General lower bounds for evolutionary algorithms
Olivier Teytaud and Sylvain Gelly
In: 10th International Conference on Parallel Problem Solving from Nature (PPSN 2006), 9-13 Sep 2006, Reykjavik.
Evolutionary optimization, among which genetic optimization, is a general framework for optimization. It is known (i) easy to use (ii) robust (iii) derivative-free (iv) unfortunately slow. Recent work in particular show that the convergence rate of some widely used evolution strategies (evolutionary optimization for continuous domains) can not b e faster than linear (i.e. the logarithm of the distance to the optimum can not decrease faster than linearly), and that the constant in the linear convergence (i.e. the constant C such that the distance to the optimum after n steps is upp er b ounded by C n ) unfortunately converges quickly to 1 as the dimension increases to infinity. We here show a very wide generalization of this result: all comparison-based algorithms have such a limitation. Note that our result also concerns methods like the Hooke &Jeeves algorithm, the simplex method, or any direct search method that only compares the values to previously seen values of the ﬁtness. But it does not cover methods that use the value of the ﬁtness (see  for cases in which the ﬁtness-values are used), even if these methods do not use gradients. The former results deal with convergence with respect to thenumber of comparisons performed, and also include a very wide family of algorithms with resp ect to the numb er of function-evaluations. However, there is still place for faster convergence rates, for more original algorithms using the full ranking information of the population and not only selections among the population. We prove that, at least in some particular cases, using the full ranking information can improve these lower bounds, and ultimately provide superlinear convergence results.