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

• URL: http://arxiv.org/abs/2502.09884v1
• Date: Fri, 14 Feb 2025 03:20:30 GMT
• Title: Nonasymptotic CLT and Error Bounds for Two-Time-Scale Stochastic Approximation
• Authors: Seo Taek Kong, Sihan Zeng, Thinh T. Doan, R. Srikant,
• Abstract summary: We consider linear two-time-scale approximation algorithms driven by martingale noise. We derive the first non-asymptotic central limit theorem with respect to the Wasserstein-1 distance for two-time-scale approximation with PolyakRuppert averaging.
• Score: 12.69327994479157
• License:
• 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 analysis focus on either asymptotic convergence in distribution or finite-time bounds that are far from optimal. Prior work on asymptotic central limit theorems (CLTs) suggest that two-time-scale algorithms may be able to achieve $1/\sqrt{n}$ 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 non-asymptotic central limit theorem with respect to the Wasserstein-1 distance for two-time-scale stochastic approximation with Polyak-Ruppert averaging. As a corollary, we show that expected error achieved by Polyak-Ruppert averaging decays at rate $1/\sqrt{n}$, which significantly improves on the rates of convergence in prior works.

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