# Near-Optimal Hyperfast Second-Order Method for convex optimization and its Sliding

arXiv: https://arxiv.org/abs/2002.09050 · HTML: https://mlsanity.com/paper/2002.09050/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2020.

## Built on by (most cited first, 2 of 2)

- [Super-Universal Regularized Newton Method](https://mlsanity.com/paper/2208.05888/) (2022)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)

## Abstract (arXiv)

In this paper, we present a new Hyperfast Second-Order Method with convergence rate O(N-5) up to a logarithmic factor for the convex function with Lipshitz the third derivative. This method based on two ideas. The first comes from the superfast second-order scheme of Yu. Nesterov (CORE Discussion Paper 2020/07, 2020). It allows implementing the third-order scheme by solving subproblem using only the second-order oracle. This method converges with rate O(N-4). The second idea comes from the work of Kamzolov et al. (arXiv:2002.01004). It is the inexact near-optimal third-order method. In this work, we improve its convergence and merge it with the scheme of solving subproblem using only the second-order oracle. As a result, we get convergence rate O(N-5) up to a logarithmic factor. This convergence rate is near-optimal and the best known up to this moment. Further, we investigate the situation when there is a sum of two functions and improve the sliding framework from Kamzolov et al. (arXiv:2002.01004) for the second-order methods.