Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, March 26, 2024, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Johannes Lengler

Population Diversity in a Neutral Fitness Landscape

One one the major advantages of population-based optimization heuristics like genetic algorithms is that we can recombine two or more solutions into a new. This operation is called crossover. In practice, crossover is known to be extremely helpful, and we would also like to understand how much it helps in theoretical benchmarks. However, there is one obstacle: The effectiveness of crossover depends on the population diversity (e.g., measured by average Hamming distance of the solutions), so we need to understand how the diversity of a population evolves over time.

We answer this question under a seemingly very strong assumption: for a flat objective function, i.e., in absence of fitness signals. We show that in this case, surprisingly, the details of algorithm have almost no influence on the diversity. Specifically, for the (\mu+1) Genetic Algorithm we show that diversity approaches an equilibrium which (almost) does not depend on the used mutation or crossover operators. The equilibrium point increases linearly with the population size.

Although flat objective functions are seemingly uninteresting, the result turned out to be surprisingly useful. I will give one application: for Jump, a standard benchmark for local optima, the runtime of the (\mu+1) GA is reduced massively by crossover, because the natural diversity in this range is large enough to speed up optimization.

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