On the Choice of the Update Strength in Estimation-of-Distribution Algorithms and Ant Colony Optimization

Probabilistic model-building Genetic Algorithms (PMBGAs) are a class of metaheuristics that evolve probability distributions favoring optimal solutions in the underlying search space by repeatedly sampling from the distribution and updating it according to promising samples. We provide a rigorous runtime analysis concerning the update strength, a vital parameter in PMBGAs such as the step size 1 / K in the so-called compact Genetic Algorithm (cGA) and the evaporation factor $$\rho $$ρ in ant colony optimizers (ACO). While a large update strength is desirable for exploitation, there is a general trade-off: too strong updates can lead to unstable behavior and possibly poor performance. We demonstrate this trade-off for the cGA and a simple ACO algorithm on the well-known OneMax function. More precisely, we obtain lower bounds on the expected runtime of $${\varOmega }(K\sqrt{n} + n \log n)$$Ω(Kn+nlogn) and $${\varOmega }(\sqrt{n}/\rho + n \log n)$$Ω(n/ρ+nlogn), respectively, suggesting that the update strength should be limited to $$1/K, \rho = O(1/(\sqrt{n} \log n))$$1/K,ρ=O(1/(nlogn)). In fact, choosing $$1/K, \rho \sim 1/(\sqrt{n}\log n)$$1/K,ρ∼1/(nlogn) both algorithms efficiently optimize OneMax in expected time $${\varTheta }(n \log n)$$Θ(nlogn). Our analyses provide new insights into the stochastic behavior of PMBGAs and propose new guidelines for setting the update strength in global optimization.