# Better SGD using Second-order Momentum

arXiv: https://arxiv.org/abs/2103.03265 · HTML: https://mlsanity.com/paper/2103.03265/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 40 references that are in the map.
- First posted on arXiv 4 Mar 2021.

## Builds on (most cited first, 21 of 21)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [fairseq: A Fast, Extensible Toolkit for Sequence Modeling](https://mlsanity.com/paper/1904.01038/) (2019)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [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)
- [ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning](https://mlsanity.com/paper/2006.00719/) (2020)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [Stochastic Nested Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1806.07811/) (2018)
- [Stochastic Cubic Regularization for Fast Nonconvex Optimization](https://mlsanity.com/paper/1711.02838/) (2017)
- [On the Ineffectiveness of Variance Reduced Optimization for Deep Learning](https://mlsanity.com/paper/1812.04529/) (2018)
- [A Progressive Batching L-BFGS Method for Machine Learning](https://mlsanity.com/paper/1802.05374/) (2018)
- [Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information](https://mlsanity.com/paper/1708.07164/) (2017)
- [Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations](https://mlsanity.com/paper/2006.13476/) (2020)
- [Second-Order Optimization for Non-Convex Machine Learning: An Empirical Study](https://mlsanity.com/paper/1708.07827/) (2017)
- [Apollo: An Adaptive Parameter-wise Diagonal Quasi-Newton Method for Nonconvex Stochastic Optimization](https://mlsanity.com/paper/2009.13586/) (2020)
- [A Limited-Memory Quasi-Newton Algorithm for Bound-Constrained Nonsmooth Optimization](https://mlsanity.com/paper/1612.07350/) (2016)

## Built on by (most cited first, 15 of 15)

- [Stochastic Policy Gradient Methods: Improved Sample Complexity for Fisher-non-degenerate Policies](https://mlsanity.com/paper/2302.01734/) (2023)
- [Order-Optimal Regret with Novel Policy Gradient Approaches in Infinite-Horizon Average Reward MDPs](https://mlsanity.com/paper/2404.02108/) (2024)
- [Momentum-Based Policy Gradient with Second-Order Information](https://mlsanity.com/paper/2205.08253/) (2022)
- [Momentum Aggregation for Private Non-convex ERM](https://mlsanity.com/paper/2210.06328/) (2022)
- [Global Convergence Guarantees for Federated Policy Gradient Methods with Adversaries](https://mlsanity.com/paper/2403.09940/) (2024)
- [Improving Stochastic Cubic Newton with Momentum](https://mlsanity.com/paper/2410.19644/) (2024)
- [High Probability Analysis for Non-Convex Stochastic Optimization with Clipping](https://mlsanity.com/paper/2307.13680/) (2023)
- [Reevaluating Theoretical Analysis Methods for Optimization in Deep Learning](https://mlsanity.com/paper/2407.01825/) (2024)
- [Second-order Optimization under Heavy-Tailed Noise: Hessian Clipping and Sample Complexity Limits](https://mlsanity.com/paper/2510.10690/) (2025)
- [Accelerated Stochastic Min-Max Optimization Based on Bias-corrected Momentum](https://mlsanity.com/paper/2406.13041/) (2024)
- [Torque-Aware Momentum](https://mlsanity.com/paper/2412.18790/) (2024)
- [Improved Convergence in Parameter-Agnostic Error Feedback through Momentum](https://mlsanity.com/paper/2511.14501/) (2025)
- [RanSOM: Second-Order Momentum with Randomized Scaling for Constrained and Unconstrained Optimization](https://mlsanity.com/paper/2602.06824/) (2026)
- [NysAct: A Scalable Preconditioned Gradient Descent using Nystrom Approximation](https://mlsanity.com/paper/2506.08360/) (2025)
- [Randomized Momentum Transport with Cheap Predictors](https://mlsanity.com/paper/2610.13789/) (2026)

## Abstract (arXiv)

We develop a new algorithm for non-convex stochastic optimization that finds an ε-critical point in the optimal O(ε-3) stochastic gradient and Hessian-vector product computations. Our algorithm uses Hessian-vector products to "correct" a bias term in the momentum of SGD with momentum. This leads to better gradient estimates in a manner analogous to variance reduction methods. In contrast to prior work, we do not require excessively large batch sizes, and are able to provide an adaptive algorithm whose convergence rate automatically improves with decreasing variance in the gradient estimates. We validate our results on a variety of large-scale deep learning architectures and benchmarks tasks.