218 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 5, NO. 3, JUNE 2001 On the Performance of -Evolution Strategies for the Ridge Function Class Hans-Georg Beyer Abstract—This paper presents the -dependent analysis of the -evolution strategy (ES) with isotropic mutations for the ridge functions including the special cases of sharp and parabolic ridges. The new approach presented allows for the prediction of the dynamics in ridge direction as well as in radial direction. The central quantities are the corresponding progress rates which are determined in terms of analytical expressions. Its predictive quality is evaluated by ES simulations and the steady-state behavior is discussed in detail. Index Terms—Evolution strategies, induced order statistics, per- formance measures, progress rates, ridge functions, scalability. I. INTRODUCTION U P UNTIL NOW, the theoretical analysis of the per- formance of evolution strategies (ESs) with Gaussian mutations has been concentrated mainly on the sphere model test functions [1]–[4]. Even though the results obtained provide valuable insight into the working of ES algorithms [5], [6], there is still a need for considering further model classes in order to acquire to a deeper understanding of why and how these algorithms really work. Especially when self-adaptation is considered, the sphere model does not cover all essential aspects of the local evolution process. That is, such algorithms can even locally exhibit qualitatively different behavior on test functions which cannot be well approximated by the sphere model. Such a class of simple test functions has been investigated empirically by Herdy [7]: the so-called “parabolic ridge” and the “sharp ridge” (see also [8]). The ridge models may be regarded as extensions of the sphere model breaking its total rotational symmetry in one dimension of the parameter space (for definitions, see below). One might expect that such a small change in the functional structure would not have a severe influence on the performance of the ES. How- ever, the change is of such a kind that each level set of the fitness landscape is an open success domain, and the ridge axis direc- tion appears as a progress direction in which the population can evolve indefinitely. From the technical point of view, the ridge function class is the one that should be considered after the sphere model. In the latter, all dimensions can be lumped together, thus opening up the possibility for a one-dimensional (1-D) description of the ES Manuscript received October 26, 1999; revised May 12, 2000. This work was supported by the Deutsche Forschungsgemeinschaft under Grant Be1578/4-1. The author is with the Department of Computer Science XI, University of Dortmund, D-44221 Dortmund, Germany (e-mail: beyer@zappa.cs.uni-dort- mund.de). Publisher Item Identifier S 1089-778X(01)05373-5. dynamics. The logical next step is, therefore, to consider models whose dynamics must be described by two state variables in the parameter space (search space). While this appears logically cogent, a first paper on this topic dates back to 1998 [9]. In that paper, Oyman et al. developed a simple local geometrical model in order to calculate the expected progress, the progress rate , in ridge direction for the -ES given the state of the parent. However, this ad hoc approach lacks in some aspects. First, it does not provide any information on the approximation error made. The influence of the parameter space dimension remained obscure. Second, as a more severe aspect, this model is not able to address the problem of the radial dynamics, i.e., the evolution of the parental distance to the ridge axis. The analysis of the dynamics of for the special case “para- bolic ridge” succeeded thereafter in [10] and a thorough analysis of the parabolic ridge for and different ES versions has been done by Oyman [11]. Still, there remains the treatment of the radial dynamics for general ridge functions. This paper provides an approach for calculating the radial as well as the longitudinal (ridge direction) dynamics for -ES on general ridge functions, thereby closing the remaining gap. The paper is organized as follows. Section II defines the general rotated ridge function class and its transformation to the normal form. In Section II-C, the me- lioration process on the ridge is discussed and the connection to optimization is established. Section II-D gives a short descrip- tion of the ES algorithm and in Section II-E, the local perfor- mance measures are introduced. Section III is devoted to the progress rate in ridge direc- tion and Section IV to the progress rate toward the ridge axis. Both sections are of technical nature. First, the integral representations for and , respectively, are derived. Unfor- tunately, these integrals are not tractable. Therefore, analytical expressions based on normal approximations and linearization techniques must be calculated. The applicability of the approx- imations used is shown for some examples by comparison with experiments in Section IV-C. The dynamical aspects and the steady-state behavior of the -ES are discussed in Section V. In the first part, the pre- dictions regarding the dynamics are compared with real ES runs. The most important observation is the appearance of a steady- state behavior keeping the population at a certain (expected) dis- tance to the ridge axis. This distance is investigated fur- ther and the transient time for its appearance is estimated. Fi- nally, the steady-state progress rate is investigated and compared with experiments. Section VI gives a short outlook at what should be done next. 1089–778X/01$10.00 © 2001 IEEE