# A stochastic first-order method with multi-extrapolated momentum for highly smooth unconstrained optimization

arXiv: https://arxiv.org/abs/2412.14488 · HTML: https://mlsanity.com/paper/2412.14488/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 34 references that are in the map.
- First posted on arXiv 19 Dec 2024.

## Builds on (most cited first, 28 of 28)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization](https://mlsanity.com/paper/2008.10898/) (2020)
- [Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming](https://mlsanity.com/paper/1310.3787/) (2013)
- [Non-convex Finite-Sum Optimization Via SCSG Methods](https://mlsanity.com/paper/1706.09156/) (2017)
- ["Convex Until Proven Guilty": Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions](https://mlsanity.com/paper/1705.02766/) (2017)
- [Sharp Analysis for Nonconvex SGD Escaping from Saddle Points](https://mlsanity.com/paper/1902.00247/) (2019)
- [Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations](https://mlsanity.com/paper/2006.13476/) (2020)
- [Towards understanding how momentum improves generalization in deep learning](https://mlsanity.com/paper/2207.05931/) (2022)
- [ProxSARAH: An Efficient Algorithmic Framework for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/1902.05679/) (2019)
- [A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/1907.03793/) (2019)
- [Stochastic Inexact Augmented Lagrangian Method for Nonconvex Expectation Constrained Optimization](https://mlsanity.com/paper/2212.09513/) (2022)
- [Complexity of Single Loop Algorithms for Nonlinear Programming with Stochastic Objective and Constraints](https://mlsanity.com/paper/2311.00678/) (2023)
- [Momentum-based variance-reduced proximal stochastic gradient method for composite nonconvex stochastic optimization](https://mlsanity.com/paper/2006.00425/) (2020)
- [Variance-reduced first-order methods for deterministically constrained stochastic nonconvex optimization with strong convergence guarantees](https://mlsanity.com/paper/2409.09906/) (2024)
- [Faster Stochastic Alternating Direction Method of Multipliers for Nonconvex Optimization](https://mlsanity.com/paper/2008.01296/) (2020)
- [On the Last Iterate Convergence of Momentum Methods](https://mlsanity.com/paper/2102.07002/) (2021)
- [Non-convex Stochastic Composite Optimization with Polyak Momentum](https://mlsanity.com/paper/2403.02967/) (2024)
- [Optimization Methods for Supervised Machine Learning: From Linear Models to Deep Learning](https://mlsanity.com/paper/1706.10207/) (2017)
- [A Newton-CG based augmented Lagrangian method for finding a second-order stationary point of nonconvex equality constrained optimization with complexity guarantees](https://mlsanity.com/paper/2301.03139/) (2023)
- [Quickly Finding a Benign Region via Heavy Ball Momentum in Non-Convex Optimization](https://mlsanity.com/paper/2010.01449/) (2020)
- [Stochastic interior-point methods for smooth conic optimization with applications](https://mlsanity.com/paper/2412.12987/) (2024)

## Abstract (arXiv)

In this paper, we consider an unconstrained stochastic optimization problem where the objective function exhibits high-order smoothness. Specifically, we propose a new stochastic first-order method (SFOM) with multi-extrapolated momentum, in which multiple extrapolations are performed in each iteration, followed by a momentum update based on these extrapolations. We demonstrate that the proposed SFOM can accelerate optimization by exploiting the high-order smoothness of the objective function f. Assuming that the pth-order derivative of f is Lipschitz continuous for some p≥2, and under additional mild assumptions, we establish that our method achieves a sample complexity of widetildeO(ε-(3p+1)/p) for finding a point x such that mathbbE[|nabla f(x)|]≤ε. To the best of our knowledge, this is the first SFOM to leverage arbitrary-order smoothness of the objective function for acceleration, resulting in a sample complexity that improves upon the best-known results without assuming the mean-squared smoothness condition. Preliminary numerical experiments validate the practical performance of our method and support our theoretical findings.