Convergence of Batch Asynchronous Stochastic Approximation With Applications to Reinforcement Learning

The stochastic approximation (SA) algorithm is a widely used probabilistic method for finding a zero or a fixed point of a vector-valued funtion, when only noisy measurements of the function are available. In the literature to date, one makes a distinction between ``synchronous'' updating, whereby every component of the current guess is updated at each time, and ``asynchronous'' updating, whereby only one component is updated. In this paper, we study an intermediate situation that we call ``batch asynchronous stochastic approximation'' (BASA), in which, at each time instant, \textit{some but not all} components of the current estimated solution are updated. BASA allows the user to trade off memory requirements against time complexity. We develop a general methodology for proving that such algorithms converge to the fixed point of the map under study. These convergence proofs make use of weaker hypotheses than existing results. Specifically, existing convergence proofs require that the measurement noise is a zero-mean i.i.d\ sequence or a martingale difference sequence. In the present paper, we permit biased measurements, that is, measurement noises that have nonzero conditional mean. Also, all convergence results to date assume that the stochastic step sizes satisfy a probabilistic analog of the well-known Robbins-Monro conditions. We replace this assumption by a purely deterministic condition on the irreducibility of the underlying Markov processes. As specific applications to Reinforcement Learning, we analyze the temporal difference algorithm $TD(\lambda)$ for value iteration, and the $Q$-learning algorithm for finding the optimal action-value function. In both cases, we establish the convergence of these algorithms, under milder conditions than in the existing literature.