NeurIPS 2020

Can Temporal-Difference and Q-Learning Learn Representation? A Mean-Field Theory

Review 1

Summary and Contributions: This paper studies Temporal Difference (TD) learning with the value function represented by overparameterized neural networks. From the mean-field perspective, it shows that TD(0) for both policy evaluation and Q learning converges to the optimal solutions at a sublinear (1/T) rate. Moreover, the mean-field theory shows that, as the TD learning algorithm proceeds, the feature representation learned by the neural networks also evolves, which shows that neural network based TD learning algorithms actually are able to learn good features automatically.

Strengths: 1. The theoretical results seem solid. 2. TD(0) with general nonlinear function approximation is known to have divergent counterexamples. This paper address such a fundamental problem for a specific nonlinear function -- overparameterized neural networks, and establishes a sublinear convergence guarantee. This is similar to the prior work [21] which studies the convergence of neural TD learning via linearization and neural tangent kernel. [21] also establishes a sublinear convergence rate. 3. Using the mean-field analysis, this work also depicts the evolution of the kernel function induced by the neural network value function. This result shows that, under the mean-field regime, neural TD learning also learns a good feature representation. This phenomenon is not captured by the neural tangent kernel based analysis (e.g. [21]) as the kernel is always fixed to the NTK in that line of work. Representation learning for RL seems a less explored direction in theory while it seems to matter a lot in practice. Thus this work seems made a contribution to this important aspect.

Weaknesses: 1. This paper lacks motivation for why we should care about the mean field regime of TD learning or why representation learning is important. Given that there is previous work [21] that has already established the finite-time convergence for neural TD learning, it would be nice to first inform the readers of the limitations of the previous NTK analysis. 2. In the title, this paper seems to claim that “Representation learning” is the main contribution. However, it seems that the related discussion is only in the paragraph after Theorem 4.3. It would be nice to further elaborate on this discussion so as to strengthen the argument. 3. Technically, one limitation of this work is that the convergence analysis is only in a continuous-time fashion. In contrast, [21] has a finite-time convergence guarantee. I understand that this is due to using different technical tools and cannot be directly compared. This paper uses mean-field theory while [21] uses NTK theory which approximates the neural network value functions by linear functions and the convergence analysis is based on convex optimization tools.

Correctness: I didn't check the proof line by line but the theory seems reasonable and the proof seems correct.

Clarity: The paper is overall very well-written. However, as I mentioned above, the presentation can be further improved by adding: (1) the motivation for the mean-field setting and the representation learning problem for neural TD learning; (2) discussions regarding the features learned from the algorithm; (3) comparison with NTK-based analysis for neural TD learning in terms of analysis.

Relation to Prior Work: Yes. This paper discusses and distinguishes from previous work properly. A missing related work: A Finite-Time Analysis of Q-Learning with Neural Network Function Approximation by Xu and Gu, 2019.

Reproducibility: Yes

Additional Feedback: [1] It would be nice to compare with [21] in terms of proof techniques. This would perhaps make the readers more clear about why the mean-field regime is able to learn representations while NTK is not. [2] How do the sublinear rates in this work and [21] compare? Can they directly compare with each other? Since one is continuous-time and the other one is discrete-time, we might need to be more careful when comparing these rates. [3] Is it possible to establish a finite-time convergence rate by discretizing the trajectory of ODE? [4] I understand that this work is a theoretical work, but it would be nice to see how far the mean-field regime is from the practical neural TD learning methods.

Review 2

Summary and Contributions: This paper studies the problem of global convergence of Temporal Difference learning with a specific (but rich) class of non-linear function approximation. The study leans on the mean-field limit theory and focuses on the behavior of the feature representation learned by a two-layer neural network. Authors show the fixed point of the projected Bellman error, an objective function proposed in previous work for producing gradient-based TD learning, and shows that the solution found by TD decays to zero at a sub-linear rate. -- Post rebuttal: I was really hoping to see the numerical results on famous counterexamples. Though this was not provided, I am still happy to see this paper getting in. I really hope that such results will be included in camera ready, and I suggest a format akin to that of Figure 1 from Maei et al., 2010.

Strengths: The work proposes an interesting and novel application of mean-field limit theory for understand TD with non-linear function approximation. This is definitely an interesting research avenue. I have issues with some of the claims made by authors, as well as some of the underlying assumptions, which undermines the soundness of the claims to a limited extent.

