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Asymptotic and Finite Sample Analysis of Nonexpansive Stochastic Approximations with Markovian Noise

Ethan Blaser, Shangtong Zhang

TL;DR

This work investigates stochastic approximations with merely nonexpansive operators, and proves for the first time that classical tabular average reward temporal difference learning converges to a sample-path dependent fixed point.

Abstract

Stochastic approximation is a powerful class of algorithms with celebrated success. However, a large body of previous analysis focuses on stochastic approximations driven by contractive operators, which is not applicable in some important reinforcement learning settings like the average reward setting. This work instead investigates stochastic approximations with merely nonexpansive operators. In particular, we study nonexpansive stochastic approximations with Markovian noise, providing both asymptotic and finite sample analysis. Key to our analysis are novel bounds of noise terms resulting from the Poisson equation. As an application, we prove for the first time that classical tabular average reward temporal difference learning converges to a sample-path dependent fixed point.

Asymptotic and Finite Sample Analysis of Nonexpansive Stochastic Approximations with Markovian Noise

TL;DR

This work investigates stochastic approximations with merely nonexpansive operators, and proves for the first time that classical tabular average reward temporal difference learning converges to a sample-path dependent fixed point.

Abstract

Stochastic approximation is a powerful class of algorithms with celebrated success. However, a large body of previous analysis focuses on stochastic approximations driven by contractive operators, which is not applicable in some important reinforcement learning settings like the average reward setting. This work instead investigates stochastic approximations with merely nonexpansive operators. In particular, we study nonexpansive stochastic approximations with Markovian noise, providing both asymptotic and finite sample analysis. Key to our analysis are novel bounds of noise terms resulting from the Poisson equation. As an application, we prove for the first time that classical tabular average reward temporal difference learning converges to a sample-path dependent fixed point.
Paper Structure (29 sections, 29 theorems, 190 equations, 1 table)

This paper contains 29 sections, 29 theorems, 190 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notations
  3. Asymptotic Analysis of SKM Iterations
  4. Finite Sample Analysis of SKM Iterations
  5. Application in Average Reward Temporal Difference Learning
  6. Reinforcement Learning Background
  7. Average Reward Temporal Difference Learning
  8. Significance of Theorem 4.2
  9. Related Work
  10. ODE and Lyapunov Methods for Asymptotic Convergence
  11. Average Reward RL
  12. Conclusion
  13. Acknowledgments
  14. Mathematical Background
  15. Additional Lemmas from Section \ref{['sec:SA']}
  16. ...and 14 more sections

Key Result

Theorem 2.6

Let Assumptions as:steadystate - as:e1 hold. Then the iterates $\qty{x_n}$ generated by eq:skm_markov satisfy where $x_* \in \mathcal{X}_*$ is a possibly sample-path dependent fixed point. Or more precisely speaking, let $\omega$ denote a sample path $(w_0, Y_0, Y_1, \dots)$ and write $x_n(\omega)$ to emphasize the dependence of $x_n$ on $\omega$. Then there exists a set $\Omega$ of sample paths

Theorems & Definitions (63)

  • Theorem 2.6
  • proof
  • Remark 2.7
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Lemma A.1: Theorem 2.1 from bravo2024stochastic
  • ...and 53 more