Geometric Moment Contraction for Stochastic Nesterov Acceleration

Abstract: We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion [ Y_k=Θk+β(Θ_k-Θ{k-1}),\qquad Θ{k+1}=Y_k-γG(Y_k,X{k+1}). ] Under mean strong monotonicity and stochastic $L<sup>p$ Lipschitz continuity, an explicit Perron comparison proves synchronous $L<sup>p$ contraction when $βγL_p&lt;(1-β)(1-q_{γ,p})$. This direct criterion includes infinite-variance gradients for $1&lt;p\&lt;2$, but its small-step regime requires $β&lt;μ/(μ+L_p)$. A complementary power-Lyapunov argument establishes a positive, generally much smaller, step-size interval for every fixed $$β\&lt;1$$ and every $$p\&gt;1$$, 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.