Saving Gradient and Negative Curvature Computations: Finding Local Minima More Efficiently

TL;DR

This paper introduces a family of nonconvex optimization algorithms that efficiently find local minima by reducing gradient and negative curvature computations, improving runtime over existing methods.

Contribution

The algorithms divide the domain into large and small gradient regions, performing targeted descent steps, and can escape small gradient regions with minimal negative curvature computations.

Findings

Can escape small gradient regions in one negative curvature step

Potentially outperform state-of-the-art local minima algorithms

Effective in both deterministic and stochastic settings

Abstract

We propose a family of nonconvex optimization algorithms that are able to save gradient and negative curvature computations to a large extent, and are guaranteed to find an approximate local minimum with improved runtime complexity. At the core of our algorithms is the division of the entire domain of the objective function into small and large gradient regions: our algorithms only perform gradient descent based procedure in the large gradient region, and only perform negative curvature descent in the small gradient region. Our novel analysis shows that the proposed algorithms can escape the small gradient region in only one negative curvature descent step whenever they enter it, and thus they only need to perform at most $N_{\epsilon}$ negative curvature direction computations, where $N_{\epsilon}$ is the number of times the algorithms enter small gradient regions. For both deterministic and stochastic settings, we show that the proposed algorithms can potentially beat the state-of-the-art local minima finding algorithms. For the finite-sum setting, our algorithm can also outperform the best algorithm in a certain regime.