A Theoretical Analysis of Deep Q-Learning

TL;DR

This paper provides the first theoretical analysis of deep Q-learning, establishing convergence rates and justifying key techniques like experience replay and target networks, thus deepening understanding of its empirical success.

Contribution

It offers a theoretical framework for understanding deep Q-learning, including convergence rates and analysis of techniques like experience replay and target networks.

Findings

Algorithmic convergence to zero at a geometric rate.

Statistical error captures bias and variance from neural network approximation.

Extension to Minimax-DQN for zero-sum Markov games.

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. Borrowing the analysis of DQN, we also quantify the difference between the policies obtained by Minimax-DQN and the Nash equilibrium of the Markov game in terms of both the algorithmic and statistical rates of convergence.