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

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

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

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

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

The claim to be assessed is:

We further, show that PPE is able to recover the latent endogenous model in a visually complex navigation problem, in accordance with the theory.

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 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).

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).

```**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.

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.

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

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

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

However, the recovered policy cover and dynamics can be directly used to optimize any given reward function with existing methods.

If the reward function only depends on the endogenous state, we can use a computationally cheaper value-iteration VI that uses the recovered transition dynamics. VI is a model-based algorithm that estimates the reward for each state and action, and performs dynamic programming on the model (see Appendix D.2 for details). 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 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. **Assumption 2** (Realizability).: _For any \(h\in[H]\), and any set of paths \(\Upsilon\subseteq\mathcal{A}^{h}\) with \(|\Upsilon|\leq SA\) and where \(\mathcal{A}^{h}\) denotes the set of all paths of length \(h\), there exists \(f^{\star}_{\Upsilon,h}\in\mathcal{F}\) such that: \(f^{\star}_{\Upsilon,h}(\mathtt{idx}(\upsilon)\mid x)=\frac{\mathbb{P}_{h}( \phi^{\star}(x))|\upsilon)}{\sum_{\upsilon^{\prime}\in\Upsilon}\mathbb{P}_{h}( \phi^{\star}(x))|\upsilon^{\prime})}\), for all \(\upsilon\in\Upsilon\) and \(x\in\mathcal{X}\) with \(\sum_{\upsilon^{\prime}\in\Upsilon}\mathbb{P}_{h}(\phi^{\star}(x))\mid \upsilon^{\prime})>0\). _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. Note that there are at most \(A^{S(H+1)}\) Bayes classifiers for different prediction problems. However, this is acceptable since our guarantees will scale as \(\ln|\mathcal{F}|\) and, therefore, the function class \(\mathcal{F}\) can be exponentially large to accommodate all of them.

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.

We visualize the latent endogenous dynamics in Figure 1(a).

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

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.

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.

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

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.

Similarly, PPE is able to reduce type 2 errors and is able to get them down to 56 with \(5\times 10^{5}\) episodes.

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

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

We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores. We derive theoretical guarantees for PPE in near-deterministic setting and provide encouraging experimental evidence in support of our arguments.

Our work also raises interesting future questions such as how to address the general setting with stochastic transitions, or handle more complex endogenous state representations.

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

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

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

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

Output:
Based on my review of the evidence, the claim "PPE is able to recover the latent endogenous model in a visually complex navigation problem, in accordance with the theory" appears to be well-supported. The evidence shows:

- Clear experimental results in a visual grid-world demonstrating PPE's ability to learn the latent dynamics (Figure 3)
- Quantitative metrics showing PPE reduces both Type 1 and Type 2 errors in recovering the latent model
- Successful state decoding accuracy compared to baselines
- Alignment with theoretical foundations presented in the paper

While the reviewer notes some gaps in experimental details, the core claim about PPE's capability to recover the latent model is demonstrated through multiple metrics and visualizations. The claim is measured in its scope and directly supported by the evidence presented.

<score>1</score>


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

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

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

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

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

The claim to be assessed is:

We further, show that PPE is able to recover the latent endogenous model in a visually complex navigation problem, in accordance with the theory.

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 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).

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).

```**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.

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.

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

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

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

However, the recovered policy cover and dynamics can be directly used to optimize any given reward function with existing methods.

If the reward function only depends on the endogenous state, we can use a computationally cheaper value-iteration VI that uses the recovered transition dynamics. VI is a model-based algorithm that estimates the reward for each state and action, and performs dynamic programming on the model (see Appendix D.2 for details). 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 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. **Assumption 2** (Realizability).: _For any \(h\in[H]\), and any set of paths \(\Upsilon\subseteq\mathcal{A}^{h}\) with \(|\Upsilon|\leq SA\) and where \(\mathcal{A}^{h}\) denotes the set of all paths of length \(h\), there exists \(f^{\star}_{\Upsilon,h}\in\mathcal{F}\) such that: \(f^{\star}_{\Upsilon,h}(\mathtt{idx}(\upsilon)\mid x)=\frac{\mathbb{P}_{h}( \phi^{\star}(x))|\upsilon)}{\sum_{\upsilon^{\prime}\in\Upsilon}\mathbb{P}_{h}( \phi^{\star}(x))|\upsilon^{\prime})}\), for all \(\upsilon\in\Upsilon\) and \(x\in\mathcal{X}\) with \(\sum_{\upsilon^{\prime}\in\Upsilon}\mathbb{P}_{h}(\phi^{\star}(x))\mid \upsilon^{\prime})>0\). _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. Note that there are at most \(A^{S(H+1)}\) Bayes classifiers for different prediction problems. However, this is acceptable since our guarantees will scale as \(\ln|\mathcal{F}|\) and, therefore, the function class \(\mathcal{F}\) can be exponentially large to accommodate all of them.

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.

We visualize the latent endogenous dynamics in Figure 1(a).

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

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.

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.

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

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.

Similarly, PPE is able to reduce type 2 errors and is able to get them down to 56 with \(5\times 10^{5}\) episodes.

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

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

We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores. We derive theoretical guarantees for PPE in near-deterministic setting and provide encouraging experimental evidence in support of our arguments.

Our work also raises interesting future questions such as how to address the general setting with stochastic transitions, or handle more complex endogenous state representations.

