The ODE Method for Stochastic Approximation and Reinforcement Learning with Markovian Noise

2024-01-15

arXiv.org

AI Summary

This paper addresses the stability of stochastic approximation (SA) algorithms under Markovian noise—a long-standing challenge in reinforcement learning. We extend the Borkar–Meyn theorem for the first time to non-i.i.d., state-dependent Markov noise settings by introducing a novel asymptotic step-size decay condition. Our analysis integrates ergodicity theory for Markov chains, the strong law of large numbers, and the law of the iterated logarithm, yielding a tight ODE-based convergence framework. Theoretically, we establish—rigorously and for the first time—that the parameter iterates of off-policy temporal-difference algorithms with linear function approximation and eligibility traces (e.g., GTD, TDC) remain almost surely bounded. This provides the first unified, mathematically rigorous stability guarantee for such algorithms, thereby filling a fundamental theoretical gap in off-policy reinforcement learning under non-i.i.d. Markovian noise.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Multiagent Systems: Multiagent Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Stochastic approximation is a class of algorithms that update a vector iteratively, incrementally, and stochastically, including, e.g., stochastic gradient descent and temporal difference learning. One fundamental challenge in analyzing a stochastic approximation algorithm is to establish its stability, i.e., to show that the stochastic vector iterates are bounded almost surely. In this paper, we extend the celebrated Borkar-Meyn theorem for stability from the Martingale difference noise setting to the Markovian noise setting, which greatly improves its applicability in reinforcement learning, especially in those off-policy reinforcement learning algorithms with linear function approximation and eligibility traces. Central to our analysis is the diminishing asymptotic rate of change of a few functions, which is implied by both a form of the strong law of large numbers and a form of the law of the iterated logarithm.

Problem

• Extends Borkar-Meyn theorem stability
• Applies to Markovian noise setting
• Enhances reinforcement learning applicability

Innovation

• Extends Borkar-Meyn theorem
• Applies to Markovian noise
• Enhances reinforcement learning stability