Provable Causal State Representation under Asynchronous Diffusion Model for POMDPs
diffusion model;causal state representation;model uncertainty;bisimulaion;POMDP
Yes
No ethics review needed.
Upon initial reading I was unclear what was meant by “POMDPS with perturbed inputs” in the abstract. Isn’t the whole point of any POMDP that its inputs (observations of the environment) are subject to noise? 
Line 151: “the action a_{t-1} directly affects the state s_t rather than the observation signal o_t”: Why then is the probability of o_t defined conditional on a_{t-1} in eq. 1a. 
Algorithm 1: Why does \zeta denote both the reward noise model and the bisimulation model?
This is a generally well-written paper based on a novel idea (as far as I can judge). It combines a proposal of an algorithm with theoretical analysis and reasonable, well-designed experiments
The paper considers causal state representations of partially observable environments for approximately solving POMDPs. Specifically, the authors propose an approach to find bisimulation-based causal state models by denoising observations and rewards via an asynchronous diffusion model. The offers also do analysis and give some theoretical guarantees under some assumptions.
A few points could be clearer in the manuscript, some of which I listed in the questions below. Part of my confusion could be my lack of in-depth knowledge of Causal State Representations.
Yes
No ethics review needed.
* In Eqs. 1a-1c, what exactly are the assumptions you make about the dynamics of the model? For example, must the state-, action- and observation spaces be continuous, or can they be discrete? What about the functions $f,g$ and $h$? Can two states give the same observation?
* In eq. 1, $f$ denotes the observation function. However, in Def. 1 and in Assumption 2, $f$ is also used to denote something that looks like an observation function but has a different number of inputs. How do these relate?
* $\zeta$ is overloaded in a confusing way: it is used to describe the predicted state (line 234), the noisy state (line 269), as well as the bisimulation model (Alg 1, line 2). Do these all represent the same thing?
* In Eqs. 7 and 8 and Alg 1, $\theta$ and $\zeta$ seem to be switched. Is this a typo?
* After eq. 6, the paper mentions a variables $n$, $\hat{s}_{t+1}$ and $r_{t+1}$. What do these refer to?
* In Alg. 1, what is the function of line 7? What do we use the sampled transitions for?
* In line 297, how can the diffusion model predict a future state? I thought it only removed noise?
* In Assumption 3, what does this assumption intuitively mean? Are $s_i, s_j \in S$, or in $S \cup O$ ?
* I do not understand Thm 1: it seems to me that if we pick $c_T \approx 0$ and $c_R \approx 1$, then any states that have the same immediate reward would have a value gap of $\approx 0$ as well, which clearly is not the case. Can you explain why it holds?
* In Fig 4 (App. A), how is $P(\hat{s}_t|o_t)$ computed? I thought this was what the diffusion model was used for.
* The topic of the paper is interesting and significant: dealing with partial observability is a key problem when applying RL in the real world. 
* The paper combines both theory and application nicely. Moreover, the proposed method can easily be incorporated into off-the-shelf RL methods, which makes it easier to apply in practice. 
* The authors compare their method with relevant baseline algorithms and use an ablation study to show the relevance of all proposed components.
In this paper, the authors propose a method of dealing with noisy observations in RL, which they call CSR-ADM. Intuitively, the algorithm uses both a denoise model and a bisimulation metric to find 'causal state representations' for a given observation, which can be used by off-the-shelf RL algorithms to compute policies. The authors provide a sub-optimality bound for their method under some (reasonable) assumptions on the dynamics of the environment. Empirically, the authors show that incorporating their method into SAC improves performance and outperforms methods that only consider denoising or finding causal representations.
The main weakness of this paper is its presentation. The intuition behind the methods is easy to follow, but details are often unclear: see some of my questions below. Because of this, I find it hard to determine the quality of the proposed method.

I'll note some other minor weaknesses:
* The method assumes Gaussian noise. Thus, the method may struggle with other noise (such as raindrops or colour shifts), and does not help with other types of partial observability (such as missing data).
* The paper does not quantify the additional computational cost of the method: this would be good to add.
Yes
No ethics review needed.
See my discussion on the weakness.
N/A

Caveat: As a researcher in learning theory, I am not familiar with the current line of research on RL with diffusion models. In below, I only evaluate this paper based on its math and theory, and my review should be taken with AC's discretion.
This paper proposes a method that incorporates the diffusion model in RL algorithms, along with the theoretical guarantees. Empirical evaluation is also included.
I find this paper difficult to understand.

