scholarly journals Mirrored Orthogonal Sampling for Covariance Matrix Adaptation Evolution Strategies

2019 ◽  
Vol 27 (4) ◽  
pp. 699-725 ◽  
Author(s):  
Hao Wang ◽  
Michael Emmerich ◽  
Thomas Bäck
Keyword(s):  
Convergence Rate ◽  
Covariance Matrix ◽  
Sampling Technique ◽  
Black Box ◽  
Evolution Strategies ◽  
High Dimensional ◽  
Major Purpose ◽  
Search Spaces ◽  
Covariance Matrix Adaptation ◽  
Adaptation Evolution

Generating more evenly distributed samples in high dimensional search spaces is the major purpose of the recently proposed mirrored sampling technique for evolution strategies. The diversity of the mutation samples is enlarged and the convergence rate is therefore improved by the mirrored sampling. Motivated by the mirrored sampling technique, this article introduces a new derandomized sampling technique called mirrored orthogonal sampling. The performance of this new technique is both theoretically analyzed and empirically studied on the sphere function. In particular, the mirrored orthogonal sampling technique is applied to the well-known Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The resulting algorithm is experimentally tested on the well-known Black-Box Optimization Benchmark (BBOB). By comparing the results from the benchmark, mirrored orthogonal sampling is found to outperform both the standard CMA-ES and its variant using mirrored sampling.

scholarly journals Diagonal Acceleration for Covariance Matrix Adaptation Evolution Strategies

2020 ◽  
Vol 28 (3) ◽  
pp. 405-435 ◽  
Author(s):  
Y. Akimoto ◽  
N. Hansen
Keyword(s):  
Covariance Matrix ◽  
Large Population ◽  
Positive Definiteness ◽  
Diagonal Matrix ◽  
Evolution Strategies ◽  
Covariance Matrix Adaptation ◽  
Population Sizes ◽  
Matrix Formula ◽  
Adaptation Evolution ◽  
Landscape Feature

We introduce an acceleration for covariance matrix adaptation evolution strategies (CMA-ES) by means of adaptive diagonal decoding (dd-CMA). This diagonal acceleration endows the default CMA-ES with the advantages of separable CMA-ES without inheriting its drawbacks. Technically, we introduce a diagonal matrix [Formula: see text] that expresses coordinate-wise variances of the sampling distribution in DCD form. The diagonal matrix can learn a rescaling of the problem in the coordinates within a linear number of function evaluations. Diagonal decoding can also exploit separability of the problem, but, crucially, does not compromise the performance on nonseparable problems. The latter is accomplished by modulating the learning rate for the diagonal matrix based on the condition number of the underlying correlation matrix. dd-CMA-ES not only combines the advantages of default and separable CMA-ES, but may achieve overadditive speedup: it improves the performance, and even the scaling, of the better of default and separable CMA-ES on classes of nonseparable test functions that reflect, arguably, a landscape feature commonly observed in practice. The article makes two further secondary contributions: we introduce two different approaches to guarantee positive definiteness of the covariance matrix with active CMA, which is valuable in particular with large population size; we revise the default parameter setting in CMA-ES, proposing accelerated settings in particular for large dimension. All our contributions can be viewed as independent improvements of CMA-ES, yet they are also complementary and can be seamlessly combined. In numerical experiments with dd-CMA-ES up to dimension 5120, we observe remarkable improvements over the original covariance matrix adaptation on functions with coordinate-wise ill-conditioning. The improvement is observed also for large population sizes up to about dimension squared.

Adaptive Niche Radii and Niche Shapes Approaches for Niching with the CMA-ES

2010 ◽  
Vol 18 (1) ◽  
pp. 97-126 ◽  
Author(s):  
Ofer M. Shir ◽  
Michael Emmerich ◽  
Thomas Bäck
Keyword(s):  
Covariance Matrix ◽  
High Dimensional ◽  
Step Size ◽  
Distance Calculation ◽  
Covariance Matrix Adaptation ◽  
Niching Methods ◽  
Geometrical Shapes ◽  
Matrix Mechanism ◽  
Adaptation Evolution ◽  
Radius Problem

While the motivation and usefulness of niching methods is beyond doubt, the relaxation of assumptions and limitations concerning the hypothetical search landscape is much needed if niching is to be valid in a broader range of applications. Upon the introduction of radii-based niching methods with derandomized evolution strategies (ES), the purpose of this study is to address the so-called niche radius problem. A new concept of an adaptive individual niche radius is applied to niching with the covariance matrix adaptation evolution strategy (CMA-ES). Two approaches are considered. The first approach couples the radius to the step size mechanism, while the second approach employs the Mahalanobis distance metric with the covariance matrix mechanism for the distance calculation, for obtaining niches with more complex geometrical shapes. The proposed approaches are described in detail, and then tested on high-dimensional artificial landscapes at several levels of difficulty. They are shown to be robust and to achieve satisfying results.

Optimization Design of Trusses Based on Covariance Matrix Adaptation Evolution Strategy Algorithm

2012 ◽  
Vol 215-216 ◽  
pp. 133-137
Author(s):  
Guo Shao Su ◽  
Yan Zhang ◽  
Zhen Xing Wu ◽  
Liu Bin Yan
Keyword(s):  
Stochastic Optimization ◽  
Covariance Matrix ◽  
Optimization Problems ◽  
Fitness Function ◽  
Optimization Methods ◽  
Evolution Strategy ◽  
Covariance Matrix Adaptation ◽  
Study Results ◽  
Highly Nonlinear ◽  
Adaptation Evolution

Covariance matrix adaptation evolution strategy algorithm (CMA-ES) is a newly evolution algorithm. It has become a powerful tool for solving highly nonlinear multi-peak optimization problems. In many real-world optimization problems, the location of multiple optima is often required in a search space. In order to evaluate the solution, thousands of fitness function evaluations are involved that is a time consuming or expensive processes. Therefore, conventional stochastic optimization methods meet a special challenge for a very large number of problem function evaluations. Aiming to overcome the shortcoming of stochastic optimization methods in the high calculation cost, a truss optimal method based on CMA-ES algorithm is proposed and applied to solve the section and shape optimization problems of trusses. The study results show that the method is feasible and has the advantages of high accuracy, high efficiency and easy implementation.

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