Prelimit Coupling and Steady-State Convergence of Constant-stepsize Nonsmooth Contractive SA
This work addresses convergence and bias issues in reinforcement learning algorithms like Q-learning, providing theoretical insights for improving performance, though it is incremental as it builds on existing SA frameworks.
The paper tackles the problem of analyzing nonsmooth contractive stochastic approximation (SA) with constant stepsize, motivated by Q-learning, and establishes weak convergence to a stationary limit distribution, showing that the asymptotic bias is proportional to the square root of the stepsize, enabling bias reduction via Richardson-Romberg extrapolation.
Motivated by Q-learning, we study nonsmooth contractive stochastic approximation (SA) with constant stepsize. We focus on two important classes of dynamics: 1) nonsmooth contractive SA with additive noise, and 2) synchronous and asynchronous Q-learning, which features both additive and multiplicative noise. For both dynamics, we establish weak convergence of the iterates to a stationary limit distribution in Wasserstein distance. Furthermore, we propose a prelimit coupling technique for establishing steady-state convergence and characterize the limit of the stationary distribution as the stepsize goes to zero. Using this result, we derive that the asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize, which stands in sharp contrast to smooth SA. This bias characterization allows for the use of Richardson-Romberg extrapolation for bias reduction in nonsmooth SA.