Tight Bounds on the Optimization Time of the (1+1) EA on Linear Functions

TL;DR

This paper tightens the bounds on the expected optimization time of the (1+1) EA on linear functions, showing the standard mutation rate is optimal and the algorithm is near-optimal in this setting.

Contribution

It improves the known bounds on the (1+1) EA's optimization time for linear functions and establishes the optimality of the standard mutation rate.

Findings

Expected optimization time is tightly bounded by en ln n + O(n).

Standard mutation probability p=1/n is optimal for linear functions.

The (1+1) EA is an optimal mutation-based algorithm for linear functions.

Abstract

The analysis of randomized search heuristics on classes of functions is fundamental for the understanding of the underlying stochastic process and the development of suitable proof techniques. Recently, remarkable progress has been made in bounding the expected optimization time of the simple (1+1) EA on the class of linear functions. We improve the best known bound in this setting from $(1.39+o(1))en\ln n$ to $en\ln n+O(n)$ in expectation and with high probability, which is tight up to lower-order terms. Moreover, upper and lower bounds for arbitrary mutations probabilities $p$ are derived, which imply expected polynomial optimization time as long as $p=O((\ln n)/n)$ and which are tight if $p=c/n$ for a constant $c$. As a consequence, the standard mutation probability $p=1/n$ is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm. The proofs are based on adaptive drift functions and the recent multiplicative drift theorem.