A Finite-Sample Analysis of an Actor-Critic Algorithm for Mean-Variance Optimization in a Discounted MDP

Tejaram Sangadi, Prashanth L. A., Krishna Jagannathan

Keywords: Temporal Difference (TD) Learning, Reinforcement Learning (RL), Markov Decision Process (MDP), Variance, Risk-Sensitive RL, Actor-Critic Methods, Simultaneous Perturbation Stochastic Approximation (SPSA), Finite-Sample Analysis, Mean Square Error (MSE) Bounds, High-Probability Bounds

TL;DR: We establish finite-sample bounds for a TD-based critic and an SPSA-based actor in mean-variance optimization for discounted MDPs.

Abstract: Motivated by applications in risk-sensitive reinforcement learning, we study mean-variance optimization in a discounted reward Markov Decision Process (MDP). Specifically, we analyze a Temporal Difference (TD) learning algorithm with linear function approximation (LFA) for policy evaluation. We derive finite-sample bounds that hold (i) in the mean-squared sense and (ii) with high probability under tail iterate averaging, both with and without regularization. Our bounds exhibit an exponentially decaying dependence on the initial error and a convergence rate of $O(1/t)$ after $t$ iterations. Moreover, for the regularized TD variant, our bound holds for a universal step size. Next, we integrate a Simultaneous Perturbation Stochastic Approximation (SPSA)-based actor update with an LFA critic and establish an $O(n^{-1/4})$ convergence guarantee, where $n$ denotes the iterations of the SPSA-based actor-critic algorithm. These results establish finite-sample theoretical guarantees for risk-sensitive actor-critic methods in reinforcement learning, with a focus on variance as a risk measure.

Submission Number: 107