Drift Analysis and Evolutionary Algorithms Revisited

TL;DR

This paper reviews and strengthens existing analyses of a simple evolutionary algorithm's runtime for optimizing boolean functions, focusing on drift analysis and its applications.

Contribution

It provides new, self-contained proofs and partly stronger results for the runtime analysis of a basic evolutionary algorithm on monotone and linear functions.

Findings

Reviewed known results on the algorithm's runtime.

Provided new, self-contained proofs of existing results.

Achieved partly stronger bounds on the algorithm's performance.

Abstract

One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a boolean function $f:\{0,1\}^n \to {\mathbb R}$. The algorithm starts with a random search point $\xi \in \{0,1\}^n$, and in each round it flips each bit of $\xi$ with probability $c/n$ independently at random, where $c>0$ is a fixed constant. The thus created offspring $\xi'$ replaces $\xi$ if and only if $f(\xi') \ge f(\xi)$. The analysis of the runtime of this simple algorithm on monotone and on linear functions turned out to be highly non-trivial. In this paper we review known results and provide new and self-contained proofs of partly stronger results.