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

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

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

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

Output:
After reviewing the evidence, the claim "PPE is able to recover the latent endogenous model in a visually complex navigation problem, in accordance with the theory" appears to be well-stated. The evidence shows:

1. In the visual grid-world experiments, PPE successfully:
- Reduced Type 1 errors to 0 with sufficient samples
- Significantly reduced Type 2 errors
- Achieved high state decoding accuracy compared to baselines
- Enabled optimal planning using the learned transition model

2. These results align with the theoretical framework and assumptions presented in the paper.

3. The limitations and experimental conditions are clearly stated, avoiding overreaching claims.

The empirical results strongly support the claim while staying within the bounds of what was actually demonstrated.

<score>1</score>


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

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

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

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

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

The claim to be assessed is:

We further, show that PPE is able to recover the latent endogenous model in a visually complex navigation problem, in accordance with the theory.

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 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).

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).

```**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.

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.

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

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

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

However, the recovered policy cover and dynamics can be directly used to optimize any given reward function with existing methods.

If the reward function only depends on the endogenous state, we can use a computationally cheaper value-iteration VI that uses the recovered transition dynamics. VI is a model-based algorithm that estimates the reward for each state and action, and performs dynamic programming on the model (see Appendix D.2 for details). 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 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. **Assumption 2** (Realizability).: _For any \(h\in[H]\), and any set of paths \(\Upsilon\subseteq\mathcal{A}^{h}\) with \(|\Upsilon|\leq SA\) and where \(\mathcal{A}^{h}\) denotes the set of all paths of length \(h\), there exists \(f^{\star}_{\Upsilon,h}\in\mathcal{F}\) such that: \(f^{\star}_{\Upsilon,h}(\mathtt{idx}(\upsilon)\mid x)=\frac{\mathbb{P}_{h}( \phi^{\star}(x))|\upsilon)}{\sum_{\upsilon^{\prime}\in\Upsilon}\mathbb{P}_{h}( \phi^{\star}(x))|\upsilon^{\prime})}\), for all \(\upsilon\in\Upsilon\) and \(x\in\mathcal{X}\) with \(\sum_{\upsilon^{\prime}\in\Upsilon}\mathbb{P}_{h}(\phi^{\star}(x))\mid \upsilon^{\prime})>0\). _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. Note that there are at most \(A^{S(H+1)}\) Bayes classifiers for different prediction problems. However, this is acceptable since our guarantees will scale as \(\ln|\mathcal{F}|\) and, therefore, the function class \(\mathcal{F}\) can be exponentially large to accommodate all of them.

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.

We visualize the latent endogenous dynamics in Figure 1(a).

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

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.

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.

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

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.

Similarly, PPE is able to reduce type 2 errors and is able to get them down to 56 with \(5\times 10^{5}\) episodes.

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

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

We present PPE that learns a multi-step inverse dynamics to filter exogenous noise and successfully explores. We derive theoretical guarantees for PPE in near-deterministic setting and provide encouraging experimental evidence in support of our arguments.

Our work also raises interesting future questions such as how to address the general setting with stochastic transitions, or handle more complex endogenous state representations.

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

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

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

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

Output:
1. The evidence provides strong support for the claim that "PPE is able to recover the latent endogenous model in a visually complex navigation problem." The visual grid world experiments demonstrate that:
- PPE reduces Type 1 errors (incorrectly merging different states) to 0 with sufficient episodes
- Type 2 errors are significantly reduced
- PPE achieves near-perfect state decoding accuracy (~100%)
- The agent successfully plans using the learned transition model
- Results align with theoretical predictions

The claim is well-stated and appropriately reflects the experimental evidence without exaggeration.

<score>1</score>

Justification:
- The claim is specific and bounded to the visual navigation task
- Figure 3 provides direct empirical evidence of successful model recovery
- Results show clear convergence in error reduction and high accuracy
- The claim doesn't overstate the generality or make unsupported assertions
- The review comments don't raise concerns about this specific claim
- The experimental details and limitations are well documented


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 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 careful review, I find that the claim about PPE's provable sample and computational efficiency in EX-BMDPs with near-deterministic endogenous dynamics is well-supported by the evidence. The paper provides:
- Theoretical guarantees via Theorem 1 showing sample complexity bounds
- Empirical validation through combination lock and visual grid-world experiments
- Detailed algorithm description with computational complexity analysis showing improvements over prior work (\(\mathcal{O}(S^3A^3H)\) vs \(\mathcal{O}(S^4A^4H)\))
- Clear limitations acknowledgment by specifying "near deterministic" dynamics requirement
- Experimental results demonstrating both sample efficiency and accuracy

The claim appears to accurately reflect the evidence without exaggeration.

<score>1</score>


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

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

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

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

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

The claim to be assessed is:

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:

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

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

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

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

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

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

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

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 evidence, the claim about PPE appears to be relatively well-stated but has some minor overstatements. The evidence demonstrates that PPE does learn a generalization of inverse dynamics and shows sample/computational efficiency in near-deterministic EX-BMDPs. The theoretical guarantees (Theorem 1) and experimental results support the algorithm's effectiveness. However, the reviewer correctly points out limitations regarding the restrictive assumptions (single starting states, near-deterministic dynamics) that aren't fully acknowledged in the claim. The evidence shows PPE works well within these constraints, but the claim could better qualify these limitations.

<score>2</score>

Justification:
1. The claim is supported by:
- Theoretical proof of sample complexity in Theorem 1
- Experimental results showing PPE successfully learns accurate state decoders
- Clear demonstration of computational efficiency (O(S³A³H) vs O(S⁴A⁴H))

2. Minor overstatements:
- Does not explicitly mention the restrictive assumption of near-deterministic dynamics (η ≤ 1/(4SH))
- Does not acknowledge limitations of single starting state requirement
- Could better qualify that efficiency is proven specifically for the near-deterministic case

3. The reviewer's comments about limited applicability and restrictive assumptions are valid but don't completely invalidate the core claims about the algorithm's capabilities within its specified domain.