The Efficiency Threshold for the Offspring Population Size of the ($\mu$, $\lambda$) 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 $(\mu,\lambda)$ EA. Previous works, roughly speaking, indicate that for $\lambda \ge (1+\varepsilon) e \mu$, this EA easily optimizes the OneMax function, whereas an offspring population size $\lambda \le (1 -\varepsilon) e \mu$ leads to an exponential runtime. Motivated also by the observation that in the efficient regime the $(\mu,\lambda)$ EA loses its ability to escape local optima, we take a closer look into this phase transition. Among other results, we show that when $\mu \le n^{1/2 - c}$ for any constant $c > 0$, then for any $\lambda \le e \mu$ we have a super-polynomial runtime. However, if $\mu \ge n^{2/3 + c}$, then for any $\lambda \ge e \mu$, the runtime is polynomial. For the latter result we observe that the $(\mu,\lambda)$ 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 $\lambda = e \mu$, and that the reasons for efficiency or not are more complex than what was known so far.