Approximation to Deep Q-Network by Stochastic Delay Differential Equations

Despite the significant breakthroughs that the Deep Q-Network (DQN) has brought to reinforcement learning, its theoretical analysis remains limited. In this paper, we construct a stochastic differential delay equation (SDDE) based on the DQN algorithm and estimate the Wasserstein-1 distance between them. We provide an upper bound for the distance and prove that the distance between the two converges to zero as the step size approaches zero. This result allows us to understand DQN's two key techniques, the experience replay and the target network, from the perspective of continuous systems. Specifically, the delay term in the equation, corresponding to the target network, contributes to the stability of the system. Our approach leverages a refined Lindeberg principle and an operator comparison to establish these results.