# Geometric Moment Contraction for Stochastic Nesterov Acceleration

arXiv: https://arxiv.org/abs/2609.31303 · HTML: https://mlsanity.com/paper/2609.31303/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 109 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimization](https://mlsanity.com/paper/1604.03257/) (2016)
- [Convergence and concentration properties of constant step-size SGD through Markov chains](https://mlsanity.com/paper/2306.11497/) (2023)
- [Acceleration of stochastic gradient descent with momentum by averaging: finite-sample rates and asymptotic normality](https://mlsanity.com/paper/2305.17665/) (2023)
- [Revisiting the central limit theorems for the SGD-type methods](https://mlsanity.com/paper/2207.11755/) (2022)
- [Nesterov acceleration in benignly non-convex landscapes](https://mlsanity.com/paper/2410.08395/) (2024)
- [Limit Theorems for Stochastic Gradient Descent with Infinite Variance](https://mlsanity.com/paper/2410.16340/) (2024)
- [The Anytime Convergence of Stochastic Gradient Descent with Momentum: From a Continuous-Time Perspective](https://mlsanity.com/paper/2310.19598/) (2023)
- [SHANG++: Robust Stochastic Acceleration under Multiplicative Noise](https://mlsanity.com/paper/2603.09355/) (2026)
- [Algorithmic Stability of Stochastic Gradient Descent with Momentum under Heavy-Tailed Noise](https://mlsanity.com/paper/2502.00885/) (2025)
- [Statistical Inference for Stochastic Gradient Descent: Beyond Finite Variance](https://mlsanity.com/paper/2605.26000/) (2026)
- [A Polyak-Ruppert Central Limit Theorem for SA-Adam with Momentum and Non-Convergent Adaptive Preconditioning](https://mlsanity.com/paper/2606.17364/) (2026)
- [Delayed supermartingale convergence lemmas for stochastic approximation with Nesterov momentum](https://mlsanity.com/paper/2406.06018/) (2024)
- [Generalized Stochastic Gradient Descent with Momentum Methods for Smooth Optimization](https://mlsanity.com/paper/2602.23444/) (2026)
- [Stochastic Gradient Descent with Momentum is Algorithmically Stable](https://mlsanity.com/paper/2605.28517/) (2026)
- [Strong error analysis for the stochastic momentum optimizer](https://mlsanity.com/paper/2608.04245/) (2026)

## Abstract (arXiv)

We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion [ Yk=Thetak+β(Thetak-Thetak-1),qquad Thetak+1=Yk-γ G(Yk,Xk+1). ] Under mean strong monotonicity and stochastic Lp Lipschitz continuity, an explicit Perron comparison proves synchronous Lp contraction when βγ Lp<(1-β)(1-qγ,p). This direct criterion includes infinite-variance gradients for 1<p<2, but its small-step regime requires β<μ/(μ+Lp). 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 pth gradient moment. At p=2, a simpler explicit certificate gives [ 0<γ<frac2μ(1-β)2L22(1-β+2β2). ] 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.