# Second-Order Methods with Cubic Regularization Under Inexact Information

arXiv: https://arxiv.org/abs/1710.05782 · HTML: https://mlsanity.com/paper/1710.05782/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2017.

## Built on by (most cited first, 5 of 5)

- [Unified Convergence Theory of Stochastic and Variance-Reduced Cubic Newton Methods](https://mlsanity.com/paper/2302.11962/) (2023)
- [Improving Stochastic Cubic Newton with Momentum](https://mlsanity.com/paper/2410.19644/) (2024)
- [A Cubic Regularization Approach for Finding Local Minimax Points in Nonconvex Minimax Optimization](https://mlsanity.com/paper/2110.07098/) (2021)
- [Second-order Optimization under Heavy-Tailed Noise: Hessian Clipping and Sample Complexity Limits](https://mlsanity.com/paper/2510.10690/) (2025)
- [Simple Stepsize for Quasi-Newton Methods with Global Convergence Guarantees](https://mlsanity.com/paper/2508.19712/) (2025)

## Abstract (arXiv)

In this paper, we generalize (accelerated) Newton's method with cubic regularization under inexact second-order information for (strongly) convex optimization problems. Under mild assumptions, we provide global rate of convergence of these methods and show the explicit dependence of the rate of convergence on the problem parameters. While the complexity bounds of our presented algorithms are theoretically worse than those of their exact counterparts, they are at least as good as those of the optimal first-order methods. Our numerical experiments also show that using inexact Hessians can significantly speed up the algorithms in practice.