Population Size vs. Mutation Strength for the (1+λ) EA on OneMax

Abstract

The (1+λ) EA with mutation probability c/n, where c > 0 is an arbitrary constant, is studied for the classical OneMax function. Its expected optimization time is analyzed exactly (up to lower order terms) as a function of c and λ. It turns out that 1/n is the only optimal mutation probability if λ = o(ln n ln ln n/ln ln ln n), which is the cut-off point for linear speed-up. However, if λ is above this cut-off point then the standard mutation probability 1/n is no longer the only optimal choice. Instead, the expected number of generations is (up to lower order terms) independent of c, irrespectively of it being less than 1 or greater.The results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds.

Optimal Static and Self-Adjusting Parameter Choices for the $$(1+(\lambda ,\lambda ))$$ ( 1 + ( λ , λ ) ) Genetic Algorithm

“…We observe that the parameters do not have an independent influence on the runtime, but that they interact in a difficult to foresee manner. A similar observation was made in [35], where it is proven for the (1 + λ) EA that the mutation probability has a decisive influence on the performance when the population size λ is asymptotically smaller than the cut-off point log(n) log log(n)/ log log log(n), whereas it has almost no influence when λ = ω(log(n) log log(n)/ log log log(n)). Such non-separable parameter influences, naturally, make the analysis of a multi-dimensional parameter space more difficult.…”

Optimal Mutation Rates for the (1+ $$\lambda $$ λ ) EA on OneMax Through Asymptotically Tight Drift Analysis

“…The approximated optimal values of c are given in Table 1. As expected, we can see that for fixed λ, the approximated mutation rate parameter c is decreasing with n, approaching 1 as predicted by the tight analysis in [GW15]. Furthermore, the approximate optimal c for n = 100 and λ = 1 is 1.19, which is only slightly higher than the exact value of 1.17 given in [CSWA15], hence justifying that the approximation does not suffer from large constants in the lower order term that might corrupt the approximation for small values of n. Interestingly, for fixed n the approximate optimal mutation rate grows with λ.…”

“…Here, the leading constant e c c is minimized for c = 1, thus justifying the unwritten rule of setting the mutation rate to 1/n in many applications. This result was refined and extended to populations by Gießen and Witt [GW15], who gave the asymptotically tight bound…”

“…Furthermore, the approximate optimal c for n = 100 and λ = 1 is 1.19, which is only slightly higher than the exact value of 1.17 given in [CSWA15], hence justifying that the approximation does not suffer from large constants in the lower order term that might corrupt the approximation for small values of n. Interestingly, for fixed n the approximate optimal mutation rate grows with λ. However, this behaviour cannot be explained by the bound from [GW15] which means that the reason for this behaviour is hidden in the lower order term. Intuitively, employing a larger population stabilizes the explorative character of allowing a higher mutation rate by reducing the chance that none of the individuals makes progress at all.…”

A Lower Bound Analysis of Population-Based Evolutionary Algorithms for Pseudo-Boolean Functions

“…In the recent decade, there has been a significant rise on the running time analysis (one essential theoretical aspect) of EAs [AD11,NW10]. For example, Droste et al [DJW02] proved that the expected running time of the (1+1)-EA on linear pseudo-Boolean functions is Θ(n ln n); for the (µ+1)-EA solving several artificially designed functions, a large parent population size µ was shown to be able to reduce the running time from exponential to polynomial [JW01,Sto08,Wit06,Wit08]; for the (1+λ)-EA solving linear functions, the expected running time was proved to be O(n ln n+λn) [DK15], and a tighter bound up to lower order terms was derived on the specific linear function OneMax [GW15].…”