Denis Antipov — Page 2

arXiv2020

A Tight Runtime Analysis for the $(μ+ λ)$ EA

Despite significant progress in the theory of evolutionary algorithms, the theoretical understanding of evolutionary algorithms which use non-trivial populations remains challenging and only few rigorous results exist. Already for the most basic problem, the determination of the asymptotic runtime of the $(μ+λ)$ evolutionary algorithm on the simple OneMax benchmark function, only the special cases $μ=1$ and $λ=1$ have been solved. In this work, we analyze this long-standing problem and show the asymptotically tight result that the runtime $T$, the number of iterations until the optimum is found, satisfies \[E[T] = Θ\bigg(\frac{n\log n}λ+\frac{n}{λ/ μ} + \frac{n\log^+\log^+ λ/ μ}{\log^+ λ/ μ}\bigg),\] where $\log^+ x := \max\{1, \log x\}$ for all $x > 0$. The same methods allow to improve the previous-best $O(\frac{n \log n}λ + n \log λ)$ runtime guarantee for the $(λ+λ)$~EA with fair parent selection to a tight $Θ(\frac{n \log n}λ + n)$ runtime result.

cs.NE↗

arXiv2019

The Efficiency Threshold for the Offspring Population Size of the ($μ$, $λ$) EA

Understanding when evolutionary algorithms are efficient or not, and how they efficiently solve problems, is one of the central research tasks in evolutionary computation. In this work, we make progress in understanding the interplay between parent and offspring population size of the $(μ,λ)$ EA. Previous works, roughly speaking, indicate that for $λ\ge (1+\varepsilon) e μ$, this EA easily optimizes the OneMax function, whereas an offspring population size $λ\le (1 -\varepsilon) e μ$ leads to an exponential runtime. Motivated also by the observation that in the efficient regime the $(μ,λ)$ EA loses its ability to escape local optima, we take a closer look into this phase transition. Among other results, we show that when $μ\le n^{1/2 - c}$ for any constant $c > 0$, then for any $λ\le e μ$ we have a super-polynomial runtime. However, if $μ\ge n^{2/3 + c}$, then for any $λ\ge e μ$, the runtime is polynomial. For the latter result we observe that the $(μ,λ)$ EA profits from better individuals also because these, by creating slightly worse offspring, stabilize slightly sub-optimal sub-populations. While these first results close to the phase transition do not yet give a complete picture, they indicate that the boundary between efficient and super-polynomial is not just the line $λ= e μ$, and that the reasons for efficiency or not are more complex than what was known so far.

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