The (1+1)-ES Reliably Overcomes Saddle Points

TL;DR

This paper proves that the simple (1+1)-ES optimization algorithm reliably escapes most saddle points in smooth functions, under mild conditions, using a novel drift analysis approach that ensures finite escape time with probability one.

Contribution

It demonstrates that the (1+1)-ES reliably overcomes saddle points, providing a new drift-based analysis that does not rely on estimating hitting times.

Findings

(1+1)-ES escapes saddle points with probability one.

The analysis uses non-standard drift methods.

Saddle point escape is guaranteed under mild regularity conditions.

Abstract

It is known that step size adaptive evolution strategies (ES) do not converge (prematurely) to regular points of continuously differentiable objective functions. Among critical points, convergence to minima is desired, and convergence to maxima is easy to exclude. However, surprisingly little is known on whether ES can get stuck at a saddle point. In this work we establish that even the simple (1+1)-ES reliably overcomes most saddle points under quite mild regularity conditions. Our analysis is based on drift with tail bounds. It is non-standard in that we do not even aim to estimate hitting times based on drift. Rather, in our case it suffices to show that the relevant time is finite with full probability.