Geometric Moment Contraction for Stochastic Nesterov Acceleration

arXiv:2609.31303v1 Announce Type: cross Abstract: We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion [ Y_k=Theta_k+eta(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 etagamma L_p1, using only a finite pth gradient moment. At p=2, a simpler explicit certificate gives [ 0<gamma<frac{2mu(1-eta)^2}{L_2^2(1-eta+2eta^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.

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Source: arXiv cs.LG | 2026-09-28

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