Convergence Rate in a Nonlinear Two-Time-Scale Stochastic Approximation with State (Time)-Dependence

Abstract

The nonlinear two-time-scale stochastic approximation is widely studied under conditions of bounded variances in noise. Motivated by recent advances that allow for variability linked to the current state or time, we consider state- and time-dependent noises. We show that the Lyapunov function exhibits polynomial convergence rates in both cases, with the rate of polynomial delay depending on the parameters of state- or time-dependent noises. Notably, if the state noise parameters fully approach their limiting value, the Lyapunov function achieves an exponential convergence rate. We provide two numerical examples to illustrate our theoretical findings in the context of stochastic gradient descent with Polyak-Ruppert averaging and stochastic bilevel optimization.