On the Convergence and Sample Complexity Analysis of Deep Q-Networks with $\epsilon$-Greedy Exploration

TL;DR

This paper offers the first theoretical convergence and sample complexity analysis of Deep Q-Networks with epsilon-greedy exploration, explaining how exploration parameters influence learning efficiency.

Contribution

It provides a novel theoretical framework analyzing DQNs with practical epsilon-greedy policies, including convergence rates and exploration effects.

Findings

Decaying epsilon leads to geometric convergence to the optimal Q-value.

Higher epsilon values expand the convergence region but slow down learning.

Experimental results support the theoretical analysis.

Abstract

This paper provides a theoretical understanding of Deep Q-Network (DQN) with the $\varepsilon$-greedy exploration in deep reinforcement learning. Despite the tremendous empirical achievement of the DQN, its theoretical characterization remains underexplored. First, the exploration strategy is either impractical or ignored in the existing analysis. Second, in contrast to conventional Q-learning algorithms, the DQN employs the target network and experience replay to acquire an unbiased estimation of the mean-square Bellman error (MSBE) utilized in training the Q-network. However, the existing theoretical analysis of DQNs lacks convergence analysis or bypasses the technical challenges by deploying a significantly overparameterized neural network, which is not computationally efficient. This paper provides the first theoretical convergence and sample complexity analysis of the practical setting of DQNs with $\epsilon$-greedy policy. We prove an iterative procedure with decaying $\epsilon$ converges to the optimal Q-value function geometrically. Moreover, a higher level of $\epsilon$ values enlarges the region of convergence but slows down the convergence, while the opposite holds for a lower level of $\epsilon$ values. Experiments justify our established theoretical insights on DQNs.