# A Nonlinear Time-Series Theory for Constant-step SGD with Momentum

arXiv: https://arxiv.org/abs/2610.16034 · HTML: https://mlsanity.com/paper/2610.16034/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 31 references that are in the map.
- First posted on arXiv 12 Oct 2026.

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

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration](https://mlsanity.com/paper/2004.04719/) (2020)
- [On the insufficiency of existing momentum schemes for Stochastic Optimization](https://mlsanity.com/paper/1803.05591/) (2018)
- [Understanding the Role of Momentum in Stochastic Gradient Methods](https://mlsanity.com/paper/1910.13962/) (2019)
- [Stochastic Heavy Ball](https://mlsanity.com/paper/1609.04228/) (2016)
- [Trajectory of Mini-Batch Momentum: Batch Size Saturation and Convergence in High Dimensions](https://mlsanity.com/paper/2206.01029/) (2022)
- [Accelerated Linear Convergence of Stochastic Momentum Methods in Wasserstein Distances](https://mlsanity.com/paper/1901.07445/) (2019)
- [Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization](https://mlsanity.com/paper/2111.06328/) (2021)
- [On the fast convergence of minibatch heavy ball momentum](https://mlsanity.com/paper/2206.07553/) (2022)
- [Provable non-accelerations of the heavy-ball method](https://mlsanity.com/paper/2307.11291/) (2023)
- [Momentum Further Constrains Sharpness at the Edge of Stochastic Stability](https://mlsanity.com/paper/2604.14108/) (2026)
- [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)
- [(Accelerated) Noise-adaptive Stochastic Heavy-Ball Momentum](https://mlsanity.com/paper/2401.06738/) (2024)

## Abstract (arXiv)

Constant-step stochastic gradient descent with momentum (SGDM) augments the parameter recursion with a momentum state that carries geometrically decaying memory of past stochastic gradients. We study this coupled recursion as a nonlinear time series and develop a framework for its stationary behavior and distributional limits. The joint parameter-momentum process forgets its initialization geometrically under a finite pth moment for p > 1, yielding a unique stationary causal solution. Under a second-moment condition, this contraction leads to an invariance principle and quenched Gaussian limit for averaged SGDM, while a small-learning-rate analysis gives a Gaussian limit for the last iterate. In a stochastic-curvature quadratic model, we further study how the rate of initialization forgetting relates to stationary variance and averaged fluctuations.