AI 中文总结
本文研究随机Nesterov加速的几何矩收缩，在均值强单调和随机Lipschitz条件下给出显式收缩准则，并利用Lyapunov论证扩展步长区间，通过端点证书降低保守性。
AI 中文摘要
我们研究了常参数随机Nesterov递归的几何矩收缩（GMC），其形式为 Y_k=Θ_k+β(Θ_k-Θ_{k-1}), Θ_{k+1}=Y_k-γG(Y_k,X_{k+1})。在均值强单调性和随机L^p Lipschitz连续性条件下，一个显式的Perron比较证明了当βγL_p<(1-β)(1-q_{γ,p})时，同步L^p收缩成立。该直接准则涵盖了1<p<2时的无限方差梯度，但其小步长机制要求β<μ/(μ+L_p)。一个互补的幂Lyapunov论证为每个固定的β<1和每个p>1建立了一个正的、通常更小的步长区间，仅使用有限的p阶梯度矩。在p=2时，一个更简单的显式证书给出 0<γ<2μ(1-β)^2/(L_2^2(1-β+2β^2))。其二次高动量缩放是所选度量的局限性，而非严格的稳定性边界。我们量化了这一损失，提供了一个一般的仅均值的二次S过程，并在更强的样本wise扇区信息下利用端点Lyapunov不等式。验证的端点证书可以比显式度量保守性低几个数量级。
英文摘要
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.
