# A Note on Inexact Condition for Cubic Regularized Newton's Method

arXiv: https://arxiv.org/abs/1808.07384 · HTML: https://mlsanity.com/paper/1808.07384/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Aug 2018.

## Built on by (most cited first, 4 of 4)

- [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)
- [Cubic regularized subspace Newton for non-convex optimization](https://mlsanity.com/paper/2406.16666/) (2024)
- [Second-Order Optimization with Asynchronous Curvature](https://mlsanity.com/paper/2510.15714/) (2025)

## Abstract (arXiv)

This note considers the inexact cubic-regularized Newton's method (CR), which has been shown in citeCartis2011a to achieve the same order-level convergence rate to a secondary stationary point as the exact CR citepNesterov2006. However, the inexactness condition in citeCartis2011a is not implementable due to its dependence on future iterates variable. This note fixes such an issue by proving the same convergence rate for nonconvex optimization under an inexact adaptive condition that depends on only the current iterate. Our proof controls the sufficient decrease of the function value over the total iterations rather than each iteration as used in the previous studies, which can be of independent interest in other contexts.