Efficient approaches for escaping higher order saddle points in non-convex optimization

TL;DR

This paper introduces an efficient algorithm that guarantees convergence to third order local optima in non-convex optimization, addressing the challenge of escaping complex saddle points using higher order derivatives.

Contribution

The paper presents the first efficient method for escaping higher order saddle points by leveraging third order derivatives, surpassing previous second order techniques.

Findings

Algorithm guarantees convergence to third order local optima.

It is NP-hard to extend to fourth order local optima.

Addresses degenerate saddle points in high-dimensional non-convex functions.

Abstract

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.