Lower Bounds from Fitness Levels Made Easy

TL;DR

This paper introduces simplified variants of the fitness levels method for proving both upper and lower run time bounds of evolutionary algorithms, making the process easier and more natural.

Contribution

It presents two new variants of the fitness levels method that rely only on level visitation probabilities, simplifying the derivation of bounds.

Findings

Reproved the run time of (1+1) EA on LeadingOnes.

Established a tight lower bound for (1+1) EA on OneMax.

Proved a lower bound for (1+1) EA on jump functions with probability bounds.

Abstract

One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed by the algorithm in increasing order, possibly skipping levels. An easy, but often strong upper bound for the run time can then be derived by adding the reciprocals of the probabilities to leave the levels (or upper bounds for these). Unfortunately, a similarly effective method for proving lower bounds has not yet been established. The strongest such method, proposed by Sudholt (2013), requires a careful choice of the viscosity parameters $\gamma_{i,j}$, $0 \le i < j \le n$. In this paper we present two new variants of the method, one for upper and one for lower bounds. Besides the level leaving probabilities, they only rely on the probabilities that levels are visited at all. We show that these can be computed or estimated without greater difficulties and apply our method to reprove the following known results in an easy and natural way. (i) The precise run time of the (1+1) EA on \textsc{LeadingOnes}. (ii) A lower bound for the run time of the (1+1) EA on \textsc{OneMax}, tight apart from an $O(n)$ term. (iii) A lower bound for the run time of the (1+1) EA on long $k$-paths. We also prove a tighter lower bound for the run time of the (1+1) EA on jump functions by showing that, regardless of the jump size, only with probability $O(2^{-n})$ the algorithm can avoid to jump over the valley of low fitness.