Abstract
Previous research proposed the uniform mutation inside the sphere as a new mutation operator for evolution strategies (continuous evolutionary algorithms), with a case study of the elitist algorithm on the SPHERE. For that landscape, one-step success probability and expected progress were estimated analytically, and further proved to converge, as space dimension increases, to the corresponding asymptotics of the algorithm with normal mutation. This article takes the analysis further by considering the RIDGE, an asymmetric landscape almost uncovered in the literature. For the elitist algorithm, estimates of expected progress along the radial and longitudinal axes are derived, then tested numerically against the real behavior of the algorithm on several functions from this class. The global behavior of the algorithm is predicted correctly by iterating the one-step analytical formulas. Moreover, experiments show identical mean value dynamics for the algorithms with uniform and normal mutation, which implies that the derived formulas apply also to the normal case. Essential to the whole analysis is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> , the inclination angle of the RIDGE. The behavior of the algorithm on the SPHERE and HYPERPLANE is also obtained, at the limits of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> interval (0°, 90°].
Highlights
A general pattern of any successful optimization theory is the following
One starts with a particular algorithm and a minimally interesting fitness function, progressively increases the complexity of either algorithm or fitness function. The more it departs from the starting point, more approximations are considered, such that the analytical formulas must always be verified by numerical simulation
An astute reader of Evolution Strategy (ES) theory would be surprised by the scarcity of results outside the symmetrical SPHERE landscape
Summary
A general pattern of any successful optimization theory is the following. One starts with a particular algorithm and a minimally interesting fitness function, progressively increases the complexity of either algorithm or fitness function. Less systematic approaches considered different measures of progress [3], non-standard mutation distributions [34] or supplementary conditions on the algorithm/fitness function [17], [19], [20], [21], yielding analytical results with limited power of generalization. Random variables are approximated through their expected values, terms vanish, Taylor series expansions are truncated after one or two terms and non-symmetric random variables are replaced by normal distributions Such methods could be criticized for not being mathematically rigorous, yet they are efficient - analytical results are always verified by experiments, obtained from numerical simulation of the algorithm [8], [11], [33].
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