1 The Dynamics of Self-Adaptive Multi-Recombinant Evolution Strategies on the General Ellipsoid Model Hans-Georg Beyer and Alexander Melkozerov Abstract—The optimization behavior of the self-adaptation (SA) evolution strategy (ES) with intermediate multi- recombination (the (µ/µI ,λ)-σSA-ES) using isotropic mutations is investigated on convex-quadratic functions (referred to as ellipsoid model). An asymptotically exact quadratic progress rate formula is derived. This is used to model the dynamical ES system by a set of difference equations. The solutions of this system are used to analytically calculate the optimal learning parameter τ . The theoretical results are compared and validated by comparison with real (µ/µI ,λ)-σSA-ES runs on two ellipsoid test model cases. The theoretical results clearly indicate that using a model-independent learning parameter τ leads to suboptimal performance of the (µ/µI ,λ)-σSA-ES on objective functions with changing local condition numbers as often encountered in practical problems with complex fitness landscapes. Index Terms—Evolution strategy, ellipsoid model, progress rate, self-adaptation I. I NTRODUCTION Theoretical analyses of Evolution Strategies have a long- standing tradition starting with Rechenberg’s early work con- cerning the (1 + 1)-ES on the sphere model published in [22]. While in the last decade of the 20th century parts of more complex ES algorithms such as (µ,λ)- and (µ/µ,λ)-ES have been analyzed, the treatment of the complete algorithm including σ mutation strength control started with the turn of the century [9]. It was continued by different authors such as Arnold [1], Auger [5], and Jägersküpper [16]. Considering test functions beyond the sphere model was the next step. In [17], Jägersküpper considered the (1 + 1)-ES with 1/5-rule on a subset of positive definite quadratic forms (PDQFs). The complementing analysis of the (µ/µ I ,λ)-ES has been done in [10]. Furthermore, the Cigar as a special PDQF [3] and ridge functions [21], [19] have been analyzed so far. However, unlike the acronym PDQF suggests, the general PDQF case has not been treated so far. Since the level set of this general case defines an ellipsoid in the N -dimensional space, we refer to this kind of test function as general ellipsoid model. The analysis of the dynamics of the (µ/µ I ,λ)-ES on ellipsoid models may be regarded as a milestone on the way to H.-G. Beyer is with the Research Center Process and Product Engineering at the Vorarlberg University of Applied Sciences, Dornbirn, Austria, Email: Hans-Georg.Beyer@fhv.at A. Melkozerov is with the Department of Television and Control, Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia and the Institute of Neural Information Processing, University of Ulm, Ulm, Germany, Email: ame@tu.tusur.ru c 2012 IEEE. Personal use of this material is permitted. However, permis- sion to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. a full analysis of covariance matrix adaptation ES (CMA-ES). While these strategies are currently among the best-performing direct search methods [14], their theoretical analysis is still in its infancy. A full analysis that considers the real CMA- ES [15] or the CMSA-ES [12] requires the analysis of the covariance learning and the mutation strength adaptation. This paper provides the solution for the second problem in the case of the self-adaptive mutation control as used in CMSA-ES. A similar analysis concerning the cumulative step-size adaptation would solve the respective problem for the CMA-ES. Besides being a step towards the analysis of CMA-like strategies, the analysis to be presented finalizes the chapter of theoret- ical analyses regarding the dynamical systems approach on quadratic fitness functions started in the 1990s. Furthermore, the analysis approach extends the standard analysis method by utilizing quadratic progress rate measures. The paper is organized as follows. First, the (µ/µ I ,λ)-σSA- ES algorithm, the ellipsoid model and previous results on the topic are presented in the remaining parts of the introduction. The quadratic progress rate is introduced and derived in Section II. This new progress measure is the basis for the dynamical systems approach in this paper. In Section III, a system of discrete nonlinear difference equations is derived and solved for the steady-state limit. The obtained solutions are compared with real (µ/µ I ,λ)-σSA-ES experiments. Based on these results, in Section IV the problem regarding the optimal choice of the learning parameter τ is tackled yielding an approximate τ formula. The paper concludes with a discussion of the results and their implications for future work. A. ES Algorithm The (µ/µ I ,λ)-σSA-ES algorithm investigated in this work is presented in Fig. 1. The parental mutation strength σ (0) and the parental parameter vector, or parental centroid y (0) , are initialized in Lines 1 and 2. λ offspring individuals are gen- erated from Line 5 to Line 11 in the following way. For each offspring, the mutation of σ (g) is performed in Line 6 using the log-normal operator e τ N l (0,1) , where N l (0, 1) is a standard normally distributed random scalar. The learning parameter τ in the log-normal operator controls the self-adaptation rate. In Line 7, an isotropic mutation direction is generated by means of a random vector N l (0, I) the components of which are standard normal variates. This direction vector is scaled with the individual’s mutation strength ˜ σ l in Line 8 forming the mutation. The offspring vector ˜ y l is generated in Line 9 and used in the calculation of the objective function value ˜ F l in Line 10.