On the runtime dynamics of the compact genetic algorithm on jump functions

Jump functions were originally introduced as benchmarks on which recombinant evolutionary algorithms can provably outperform those that use mutation alone. To optimize a jump function, an algorithm must be able to execute an initial hill-climbing phase, after which a point across a large gap must be generated. Standard GAs mix mutation and crossover to achieve both behaviors. It seems likely that other techniques, such as estimation of distribution algorithms (EDAs) may exhibit such behavior, but an analysis is so far missing. We analyze an EDA called the compact Genetic Algorithm (cGA) on jump functions with gap k. We prove that the cGA initially exhibits a strong positive drift resulting in good hillclimbing behavior. Interpreting the diversity of the process as the variance of the underlying probabilistic model, we show the existence of a critical point beyond which progress slows and diversity vanishes. If k is not too large, the cGA generates with high probability an optimal solution in polynomial time before losing diversity. For k = Ω(logn), this yields a superpolynomial speedup over mutation-only approaches. We show a small modification that creates > 2 offspring boosts the critical threshold and allows the cGA to solve functions with a larger gap within the same number of fitness evaluations.