How to Escape Saddle Points Efficiently

TL;DR

This paper demonstrates that perturbed gradient descent efficiently escapes saddle points in high-dimensional non-convex optimization, with convergence rates nearly independent of dimension, benefiting machine learning applications like deep learning and matrix factorization.

Contribution

It introduces a perturbed gradient descent method that converges to second-order stationary points with near dimension-free complexity, improving understanding of escaping saddle points in non-convex optimization.

Findings

Convergence rate depends only poly-logarithmically on dimension.

Perturbed gradient descent can escape saddle points efficiently.

Applicable to deep learning and matrix factorization problems.

Abstract

This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate of this procedure matches the well-known convergence rate of gradient descent to first-order stationary points, up to log factors. When all saddle points are non-degenerate, all second-order stationary points are local minima, and our result thus shows that perturbed gradient descent can escape saddle points almost for free. Our results can be directly applied to many machine learning applications, including deep learning. As a particular concrete example of such an application, we show that our results can be used directly to establish sharp global convergence rates for matrix factorization. Our results rely on a novel characterization of the geometry around saddle points, which may be of independent interest to the non-convex optimization community.