Yet another fast variant of Newton's method for nonconvex optimization

TL;DR

This paper introduces a new second-order optimization algorithm for nonconvex functions that combines regularized Newton and negative curvature steps, with theoretical complexity bounds and initial numerical experiments.

Contribution

It proposes a novel class of algorithms that efficiently combine regularized Newton and negative curvature steps, providing complexity guarantees and practical variants for nonconvex optimization.

Findings

Complexity of $ ilde{O}( ext{log} rac{1}{ ext{epsilon}} ext{epsilon}^{-3/2})$ for first-order minimizers.

Complexity of $ ilde{O}( ext{log} rac{1}{ ext{epsilon}}) ext{epsilon}^{-3}$ for second-order minimizers.

Initial numerical experiments demonstrate effectiveness of the proposed methods.

Abstract

A class of second-order algorithms is proposed for minimizing smooth nonconvex functions that alternates between regularized Newton and negative curvature steps in an iteration-dependent subspace. In most cases, the Hessian matrix is regularized with the square root of the current gradient and an additional term taking moderate negative curvature into account, a negative curvature step being taken only exceptionally. Practical variants have been detailed where the subspaces are chosen to be the full space, or Krylov subspaces. In the first case, the proposed method only requires the solution of a single linear system at nearly all iterations. We establish that at most $\mathcal{O}\left(|\log\epsilon|\,\epsilon^{-3/2}\right)$ evaluations of the problem's objective function and derivatives are needed for algorithms in the new class to obtain an $\epsilon$-approximate first-order minimizer, and at most $\mathcal{O}\left(|\log\epsilon|\,\epsilon^{-3}\right)$ to obtain a second-order one. Encouraging initial numerical experiments with two full-space and two Krylov-subspaces variants are finally presented.