On the saddle point problem for non-convex optimization

TL;DR

This paper identifies saddle points, rather than local minima, as the main obstacle in high-dimensional non-convex optimization and introduces a saddle-free Newton method to efficiently escape these saddle points, improving neural network training.

Contribution

It proposes a novel saddle-free Newton algorithm designed to rapidly escape saddle points in high-dimensional non-convex optimization problems.

Findings

The saddle-free Newton method outperforms traditional methods in escaping saddle points.

Saddle points create high error plateaus that hinder learning.

Preliminary results show improved neural network training performance.

Abstract

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for the ability of these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, and neural network theory, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new algorithm, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep neural network training, and provide preliminary numerical evidence for its superior performance.