Finite-Time Analysis of Projected Two-Time-Scale Stochastic Approximation

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Hauptverfasser: Bai, Yitao, Doan, Thinh T., Romberg, Justin
Format: Preprint
Veröffentlicht: 2026
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author Bai, Yitao
Doan, Thinh T.
Romberg, Justin
author_facet Bai, Yitao
Doan, Thinh T.
Romberg, Justin
contents We study the finite-time convergence of projected linear two-time-scale stochastic approximation with constant step sizes and Polyak--Ruppert averaging. We establish an explicit mean-square error bound, decomposing it into two interpretable components, an approximation error determined by the constrained subspace and a statistical error decaying at a sublinear rate, with constants expressed through restricted stability margins and a coupling invertibility condition. These constants cleanly separate the effect of subspace choice (approximation errors) from the effect of the averaging horizon (statistical errors). We illustrate our theoretical results through a number of numerical experiments on both synthetic and reinforcement learning problems.
format Preprint
id arxiv_https___arxiv_org_abs_2604_00179
institution arXiv
publishDate 2026
record_format arxiv
spellingShingle Finite-Time Analysis of Projected Two-Time-Scale Stochastic Approximation
Bai, Yitao
Doan, Thinh T.
Romberg, Justin
Systems and Control
Machine Learning
62L20, 93E35
We study the finite-time convergence of projected linear two-time-scale stochastic approximation with constant step sizes and Polyak--Ruppert averaging. We establish an explicit mean-square error bound, decomposing it into two interpretable components, an approximation error determined by the constrained subspace and a statistical error decaying at a sublinear rate, with constants expressed through restricted stability margins and a coupling invertibility condition. These constants cleanly separate the effect of subspace choice (approximation errors) from the effect of the averaging horizon (statistical errors). We illustrate our theoretical results through a number of numerical experiments on both synthetic and reinforcement learning problems.
title Finite-Time Analysis of Projected Two-Time-Scale Stochastic Approximation
topic Systems and Control
Machine Learning
62L20, 93E35
url https://arxiv.org/abs/2604.00179