theoretical guarantees and experimental results without overstating the findings.

<score>1</score>