The (1+1) Elitist Black-Box Complexity of LeadingOnes

TL;DR

This paper establishes a new lower bound of a9(n^2) for the (1+1) elitist black-box complexity of the LeadingOnes problem, highlighting the impact of elitist restrictions on optimization difficulty.

Contribution

It introduces a novel information-theoretic method to derive lower bounds for black-box complexity, specifically for the permutation- and bit-invariant LeadingOnes problem under elitist constraints.

Findings

The (1+1) elitist black-box complexity of LeadingOnes is a9(n^2).

This complexity is significantly larger than the unrestricted case of a9(n f0b n a0loga0log n).

The new bound matches the performance of (1+1)-type evolutionary algorithms.

Abstract

One important goal of black-box complexity theory is the development of complexity models allowing to derive meaningful lower bounds for whole classes of randomized search heuristics. Complementing classical runtime analysis, black-box models help us understand how algorithmic choices such as the population size, the variation operators, or the selection rules influence the optimization time. One example for such a result is the $\Omega(n \log n)$ lower bound for unary unbiased algorithms on functions with a unique global optimum [Lehre/Witt, GECCO 2010], which tells us that higher arity operators or biased sampling strategies are needed when trying to beat this bound. In lack of analyzing techniques, almost no non-trivial bounds are known for other restricted models. Proving such bounds therefore remains to be one of the main challenges in black-box complexity theory. With this paper we contribute to our technical toolbox for lower bound computations by proposing a new type of information-theoretic argument. We regard the permutation- and bit-invariant version of \textsc{LeadingOnes} and prove that its (1+1) elitist black-box complexity is $\Omega(n^2)$, a bound that is matched by (1+1)-type evolutionary algorithms. The (1+1) elitist complexity of \textsc{LeadingOnes} is thus considerably larger than its unrestricted one, which is known to be of order $n\log\log n$ [Afshani et al., 2013].