Stability of Q-Learning Through Design and Optimism

TL;DR

This paper explores the stability of Q-learning algorithms, introduces new methods for ensuring stability and accelerated convergence, and presents the Zap Zero algorithm as a general stochastic approximation approach applicable to various settings.

Contribution

It demonstrates stability of Q-learning with linear function approximation using optimistic training and introduces the Zap Zero algorithm for stable, matrix-free Newton-Raphson flow approximation.

Findings

Stability of Q-learning with linear approximation established under optimistic training.

The Zap Zero algorithm is stable and convergent under mild assumptions.

Applicable to Q-learning with non-linear function approximation and oblivious training.

Abstract

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.