Almost Sure Convergence of Linear Temporal Difference Learning with Arbitrary Features

Jiuqi Wang, Shangtong Zhang
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Abstract

Temporal difference (TD) learning with linear function approximation, abbreviated as linear TD, is a classic and powerful prediction algorithm in reinforcement learning. While it is well understood that linear TD converges almost surely to a unique point, this convergence traditionally requires the assumption that the features used by the approximator are linearly independent. However, this linear independence assumption does not hold in many practical scenarios. This work is the first to establish the almost sure convergence of linear TD without requiring linearly independent features. In fact, we do not make any assumptions on the features. We prove that the approximated value function converges to a unique point and the weight iterates converge to a set. We also establish a notion of local stability of the weight iterates. Importantly, we do not need to introduce any other additional assumptions and do not need to make any modification to the linear TD algorithm. Key to our analysis is a novel characterization of bounded invariant sets of the mean ODE of linear TD.
具有任意特征的线性时差学习的几乎确定收敛性
带有线性函数近似的时差(TD)学习,简称线性 TD,是强化学习中一种经典而强大的预测算法。虽然人们都知道线性 TD 几乎肯定会收敛到一个唯一点,但这种收敛传统上需要假设近似器使用的特征是线性独立的。这项工作首次在不要求线性独立特征的情况下,建立了线性 TD 几乎确定的收敛性。事实上,我们对特征不做任何假设。我们证明了近似值函数会收敛到一个唯一点,权重迭代会收敛到一个集合。我们还建立了权重迭代的局部稳定性概念。我们分析的关键是线性 TD 平均 ODE 有界不变集的新特征。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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