Rigorous Runtime Analysis of the (1+1) ES: 1/5-Rule and Ellipsoidal Fitness Landscapes

Abstract

We consider the (1+1) Evolution Strategy, a simple evolutionary algorithm for continuous optimization problems, using so-called Gaussian mutations and the 1/5-rule for the adaptation of the mutation strength. Here, the function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ to be minimized is given by a quadratic form f(x) = x ^⊤ Qx, where Q ∈ ℝ^n×n is a positive definite diagonal matrix and x denotes the current search point. This is a natural extension of the well-known Sphere-function (Q=I). Thus, very simple unconstrained quadratic programs are investigated, and the question is addressed how Q effects the runtime. For this purpose, quadratic forms

$$ f({\mathbf x}) = \xi\cdot\left({x_{1}}^{2}+\dots+{x_{n/2}}^{2}\right)+{x_{n/2+1}}^{2}+\dots+{x_{n}}^{2} $$

with ξ=ω(1), i. e. 1/ξ→0 as n→∞, and ξ=poly(n) are investigated exemplarily. It is proved that the optimization very quickly stabilizes and that, subsequently, the runtime (defined as the number of f-evaluations) to halve the approximation error is Θ(ξ n). Though ξ n=poly(n), this result actually shows that the evolving search point indeed creeps along the “gentlest descent” of the ellipsoidal fitness landscape.