Achieving Tight $O(4^k)$ Runtime Bounds on Jump$_k$ by Proving that Genetic Algorithms Evolve Near-Maximal Population Diversity

Abstract: The JUMP$k$ benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of $O(\text{poly}(n) + 4^k/p_c)$ for the ($\mu$+1) Genetic Algorithm ($(\mu+1)$ GA), but only for unrealistically small crossover probabilities $p_c$. To this date, it remains an open problem to prove similar upper bounds for realistic $p_c$; the best known runtime bound, in terms of function evaluations, for $p_c = \Omega(1)$ is $O((n/\chi)^{k-1})$, $\chi$ a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the $(\mu+1)$ GA on JUMP$_k$. The $(\mu+1)$-$\lambda_c$-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover $\lambda_c$ times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the $(\mu+1)$-$\lambda_c$-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of $O(\mu n \log(\mu) + 4^k)$ function evaluations for a range of $k$ under the mild assumptions $p_c = O(1/k)$ and $\mu \in \Omega(kn)$. For all constant $k$, the restriction is satisfied for some $p_c = \Omega(1)$ and it implies that the expected runtime for all constant $k$ and an appropriate $\mu = \Theta(kn)$ is bounded by $O(n^2 \log n)$, irrespective of $k$. For larger $k$, the expected time of the $(\mu+1)$-$\lambda_c$-GA is $\Theta(4^k)$, which is tight for a large class of unbiased black-box algorithms and faster than the original $(\mu+1)$ GA by a factor of $\Omega(1/p_c)$. We also show that our analysis can be extended to other unitation functions such as JUMP${k, \delta}$ and HURDLE.