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

• URL: http://arxiv.org/abs/2401.07844v5
• Date: Thu, 11 Jul 2024 15:51:39 GMT
• Title: The ODE Method for Stochastic Approximation and Reinforcement Learning with Markovian Noise
• Authors: Shuze Liu, Shuhang Chen, Shangtong Zhang,
• Abstract summary: One fundamental challenge in analyzing an approximation algorithm is to establish its stability. In this paper, we extend the celebrated Borkar-Meyn theorem for stability bounded from the Martingale difference noise setting to the Markovian noise setting. Central to our analysis is the diminishing rate of change of a few functions, which is implied by both a form of strong law of large numbers and a commonly used V4 Lynov drift condition.
• Score: 17.493808856903303
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 strong law of large numbers and a commonly used V4 Lyapunov drift condition and trivially holds if the Markov chain is finite and irreducible.

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