Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximation to Deep Q-Network by Stochastic Delay Differential Equations

Jianya Lu, Yingjun Mo

TL;DR

The paper establishes a diffusion-approximation framework by linking Deep Q-Network (DQN) dynamics to a stochastic differential delay equation (SDDE). It proves a quantitative bound in the Wasserstein-1 metric between the SDDE solution and the DQN iterates that vanishes as the step size $\eta$ tends to zero, enabling a continuous-time interpretation of DQN. By employing a refined Lindeberg principle and operator comparison, the authors separate the roles of experience replay (sampling-induced diffusion) and the target network (delay) in stabilizing learning. This SDDE perspective clarifies variance reduction mechanisms in DQN and provides a foundation for analyzing deep reinforcement learning algorithms via stochastic delay systems.

Abstract

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.

Approximation to Deep Q-Network by Stochastic Delay Differential Equations

TL;DR

The paper establishes a diffusion-approximation framework by linking Deep Q-Network (DQN) dynamics to a stochastic differential delay equation (SDDE). It proves a quantitative bound in the Wasserstein-1 metric between the SDDE solution and the DQN iterates that vanishes as the step size tends to zero, enabling a continuous-time interpretation of DQN. By employing a refined Lindeberg principle and operator comparison, the authors separate the roles of experience replay (sampling-induced diffusion) and the target network (delay) in stabilizing learning. This SDDE perspective clarifies variance reduction mechanisms in DQN and provides a foundation for analyzing deep reinforcement learning algorithms via stochastic delay systems.

Abstract

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.
Paper Structure (19 sections, 11 theorems, 107 equations, 1 algorithm)

This paper contains 19 sections, 11 theorems, 107 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review and motivations
  3. Notations and organization
  4. Background and setting
  5. Main Result
  6. Primary lemmas and the proof of main Theorem
  7. Approximation of internal Markov chain.
  8. Approximation of external Markov chain.
  9. Conclusion
  10. Deep Q-Network Algorithm
  11. Proof of lemmas in section \ref{['Result']}
  12. Proof of lemma \ref{['property1']}
  13. Proof of lemma \ref{['property2']}
  14. Proof of lemmas in section \ref{['proof']}
  15. Proof of Lemma \ref{['Markov']}
  16. ...and 4 more sections

Key Result

Theorem 3.1

Assume that the Assumptions assum:1 and assum:2 hold. Choosing $0<\delta \leqslant 1$ and $\eta\leqslant \min \left\{\delta,\frac{1}{64 L},\frac{L}{8 K^2}\right\}$. Then, for any $T\in \mathbb{N}$, $T>m$, there exists a constant $C_{T,m,A,K, L, d,\beta_{max}, |b(0,0)|}$ such that

Theorems & Definitions (19)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Lemma 4.7
  • ...and 9 more