Analysis of a simple (1+1) ES for the Class of Positive Definite Quadratic Forms with Bounded Condition Number [report]

Jens Jägersküpper, Technische Universität Dortmund, Technische Universität Dortmund
2006
The (1+1) Evolution Strategy (ES), a simple, mutation-based evolutionary algorithm for continuous optimization problems, is analyzed. In particular, we consider the most common type of mutations, namely Gaussian mutations, and the 1/5-rule for mutation adaptation, and we are interested in how the runtime, which we define as the number of function evaluations, to obtain a predefined reduction of the approximation error depends on the dimension of the search space. The most discussed function in
more » ... he area of ES is the so-called Sphere-function given by Sphere: R n → R with x → x Ix (where I ∈ R n×n is the identity matrix), which also has already been the subject of a runtime analysis. This analysis is extended to arbitrary positive definite quadratic forms (PDQFs) that induce ellipsoidal fitness landscapes which are "close to being spherically symmetric. " Namely, all functions x → x Qx are covered, where Q ∈ R n×n is positive definite such that its condition number, which equals the ratio of the largest of the n eigenvalues of Q to the smallest one, is O(1). We show that indeed the order of the runtime does not change compared to Sphere. Namely, we prove that any (1+1) ES using isotropic mutations needs Ω(n) function evaluations to halve the approximation error in expectation and yet with an overwhelming probability. On the other hand, also with an overwhelming probability O(n) function evaluations suffice to halve the approximation error when a (1+1) ES uses Gaussian mutations adapted by a 1/5-rule.
doi:10.17877/de290r-14522 fatcat:g6wqctexcfgk5o5v3qonxvkaxi