Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift
Authors: Xinyu Liu, Zixuan Xie, Shangtong Zhang
Organizations: University of Virginia
Abstract
Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as Q-learning and linear temporal difference learning. Specifically, for a power-law learning rate O(n−η) with η∈(1/2,1), we obtain an almost sure convergence rate arbitrarily close to o(n1−2η). For a harmonic learning rate O(n−1), we obtain an almost sure convergence rate arbitrarily close to o(n−1), which we argue is a strong result because it is close to the optimal rate O(n−1loglogn) given by the law of the iterated logarithm (for a special case of i.i.d. noise). Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.
This work studies the convergence of two-timescale stochastic approximations (SA), a class of iterative algorithms that update two sets of parameters in fast and slow timescales respectively. Notable examples of two-timescale SA in reinforcement learning (RL) include temporal difference learning with gradient correction (TDC) and actor-critic methods. Previously, the stability (i.e., boundedness) and convergence of two-timescale SA were only established under i.i.d. noise. This work instead establishes the stability and convergence of two-timescale SA under Markovian noise, a setup that is more realistic in RL. Notably, we do not need to use any projection operator and the noise does not need to live in a compact space. Our key technical novelty is to control the fast timescale parameter with the running max of the slow timescale parameter, instead of with the current slow timescale parameter, as most prior works do. As a key application, we establish the first almost sure convergence of TDC with eligibility traces under off-policy learning with linear function approximation.
We survey Lyapunov-based techniques for the finite-time analysis of stochastic iterative algorithms, also known as stochastic approximation (SA) algorithms, for solving fixed-point equations Fˉ(x)=x, where the operator Fˉ(⋅) can only be accessed through a noisy oracle. We first focus on the standard setting in which Fˉ(⋅) is contractive with respect to some norm and the noise is i.i.d., and explain how generalized Moreau envelopes serve as universal Lyapunov functions, regardless of the underlying norm. We then show how this framework yields mean-square convergence guarantees and applies to stochastic gradient descent, linear SA, and value-based reinforcement learning algorithms such as Q-learning and temporal-difference learning. Finally, we discuss extensions to Markovian noise, seminorm-contractive operators, dissipative operators, and high-probability bounds, and conclude with open problems. The goal is to present a unified and self-contained roadmap for the finite-time analysis of SA and its applications, especially in reinforcement learning.
We establish mean-square and concentration bounds for stochastic approximation (SA) with arbitrary norm contractive mappings, under a multiplicative noise model where the noise may scale affinely with the norm of the iterates, and the iterates are potentially unbounded. These settings arise in reinforcement learning, where operators are often contractive in the ℓ∞ norm and the noise scales with the iterates. To address the arbitrary norm, earlier works replace the non-smooth squared norm with a smooth Lyapunov function constructed via the generalized Moreau envelope. For concentration analysis, these works handle multiplicative noise and unbounded iterates through a multi-stage bootstrapping argument that starts from a time-varying worst-case bound and iteratively refines it. We instead present a unified and elementary analysis that yields both bounds. Using an averaged noise sequence and corresponding auxiliary iterates, we obtain a one-step Lyapunov drift inequality for the normed error directly, without smoothing the norm or constructing an envelope. For the mean-square bound, we combine this drift inequality with an induction argument showing that the iterates remain bounded in expectation. For the concentration bound, we develop a probabilistic induction over a sequence of "good" events on which the iterates are controlled, allowing the standard Azuma-Hoeffding bound to be applied. Our approach yields the first sub-Gaussian tailed maximal (all-time) concentration bound for SA under multiplicative noise, by allowing the stepsize to depend logarithmically on the confidence level. Beyond the specific setting considered here, we discuss the generalizability of these proof techniques to other noise models and iterative algorithms.