How to escape saddle points efficiently

Conference Paper

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.

Duke Authors

Cited Authors

  • Jin, C; Ge, R; Netrapalli, P; Kakade, SM; Jordan, MI

Published Date

  • January 1, 2017

Published In

  • 34th International Conference on Machine Learning, Icml 2017

Volume / Issue

  • 4 /

Start / End Page

  • 2727 - 2752

International Standard Book Number 13 (ISBN-13)

  • 9781510855144

Citation Source

  • Scopus