Better Runtime Guarantees via Stochastic Domination

Apart from few exceptions, the mathematical runtime analysis of evolutionary algorithms is mostly concerned with expected runtimes. In this work, we argue that stochastic domination is a notion that should be used more frequently in this area. Stochastic domination allows to formulate much more informative performance guarantees than the expectation alone, it allows to decouple the algorithm analysis into the true algorithmic part of detecting a domination statement and probability theoretic part of deriving the desired probabilistic guarantees from this statement, and it allows simpler and more natural proofs. As particular results, we prove a fitness level theorem which shows that the runtime is dominated by a sum of independent geometric random variables, we prove tail bounds for several classic problems, and we give a short and natural proof for Witt's result that the runtime of any $(\mu,p)$ mutation-based algorithm on any function with unique optimum is subdominated by the runtime of a variant of the (1+1) evolutionary algorithm on the OneMax function.