Fast Two-Time-Scale Stochastic Gradient Method with Applications in Reinforcement Learning

Two-time-scale optimization is a framework introduced in Zeng et al. (2024) that abstracts a range of policy evaluation and policy optimization problems in reinforcement learning (RL). Akin to bi-level optimization under a particular type of stochastic oracle, the two-time-scale optimization framework has an upper level objective whose gradient evaluation depends on the solution of a lower level problem, which is to find the root of a strongly monotone operator. In this work, we propose a new method for solving two-time-scale optimization that achieves significantly faster convergence than the prior arts. The key idea of our approach is to leverage an averaging step to improve the estimates of the operators in both lower and upper levels before using them to update the decision variables. These additional averaging steps eliminate the direct coupling between the main variables, enabling the accelerated performance of our algorithm. We characterize the finite-time convergence rates of the proposed algorithm under various conditions of the underlying objective function, including strong convexity, convexity, Polyak-Lojasiewicz condition, and general non-convexity. These rates significantly improve over the best-known complexity of the standard two-time-scale stochastic approximation algorithm. When applied to RL, we show how the proposed algorithm specializes to novel online sample-based methods that surpass or match the performance of the existing state of the art. Finally, we support our theoretical results with numerical simulations in RL.