Nonasymptotic CLT and Error Bounds for Linear Two-Time-Scale Stochastic Approximation

Abstract:We consider linear two-time-scale stochastic approximation algorithms driven by martingale noise. Recent applications in machine learning motivate the need to understand finite-time error rates, but conventional stochastic approximation analyses focus on either asymptotic convergence in distribution or finite-time bounds that are far from optimal. Prior work on asymptotic central limit theorems (CLTs) suggests that two-time-scale algorithms may be able to achieve $1/\sqrt{K}$ error in expectation, with a constant given by the expected norm of the limiting Gaussian vector. However, the best known finite-time rates are much slower. We derive the first nonasymptotic Wasserstein-1 CLT for linear two-time-scale stochastic approximation with Polyak-Ruppert averaging driven by martingale difference noise. As a corollary, we show that the expected error achieved by Polyak-Ruppert averaging decays at rate $1/\sqrt{K}$, which significantly improves on the rates of convergence in prior works.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2502.09884 [cs.LG]

(or arXiv:2502.09884v4 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2502.09884 Focus to learn more arXiv-issued DOI via DataCite

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