Finite-Time Decoupled Convergence in Nonlinear Two-Time-Scale Stochastic Approximation
arxiv(2024)
摘要
In two-time-scale stochastic approximation (SA), two iterates are updated at
varying speeds using different step sizes, with each update influencing the
other. Previous studies in linear two-time-scale SA have found that the
convergence rates of the mean-square errors for these updates are dependent
solely on their respective step sizes, leading to what is referred to as
decoupled convergence. However, the possibility of achieving this decoupled
convergence in nonlinear SA remains less understood. Our research explores the
potential for finite-time decoupled convergence in nonlinear two-time-scale SA.
We find that under a weaker Lipschitz condition, traditional analyses are
insufficient for achieving decoupled convergence. This finding is further
numerically supported by a counterexample. But by introducing an additional
condition of nested local linearity, we show that decoupled convergence is
still feasible, contingent on the appropriate choice of step sizes associated
with smoothness parameters. Our analysis depends on a refined characterization
of the matrix cross term between the two iterates and utilizes fourth-order
moments to control higher-order approximation errors induced by the local
linearity assumption.
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