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

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

#### Authors

*Yufeng Zhang, Qi Cai, Zhuoran Yang, Yongxin Chen, Zhaoran Wang*

#### Abstract

Temporal-diﬀerence and Q-learning play a key role in deep reinforcement learning, where they are empowered by expressive nonlinear function approximators such as neural networks. At the core of their empirical successes is the learned feature representation, which embeds rich observations, e.g., images and texts, into the latent space that encodes semantic structures. Meanwhile, the evolution of such a feature representation is crucial to the convergence of temporal-diﬀerence and Q-learning.

In particular, temporal-diﬀerence learning converges when the function approximator is linear in a feature representation, which is ﬁxed throughout learning, and possibly diverges otherwise. We aim to answer the following questions: When the function approximator is a neural network, how does the associated feature representation evolve? If it converges, does it converge to the optimal one?

We prove that utilizing an overparameterized two-layer neural network, temporal-diﬀerence and Q-learning globally minimize the mean-squared projected Bellman error at a sublinear rate. Moreover, the associated feature representation converges to the optimal one, generalizing the previous analysis of Cai et al. (2019) in the neural tangent kernel regime, where the associated feature representation stabilizes at the initial one. The key to our analysis is a mean-ﬁeld perspective, which connects the evolution of a ﬁnite-dimensional parameter to its limiting counterpart over an inﬁnite-dimensional Wasserstein space. Our analysis generalizes to soft Q-learning, which is further connected to policy gradient.