Spherical Distributions Used in Evolutionary Algorithms

Alexandru Agapie

doi:10.3390/math9233098

Abstract

Performance of evolutionary algorithms in real space is evaluated by local measures such as success probability and expected progress. In high-dimensional landscapes, most algorithms rely on the normal multi-variate, easy to assemble from independent, identically distributed components. This paper analyzes a different distribution, also spherical, yet with dependent components and compact support: uniform in the sphere. Under a simple setting of the parameters, two algorithms are compared on a quadratic fitness function. The success probability and the expected progress of the algorithm with uniform distribution are proved to dominate their normal mutation counterparts by order n!!.

Highlights

Probabilistic algorithms are among the most popular optimization techniques due to their easy implementation and high efficiency. Their roots can be traced back to the first random walk problem proposed by Pearson in 1905: “A man starts from a point O, and walks yards in a straight line; he turns through any angle whatever, and walks another yards in a second straight line
We prove the result on conditional marginals of the uniform distribution in the unit sphere
We restrict the study to the case P nearby O and analyze the local performance of two evolutionary algorithms (EAs), one with uniform, the other with normal mutation, in terms of success probability and expected progress

Summary

Introduction

Probabilistic algorithms are among the most popular optimization techniques due to their easy implementation and high efficiency Their roots can be traced back to the first random walk problem proposed by Pearson in 1905: “A man starts from a point O, and walks yards in a straight line; he turns through any angle whatever, and walks another yards in a second straight line. I require the probability that after these n stretches he is at a distance between r and r + dr from his starting point O.” [1,2,3,4]. Probabilistic algorithms do not require additional information on the fitness function, they generate potential candidate solutions, select the best, and move on

Methods

Results

Discussion

Conclusion