Multiplicative Drift Analysis

TL;DR

This paper introduces multiplicative drift analysis as a new tool for analyzing the runtime of randomized search heuristics, providing simpler proofs and tighter bounds for evolutionary algorithms optimizing various functions.

Contribution

The paper develops a multiplicative drift theorem and demonstrates its effectiveness in analyzing the runtime of evolutionary algorithms on linear and combinatorial problems.

Findings

Expected optimization time for linear functions is O(n log n).

Expected time for linear functions is at most 1.39 e n ln(n).

Lower bound of e n ln(n) for functions with a unique global optimum.

Abstract

In this work, we introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. We give a multiplicative version of the classical drift theorem. This allows easier analyses in those settings where the optimization progress is roughly proportional to the current distance to the optimum. To display the strength of this tool, we regard the classical problem how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time $O(n \log n)$, where $n$ is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most ${(1+o(1)) 1.39 \euler n\ln (n)}$, again using multiplicative drift analysis. We also prove a corresponding lower bound of ${(1-o(1))e n\ln(n)}$ which actually holds for all functions with a unique global optimum. We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours.