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. 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).
The algorithm achieving the O(n log n/ log log n) bound mentioned in Section 1.1 can be implemented as a ternary unbiased one REF .
Black-Box Complexity: Breaking the $O(n \log n)$ Barrier of LeadingOnes
ArXiv
ArXiv
Mathematics Computer Science