Stochastic Variance-Reduced Cubic Regularized Newton Method

TL;DR

This paper introduces a stochastic variance-reduced cubic regularized Newton method for non-convex optimization, achieving faster convergence guarantees and demonstrating effectiveness through experiments.

Contribution

It presents a novel semi-stochastic gradient and Hessian for cubic regularization, improving convergence rates over existing methods.

Findings

Converges to an approximate local minimum within  O(n^{4/5}/psilon^{3/2}) oracle calls.

Outperforms existing cubic regularization algorithms in efficiency.

Experimental results validate theoretical improvements.

Abstract

We propose a stochastic variance-reduced cubic regularized Newton method for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an $(\epsilon,\sqrt{\epsilon})$-approximately local minimum within $\tilde{O}(n^{4/5}/\epsilon^{3/2})$ second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.