Two-Timescale Critic-Actor for Average Reward MDPs with Function Approximation

Several recent works have focused on carrying out non-asymptotic convergence analyses for AC algorithms. Recently, a two-timescale critic-actor algorithm has been presented for the discounted cost setting in the look-up table case where the timescales of the actor and the critic are reversed and only asymptotic convergence shown. In our work, we present the first two-timescale critic-actor algorithm with function approximation in the long-run average reward setting and present the first finite-time non-asymptotic as well as asymptotic convergence analysis for such a scheme. We obtain optimal learning rates and prove that our algorithm achieves a sample complexity of {$\mathcal{\tilde{O}}(\epsilon^{-(2+\delta)})$ with $\delta >0$ arbitrarily close to zero,} for the mean squared error of the critic to be upper bounded by $\epsilon$ which is better than the one obtained for two-timescale AC in a similar setting. A notable feature of our analysis is that we present the asymptotic convergence analysis of our scheme in addition to the finite-time bounds that we obtain and show the almost sure asymptotic convergence of the (slower) critic recursion to the attractor of an associated differential inclusion with actor parameters corresponding to local maxima of a perturbed average reward objective. We also show the results of numerical experiments on three benchmark settings and observe that our critic-actor algorithm performs the best amongst all algorithms.