Black-Box Complexity: Breaking the $O(n \log n)$ Barrier of LeadingOnes
This work addresses a foundational problem in theoretical computer science by improving complexity bounds for black-box optimization, though it is incremental as it refines existing results.
The authors tackled the problem of determining the black-box complexity of the LeadingOnes function class, showing it is O(n log n / log log n), which breaks the previous O(n log n) barrier and demonstrates it is not tight.
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).