Geometric Moment Contraction for Stochastic Nesterov Acceleration

Wei Biao Wu

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.