Abstract
Drift analysis aims at translating the expected progress of an evolutionary algorithm (or more generally, a random process) into a probabilistic guarantee on its run time (hitting time). So far, drift arguments have been successfully employed in the rigorous analysis of evolutionary algorithms, however, only for the situation that the progress is constant or becomes weaker when approaching the target. Motivated by questions like how fast fit individuals take over a population, we analyze random processes exhibiting a (1+delta )-multiplicative growth in expectation. We prove a drift theorem translating this expected progress into a hitting time. This drift theorem gives a simple and insightful proof of the level-based theorem first proposed by Lehre (2011). Our version of this theorem has, for the first time, the best-possible near-linear dependence on 1/delta (the previous results had an at least near-quadratic dependence), and it only requires a population size near-linear in delta (this was super-quadratic in previous results). These improvements immediately lead to stronger run time guarantees for a number of applications. We also discuss the case of large delta and show stronger results for this setting.
Highlights
In a typical situation in evolutionary search, an algorithm first makes good progress while far away from the target, since a lot can still be improved
When regarding the subpopulation of individuals having some desired property, in an algorithm using comma selection, this might die out completely in one iteration. To cover such processes, in our second drift theorem (Theorem 16) we extend our Theorem 3 to include that state 0 is reached with at most the probability that can be deduced from the up-drift and the binomial distribution conditions
The first is concerned with processes that cannot reach the value 0; the second one extends the first theorem to include the possibility of going down to 0
Summary
In a typical situation in evolutionary search, an algorithm first makes good progress while far away from the target, since a lot can still be improved. When regarding the subpopulation of individuals having some desired property, in an algorithm using comma selection, this might die out completely in one iteration (though often with small probability only) To cover such processes, in our second drift theorem (Theorem 16) we extend our Theorem 3 to include that state 0 is reached with at most the probability that can be deduced from the up-drift and the binomial distribution conditions. This is asymptotically better than the previously known bound of O(n2.5 log(n)) and shows more explicitly how optimization proceeds Beyond these particular results, our modular proof (first analyzing the multiplicative up-drift excluding 0, including 0, applying it in the context of the level-based theorem) shows the level-based theorem in a way that is more accessible than the previous versions and that gives more insight into populationbased optimization processes. We are optimistic that this increased understanding of population-based processes helps in the future design and analysis of such processes
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