# Cubic Regularization with Momentum for Nonconvex Optimization

arXiv: https://arxiv.org/abs/1810.03763 · HTML: https://mlsanity.com/paper/1810.03763/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2018.

## Built on by (most cited first, 8 of 8)

- [GT-STORM: Taming Sample, Communication, and Memory Complexities in Decentralized Non-Convex Learning](https://mlsanity.com/paper/2105.01231/) (2021)
- [Improving Stochastic Cubic Newton with Momentum](https://mlsanity.com/paper/2410.19644/) (2024)
- [Escaping Saddle Points in Distributed Newton's Method with Communication Efficiency and Byzantine Resilience](https://mlsanity.com/paper/2103.09424/) (2021)
- [A Momentum Accelerated Adaptive Cubic Regularization Method for Nonconvex Optimization](https://mlsanity.com/paper/2210.05987/) (2022)
- [Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization](https://mlsanity.com/paper/2205.03059/) (2022)
- [Approximate Secular Equations for the Cubic Regularization Subproblem](https://mlsanity.com/paper/2209.13268/) (2022)
- [Series of Hessian-Vector Products for Tractable Saddle-Free Newton Optimisation of Neural Networks](https://mlsanity.com/paper/2310.14901/) (2023)
- [AdaCubic: An Adaptive Cubic Regularization Optimizer for Deep Learning](https://mlsanity.com/paper/2604.09437/) (2026)

## Abstract (arXiv)

Momentum is a popular technique to accelerate the convergence in practical training, and its impact on convergence guarantee has been well-studied for first-order algorithms. However, such a successful acceleration technique has not yet been proposed for second-order algorithms in nonconvex optimization.In this paper, we apply the momentum scheme to cubic regularized (CR) Newton's method and explore the potential for acceleration. Our numerical experiments on various nonconvex optimization problems demonstrate that the momentum scheme can substantially facilitate the convergence of cubic regularization, and perform even better than the Nesterov's acceleration scheme for CR. Theoretically, we prove that CR under momentum achieves the best possible convergence rate to a second-order stationary point for nonconvex optimization. Moreover, we study the proposed algorithm for solving problems satisfying an error bound condition and establish a local quadratic convergence rate. Then, particularly for finite-sum problems, we show that the proposed algorithm can allow computational inexactness that reduces the overall sample complexity without degrading the convergence rate.