Run Time Bounds for Integer-Valued OneMax Functions

TL;DR

This paper analyzes the runtime of various algorithms on integer-valued OneMax functions over the space Z, providing bounds and empirical comparisons for different step operators and adaptations.

Contribution

It introduces runtime bounds for the OEA and RLS algorithms on integer-valued OneMax functions, including heavy-tailed step operators and step size adaptation, extending previous finite search space analyses.

Findings

Expected optimization time is linear in the maximum value of the optimum.

Heavy-tailed step operators can lead to infinite expected runtime.

Step size adaptation in RLS achieves near-optimal runtime bounds.

Abstract

While most theoretical run time analyses of discrete randomized search heuristics focused on finite search spaces, we consider the search space $\mathbb{Z}^n$. This is a further generalization of the search space of multi-valued decision variables $\{0,\ldots,r-1\}^n$. We consider as fitness functions the distance to the (unique) non-zero optimum $a$ (based on the $L_1$-metric) and the \ooea which mutates by applying a step-operator on each component that is determined to be varied. For changing by $\pm 1$, we show that the expected optimization time is $\Theta(n \cdot (|a|_{\infty} + \log(|a|_H)))$. In particular, the time is linear in the maximum value of the optimum $a$. Employing a different step operator which chooses a step size from a distribution so heavy-tailed that the expectation is infinite, we get an optimization time of $O(n \cdot \log^2 (|a|_1) \cdot \left(\log (\log (|a|_1))\right)^{1 + \epsilon})$. Furthermore, we show that RLS with step size adaptation achieves an optimization time of $\Theta(n \cdot \log(|a|_1))$. We conclude with an empirical analysis, comparing the above algorithms also with a variant of CMA-ES for discrete search spaces.