Geometric Variance Reduction in Markov Chains. Application to Value Function and Gradient Estimation

We study a variance reduction technique for Monte Carlo estimation of functionals in Markov chains. The method is based on designing sequential control variates using successive approximations of the function of interest V . Regular Monte Carlo estimates have a variance of O(1/N), where N is the number of sample trajectories of the Markov chain. Here, we obtain a geometric variance reduction O(ρN) (with ρ < 1) up to a threshold that depends on the approximation error V −AV , where A is an approximation operator linear in the values. Thus, if V belongs to the right approximation space (i.e. AV =V ), the variance decreases geometrically to zero. An immediate application is value function estimation in Markov chains, which may be used for policy evaluation in a policy iteration algorithm for solving Markov Decision Processes. Another important domain, for which variance reduction is highly needed, is gradient estimation, that is computing the sensitivity ∂αV of the performance measure V with respect to some parameter α of the transition probabilities. For example, in policy parametric optimization, computing an estimate of the policy gradient is required to perform a gradient optimization method. We show that, using two approximations for the value function and the gradient, a geometric variance reduction is also achieved, up to a threshold that depends on the approximation errors of both of those representations.