Better Fixed-Arity Unbiased Black-Box Algorithms

TL;DR

This paper introduces a simpler, more efficient fixed-arity unbiased black-box algorithm for OneMax, achieving better asymptotic constants and demonstrating effectiveness for small k values.

Contribution

It presents an alternative, conceptually simpler method for achieving optimal black-box complexity using $k$-ary operators, applicable beyond OneMax.

Findings

Achieves $O(n/k)$ complexity for $3 ext{-} ext{to} ext{log}_2 n$ arity.

Provides an algorithm with asymptotic complexity $(2+o(1)) imes n/(k-1)$.

Experimental results show efficiency for $k$ between 3 and 6.

Abstract

In their GECCO'12 paper, Doerr and Doerr proved that the $k$-ary unbiased black-box complexity of OneMax on $n$ bits is $O(n/k)$ for $2\le k\le O(\log n)$. We propose an alternative strategy for achieving this unbiased black-box complexity when $3\le k\le\log_2 n$. While it is based on the same idea of block-wise optimization, it uses $k$-ary unbiased operators in a different way. For each block of size $2^{k-1}-1$ we set up, in $O(k)$ queries, a virtual coordinate system, which enables us to use an arbitrary unrestricted algorithm to optimize this block. This is possible because this coordinate system introduces a bijection between unrestricted queries and a subset of $k$-ary unbiased operators. We note that this technique does not depend on OneMax being solved and can be used in more general contexts. This together constitutes an algorithm which is conceptually simpler than the one by Doerr and Doerr, and at the same time achieves better constant factors in the asymptotic notation. Our algorithm works in $(2+o(1))\cdot n/(k-1)$, where $o(1)$ relates to $k$. Our experimental evaluation of this algorithm shows its efficiency already for $3\le k\le6$.