Non-Existence of Linear Universal Drift Functions

TL;DR

This paper investigates the limitations of drift analysis in evolutionary algorithms, demonstrating that universal drift functions cannot exist for a broad class of problems unless the mutation rate is near the standard value.

Contribution

It proves the non-existence of universal drift functions for linear pseudo-Boolean functions at mutation probabilities significantly different from 1/n.

Findings

Universal drift functions only exist near mutation rate 1/n

Drift analysis limitations depend on mutation probability

Results impact runtime analysis of evolutionary algorithms

Abstract

Drift analysis has become a powerful tool to prove bounds on the runtime of randomized search heuristics. It allows, for example, fairly simple proofs for the classical problem how the (1+1) Evolutionary Algorithm (EA) optimizes an arbitrary pseudo-Boolean linear function. The key idea of drift analysis is to measure the progress via another pseudo-Boolean function (called drift function) and use deeper results from probability theory to derive from this a good bound for the runtime of the EA. Surprisingly, all these results manage to use the same drift function for all linear objective functions. In this work, we show that such universal drift functions only exist if the mutation probability is close to the standard value of $1/n$.