# Sub-sampled Cubic Regularization for Non-convex Optimization

## TL;DR

This paper introduces a sub-sampled cubic regularization method for non-convex optimization that reduces computational costs while maintaining strong convergence guarantees, supported by theoretical analysis and experiments.

## Contribution

It proposes the first global convergence guarantees for a sub-sampled cubic regularization method in non-convex optimization.

## Key findings

- Achieves computational efficiency through sub-sampling.
- Maintains strong convergence guarantees.
- Experimental results support theoretical claims.

## Abstract

We consider the minimization of non-convex functions that typically arise in machine learning. Specifically, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence guarantees than first- and second-order as well as classical trust region methods. However, it suffers from a high computational complexity that makes it impractical for large-scale learning. Here, we propose a novel method that uses sub-sampling to lower this computational cost. By the use of concentration inequalities we provide a sampling scheme that gives sufficiently accurate gradient and Hessian approximations to retain the strong global and local convergence guarantees of cubically regularized methods. To the best of our knowledge this is the first work that gives global convergence guarantees for a sub-sampled variant of cubic regularization on non-convex functions. Furthermore, we provide experimental results supporting our theory.