Q-learning has become an important part of the reinforcement learning toolkit since its introduction in the dissertation of Chris Watkins in the 1980s. The purpose of this paper is in part a tutorial on stochastic approximation and Q-learning, providing details regarding the INFORMS APS inaugural Applied Probability Trust Plenary Lecture, presented in Nancy France, June 2023. The paper also presents new approaches to ensure stability and potentially accelerated convergence for these algorithms, and stochastic approximation in other settings. Two contributions are entirely new: 1. Stability of Q-learning with linear function approximation has been an open topic for research for over three decades. It is shown that with appropriate optimistic training in the form of a modified Gibbs policy, there exists a solution to the projected Bellman equation, and the algorithm is stable (in terms of bounded parameter estimates). Convergence remains one of many open topics for research. 2. The new Zap Zero algorithm is designed to approximate the Newton-Raphson flow without matrix inversion. It is stable and convergent under mild assumptions on the mean flow vector field for the algorithm, and compatible statistical assumption on an underlying Markov chain. The algorithm is a general approach to stochastic approximation which in particular applies to Q-learning with "oblivious" training even with non-linear function approximation.

该论文介绍了Q-learning在强化学习工具中的重要性，提供了随机逼近和Q-learning的教程，并介绍了确保算法稳定性和加速收敛的新方法。其中两个新的贡献是解决了Q-learning中线性函数逼近的稳定性问题，以及设计了一种近似牛顿-拉普森流动的算法。

Q-learning设计与乐观性的稳定性