Geometric Moment Contraction for Stochastic Nesterov Acceleration
Abstract
We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion Yk=Θk+β(Θk−Θk−1), Θk+1=Yk−γG(Yk,Xk+1). Under mean strong monotonicity and stochastic Lp Lipschitz continuity, an explicit Perron comparison proves synchronous Lp contraction when βγLp<(1−β)(1−qγ,p). This direct criterion includes infinite-variance gradients for 1<p<2, but its small-step regime requires β<μ/(μ+Lp). A complementary power-Lyapunov argument establishes a positive, generally much smaller, step-size interval for every fixed β<1 and every p>1, using only a finite pth gradient moment. At p=2, a simpler explicit certificate gives 0<γ<(2μ(1−β)2)/(L22(1−β+2β2)). Its quadratic high-momentum scaling is a limitation of the chosen metric, not a sharp stability boundary. We quantify this loss, provide a general mean-only quadratic S-procedure, and exploit endpoint Lyapunov inequalities under stronger samplewise sector information. Verified endpoint certificates can be orders of magnitude less conservative than the explicit metric.
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BibTeX
@misc{wu2026geometric,
title = {{Geometric Moment Contraction for Stochastic Nesterov Acceleration}},
author = {Wei Biao Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31303},
eprint = {2609.31303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31303},
}
RIS
TY - UNPB
TI - Geometric Moment Contraction for Stochastic Nesterov Acceleration
AU - Wei Biao Wu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31303
UR - https://arxiv.org/abs/2609.31303
AB - We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion Y_k=Θ_k+β(Θ_k−Θ_k−1), Θ_k+1=Y_k−γG(Y_k,X_k+1). Under mean strong monotonicity and stochastic L^p Lipschitz continuity, an explicit Perron comparison proves synchronous L^p contraction when βγL_p<(1−β)(1−q_γ,p). This direct criterion includes infinite-variance gradients for 1<p<2, but its small-step regime requires β<μ/(μ+L_p). A complementary power-Lyapunov argument establishes a positive, generally much smaller, step-size interval for every fixed β<1 and every p>1, using only a finite pth gradient moment. At p=2, a simpler explicit certificate gives 0<γ<(2μ(1−β)^2)/(L_2^2(1−β+2β^2)). Its quadratic high-momentum scaling is a limitation of the chosen metric, not a sharp stability boundary. We quantify this loss, provide a general mean-only quadratic S-procedure, and exploit endpoint Lyapunov inequalities under stronger samplewise sector information. Verified endpoint certificates can be orders of magnitude less conservative than the explicit metric.
ER -