Geometric Moment Contraction for Stochastic Nesterov Acceleration

2026-09-25

AI Summary

This study addresses the overly conservative convergence analysis of stochastic Nesterov accelerated algorithms under gradient noise with potentially infinite variance, which typically arises from high-momentum metrics. To overcome this limitation, we investigate the geometric moment contraction of constant-parameter recursions by integrating the Perron comparison method, power Lyapunov functions, and the S-procedure technique. This framework yields the first direct contraction criteria and introduces non-conservative certificates based on endpoint Lyapunov inequalities. We establish synchronous Lp contraction conditions and derive explicit step-size intervals alongside higher-order moment convergence requirements. Notably, the proposed endpoint certificates reduce conservatism by several orders of magnitude compared to conventional approaches, substantially expanding the range of stabilizing step sizes.

Abstract

We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion \[ Y_k=Θ_k+β(Θ_k-Θ_{k-1}),\qquad Θ_{k+1}=Y_k-γG(Y_k,X_{k+1}). \] 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 $p$th gradient moment. At $p=2$, a simpler explicit certificate gives \[ 0<γ<\frac{2μ(1-β)^2}{L_2^2(1-β+2β^2)}. \] 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.

Problem

• Geometric Moment Contraction
• Stochastic Nesterov Acceleration
• $L^p$ contraction
• Lyapunov stability
• step-size bounds

Innovation

• Geometric Moment Contraction
• Stochastic Nesterov Acceleration
• Power-Lyapunov Argument
• S-procedure
• Infinite-variance Gradients