Adaptive Drift Analysis

TL;DR

This paper proves that a (1+1) evolutionary algorithm with any constant mutation rate c/n efficiently finds the optimum of linear functions in expected time Theta(n log n), extending previous results and introducing objective-dependent drift functions.

Contribution

It generalizes the known runtime bounds for the (1+1) EA to any constant mutation rate c/n and introduces new drift functions tailored to specific objective functions.

Findings

Expected optimization time is Theta(n log n) for any c>0.

Universal drift functions do not exist for c larger than a certain constant.

High probability bounds match the expected runtime.

Abstract

We show that, for any c>0, the (1+1) evolutionary algorithm using an arbitrary mutation rate p_n = c/n finds the optimum of a linear objective function over bit strings of length n in expected time Theta(n log n). Previously, this was only known for c at most 1. Since previous work also shows that universal drift functions cannot exist for c larger than a certain constant, we instead define drift functions which depend crucially on the relevant objective functions (and also on c itself). Using these carefully-constructed drift functions, we prove that the expected optimisation time is Theta(n log n). By giving an alternative proof of the multiplicative drift theorem, we also show that our optimisation-time bound holds with high probability.