  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.
