Black-Box Complexity: Breaking the $O(n \log n)$ Barrier of LeadingOnes

TL;DR

This paper improves the known upper bound on the black-box complexity of LeadingOnes functions, showing it is asymptotically smaller than previously thought, using only simple variation operators and applicable to unbiased and ranking-based models.

Contribution

It establishes a tighter upper bound of O(n log n / log log n) for the black-box complexity of LeadingOnes, surpassing the prior O(n log n) bound, and demonstrates this with simple unbiased operators.

Findings

Black-box complexity of LeadingOnes is O(n log n / log log n)

The bound applies to unbiased and ranking-based models

The algorithm uses only 3-ary unbiased variation operators

Abstract

We show that the unrestricted black-box complexity of the $n$-dimensional XOR- and permutation-invariant LeadingOnes function class is $O(n \log (n) / \log \log n)$. This shows that the recent natural looking $O(n\log n)$ bound is not tight. The black-box optimization algorithm leading to this bound can be implemented in a way that only 3-ary unbiased variation operators are used. Hence our bound is also valid for the unbiased black-box complexity recently introduced by Lehre and Witt (GECCO 2010). The bound also remains valid if we impose the additional restriction that the black-box algorithm does not have access to the objective values but only to their relative order (ranking-based black-box complexity).