A Theoretical Analysis of Deep Q-Learning
This work addresses the lack of theoretical foundations in deep reinforcement learning, providing insights for researchers and practitioners in AI and machine learning.
The authors tackled the theoretical understanding of the Deep Q-Network (DQN) algorithm by analyzing its convergence rates for action-value functions, establishing geometric algorithmic error and statistical error from neural network approximation, and justifying key techniques like experience replay and target networks. They also extended this to propose Minimax-DQN for zero-sum Markov games, quantifying its convergence to Nash equilibrium.
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.