Weaknesses: - The paper promises results on, not only the prediction case with TD, but also with Q-learning and control. Yet, the two assumptions B1 and B3 in the appendix are super strong. In fact, it took a while for me to come up with examples where these assumptions hold. What's more, the validity of the assumptions seem to be a property of the problem, and even within a problem these assumptions may not hold on a per state basis. I think this paper needs to scale back its claims when it comes to Q-learning. I think the TD case is in and of itself interesting, and I find it disappointing that the paper has rushed to extend results to the control case by making these self-serving assumptions. I am happy to increase my score if authors scale back their claims pertaining to control. - It is worthwhile to mention that the TD update 3.3 is rarely being used with deep neural networks, because practitioners often find it necessary to use a second network, referred to as target network, to stabilize the learning process.

Correctness: The claims are correct based on the given assumptions, but I have already indicated my concern about some of the assumptions in the control case.

Clarity: Notationally the paper can improve. As a concrete example, the quantity Q^{*} has a specific solution-agnostic (as opposed to solution-dependent) meaning in RL, and has been used here (for example in 4.4) to indicate something different than the classic meaning of Q^{*}. But overall, this is a well-written paper.

Relation to Prior Work: Yes.

Reproducibility: Yes

Additional Feedback: I don't see where we need to use Wasserstein 2 as opposed to its more generic form. Maybe to upper bound it easier, or maybe in Lemma 5.2? Can you clarify? It is worthwhile to mention that Assumption 4.1 may not hold in some settings regardless of how large the normalization factor is. The use of MSPBE as opposed to MSBE is being justified due to failure of function approximator to represent TQ, yet you are studying neural networks that are actually universal function approximators, and therefore capable of representing arbitrarily complicated (but I guess Lipschitz) functions. Can you clarify? How do you square the example from Figure 1 of Tsitsiklis and Van Roy with your result? Via over parameterization? And if yes, can you numerically confirm that such a neural net actually converges in the example provided by Tsitsiklis and Van Roy? And/or in Baird's counter example.

Review 3

Summary and Contributions: The paper presents a novel theoretical result regarding the convergence of TD and Q-learning when the action-value function is approximated using a two-layer neural network. Concretely, the authors show that the action-value function converges at a sublinear rate to within a constant factor of optimal, where the constant error O(1/alpha) depends on the scaling factor alpha applied to the output of the neural network. The result follows from a mean-field analysis that converts the TD dynamics in finite-dimensional parameter space to an infinite-dimensional Wasserstein space.

Strengths: To the best of my knowledge, the proofs are correct, and I believe that the result is an important contribution to the fields of reinforcement learning and deep learning, since there are relatively few positive theoretical results about deep learning.

Weaknesses: In my view (and given my background), the main weakness is that some of the proofs are difficult to follow.

Correctness: As mentioned above, I find some proofs hard to follow, but I have not been able to find any flaws, so I believe them to be correct.

Clarity: For the most part the paper is well-written, but sometimes it omits notation and definitions (see my comments below).

Relation to Prior Work: I believe that the authors do an adequate job of discussing related work.

Reproducibility: No

Additional Feedback: In the definition of the continuity equation, what does "div" stand for? And how it is defined? The definition of Q-hat in (3.1) implies that the activation function sigma is only applied in the first layer of the network. (At first I though the output was an *unweighted* linear combination of the intermediate nodes, but each \theta_i can apparently include a parameter for the second layer as well.) How much harder would the problem be to analyze if the second layer also applied an activation function? I guess dimensions D and d should be closely related, e.g. D = d + 2 since each \theta_i usually defines d + 1 parameters for d inputs, plus one parameter for the second layer. I assume the notation \delta_{\theta_i} on line 150 is the Dirac delta? As a result of not knowing how "div" is defined, the transformation from (3.3) to (3.4) is difficult to parse. This transformation is critical to understanding how the proof of Theorem 4.3 works, so I would try to make it more explicit (and this is also the reason I put "No" for reproducibility). In Theorem 4.3, the definition of \rho-bar is rather implicit (it is the distribution corresponding to the fixed point Q^* of MSPBE). Consequently it took me a while to parse what \rho-bar refers to in (4.4). Author feedback: I think the authors did a good job of responding to the concerns of reviewers, and I appreciate the efforts to make the paper and proofs more readable (by including a flowchart and clarifying several definitions).