Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift
This work simplifies and strengthens theoretical analysis for evolutionary algorithms, benefiting researchers in computational optimization by making lower bound proofs more accessible and applicable to concrete scenarios.
The paper tackles the complexity of proving lower bounds for non-elitist evolutionary algorithms by introducing a simpler negative drift theorem for multiplicative drift, which reduces the number of technical conditions needed from five to three and yields explicit, non-asymptotic bounds. It demonstrates that super-polynomial runtimes occur when the reproduction rate is slightly below a threshold, specifically a (1 - ω(n^{-1/2})) factor, and applies this to extend results for mutation operators and prove an exponential lower bound for the simple genetic algorithm on OneMax.
A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems -- general results which translate an expected progress away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) \emph{negative drift in populations} method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a $(1 - ω(n^{-1/2}))$ factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than $Θ(1/n)$, which includes the heavy-tailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally apply our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on \onemax for arbitrary population size.