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.

翻译：随机逼近是一类以迭代、增量及随机方式更新向量的算法，例如随机梯度下降和时间差分学习。分析随机逼近算法的一个基本挑战在于确立其稳定性，即证明随机向量迭代几乎必然有界。本文将著名的Borkar-Meyn稳定性定理从鞅差噪声场景拓展至马尔可夫噪声场景，这显著提升了该定理在强化学习中的适用性，尤其适用于那些采用线性函数逼近和资格迹的离策略强化学习算法。我们分析的核心在于若干函数变化率的渐近衰减性质，该性质可由某种形式的强大数定律及常用的V4李雅普诺夫漂移条件共同导出，当马尔可夫链有限且不可约时此性质自然成立。