Geometric Moment Contraction for Stochastic Nesterov Acceleration

Abstract:We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion \[ Y_k=\Theta_k+\beta(\Theta_k-\Theta_{k-1}),\qquad \Theta_{k+1}=Y_k-\gamma G(Y_k,X_{k+1}). \] Under mean strong monotonicity and stochastic $L^p$ Lipschitz continuity, an explicit Perron comparison proves synchronous $L^p$ contraction when $\beta\gamma L_p<(1-\beta)(1-q_{\gamma,p})$. This direct criterion includes infinite-variance gradients for $1<p<2$, but its small-step regime requires $\beta<\mu/(\mu+L_p)$. A complementary power-Lyapunov argument establishes a positive, generally much smaller, step-size interval for every fixed $\beta<1$ and every $p>1$, using only a finite $p$th gradient moment. At $p=2$, a simpler explicit certificate gives \[ 0<\gamma<\frac{2\mu(1-\beta)^2}{L_2^2(1-\beta+2\beta^2)}. \] 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.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2609.31303 [stat.ML]

(or arXiv:2609.31303v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.31303 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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