The Interplay of Population Size and Mutation Probability in the ( $$1+\lambda $$ 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 theoretical results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds. Experimental supplements shed light on the optimal mutation probability for small problem sizes.

Significance-based estimation-of-distribution algorithms

“…where the estimate for the binomial distribution is well-known (see [23,Lemma 3] or [7, Lemma 10.37]). Regarding the correlation of the X is , we see that either X is is a fresh random sample independent from all X i ′ s ′ with s ′ < s and i ′ ∈ [n] or, namely if the reduced parent population in iteration s has the i-th bit converged, X is = 0 with probability one.…”

Does Comma Selection Help to Cope with Local Optima?

“…Our result, an analysis of the (𝜇, 𝜆) EA on jump functions that is precise for 𝑘 ≤ 0.1𝑛, 𝜆 = 𝑜 (𝑛 𝑘−1 ), 𝜆 ≥ (1 + 𝜀)𝑒𝜇, and 𝜆 = Ω(log 𝑛) sufficiently large, is the second precise analysis for a populationbased algorithm (after [32]), is the second precise analysis for a multimodal fitness function (after [25]), and is the first precise analysis for a non-elitist algorithm (apart from fact that the result [32] could be transfered to the (1, 𝜆) EA for large 𝜆 via the argument [36] that in this case the (1 + 𝜆) EA and the (1, 𝜆) EA have essentially identical performances).…”

“…The only precise runtime analysis for an algorithm with a nontrivial population can be found in [32], where the runtime of the (1 + 𝜆) EA with mutation rate 𝑐 𝑛 , 𝑐 a constant, on OneMax was shown to be (1 + 𝑜 (1))( 𝑒 𝑐 𝑐 𝑛 ln 𝑛 + 𝑛𝜆 ln ln 𝜆 2 ln 𝜆 ). This result has the surprising implication that here the mutation rate is only important when 𝜆 is small.…”