# Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method

## TL;DR

This paper analyzes the iteration complexity of a cubic regularized Newton method for uniformly convex functions, establishing convergence rates and demonstrating its superiority over gradient methods in certain settings.

## Contribution

It introduces a second-order condition number and proves the optimal global complexity bounds for the cubic regularization of Newton method on uniformly convex functions.

## Key findings

- The method achieves linear convergence in nondegenerate cases.
- It automatically attains the best possible complexity bounds.
- Newton method outperforms gradient methods on strongly convex functions.

## Abstract

In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with H\"older continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.