1. The term "bisimulation" is never defined or well-explained. Definition 1 is also not rigorous: in POMDP, the distribution P(s_{t+1}|o_t,a) clearly depends on the history (and hence the policy). It is also not clear why there is a partition of the state space into the observation space.

2. Definition 2/Assumption 3 seems to define the bisimulation metric in terms of a fixed-point equation. While such a metric is shown to exist (at least when p=1), I find it difficult to interpret.

3. In Theorem 2, $\mathcal{E}_\zeta$ is defined twice.

4. Theorem 4 is claimed to establish the convergence of the proposed algorithm, but I can't see how. In. particular, the RHS of (15) is not vanishing when n tends to infinity. Further, the algorithm is based on gradient descent, but there is no analysis of GD here (it is vaguely mentioned that previous results can be invoked).
Yes
No ethics review needed.
Same as the weakness
1.	The authors propose novel methods to make use of the causal structure under the POMDP environment, which achieves better performance in the simulations.
2.	This work contains the solid theoretical guarantee for the proposed methods.
This work considers the decision-making problem in POMDP with diffusion as an estimation tool. The authors adopt the diffusion model to utilize the causal graph under the POMDP for better value function estimation. They provide the theoretical analysis of the proposed algorithm, and the efficacy of the algorithm is also verified by the experimental results.
1.	The implications of the assumptions adopted in this work are not clear. For Assumption 2, it is beneficial to justify when the lower boundness of $f$ holds in the real applications. For Assumption 3, I find that the statement of Definition 2 is not appropriate. When defining a mathematical notion, it is uncommon to state `` following metric exists and is unique’’, which looks like the statement of a theorem. In addition, I suggest the authors to discuss the sufficient conditions of Assumption 3. For example, if the state, action, and observation spaces are finite, will this metric exist and be unique?

2.	It will be helpful to discuss more about the results of diffusion model. In the existing analysis of diffusion models, the distribution estimation error usually consists of initialization error, score estimation error, and the discretization error. But such error decomposition structure is not presented in the current results. 

3.	In addition, this is beneficial to discuss and explain each theorem below the statement of the results.
We present a new RL framework CSR-ADM, which can accommodate and enhance any RL algorithm by capturing causal state representations for POMDPs under perturbation via a novel asynchronous diffusion model and bisimulation technology.
@inproceedings{
anonymous2024provable,
title={Provable Causal State Representation under Asynchronous Diffusion Model for {POMDP}s},
author={Anonymous},
booktitle={Submitted to The Thirteenth International Conference on Learning Representations},
year={2024},
url={https://openreview.net/forum?id=FNiqaC382D},
note={under review}
}
A major challenge in applying reinforcement learning (RL) to real-world scenarios is managing high-dimensional, noisy perception input signals. Identifying and utilizing representations that contain sufficient and essential information for decision-making tasks is key to computational efficiency and generalization of RL by reducing bias in decision-making processes. In this paper, we present a new RL framework, named *Causal State Representation under Asynchronous Diffusion Model (CSR-ADM)*, which accommodates and enhances any RL algorithm for partially observable Markov decision processes (POMDPs) with perturbed inputs. A new asynchronous diffusion model is proposed to denoise both reward and observation spaces, and integrated with the bisimulation technology to capture causal state representations in POMDPs. Notably, the causal state is the coarsest partition of the denoised observations. We link the causal state to a causal feature set and provide theoretical guarantees by deriving the upper bound on value function approximation between the noisy observation space and the causal state space, demonstrating equivalence to bisimulation under the Lipschitz assumption. To the best of our knowledge, CSR-ADM is the first framework to approximate causal states with diffusion models, substantiated by a comprehensive theoretical foundation. Extensive experiments on Roboschool tasks show that CSR-ADM outperforms state-of-the-art methods, significantly improving the robustness of existing RL algorithms under varying scales of random noise.
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diffusion model causal state representation model uncertainty bisimulaion POMDP
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Provable Causal State Representation under Asynchronous Diffusion Model for POMDPs
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