A Theoretical Analysis of Deep Q-Learning

Zhuoran Yang, Yuchen Xie, Zhaoran Wang

Keywords: reinforcement learning, deep Q network, minimax-Q learning, zero-sum Markov Game

TL;DR: We provide a characterization of the sample complexity of Q-learning and minimax Q-learning with deep neural networks

Abstract: Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on a slight simplification of DQN that fully captures its key features. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by DQN. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players, which is deferred to the appendix due to space limitations.