Theory of (1 + 1) ES on the RIDGE

Abstract

Previous research proposed the uniform mutation inside the sphere as a new mutation operator for evolution strategies (continuous evolutionary algorithms), with case study 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 paper takes the analysis further by considering the RIDGE, an asymmetric landscape almost uncovered in 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 θ, the inclination angle of the RIDGE. The behavior of the algorithm on the SPHERE and HYPERPLANE is also obtained, at the limits of the θ interval (0 • , 90 • ].

Spherical Distributions Used in Evolutionary Algorithms

“…Rather, specific algorithms with particular operators should be analyzed separately, on different optimization landscapes. This is where the second advantage of the new uniform distribution occurs, in terms of more tractable mathematical analysis, yielding close formulas, previously not attained by normal mutation theory-see the studies of the RIDGE landscape in [28] and of the elitist evolutionary algorithm with mutation and crossover on SPHERE in [27].…”

“…One could consider only the first component of the expected value, the one pointing towards the optimum, which has been applied on a different fitness landscape, the inclined plane [21]. (Yet in another landscape, the RIDGE, it is customary to consider the progress along the perpendicular component h, see in [6,27] for an inventory of fitness functions used in EA testing, the reader is referred to the work in [32].) For UNIFORM mutation, a simplified version of the expected progress may be defined as the centroid of the corresponding success region [9,10].…”

“…A theory of continuous evolutionary algorithms could not be complete without the analysis of global behavior and adaptive mutation parameter. These cases have already been treated in [27,28]-under a normalization of mutation sphere radius which makes algorithm behave similarly to the one with normal mutation, in terms of difference and differential equations, following the works in [6,29]. This opens the way for the challenging task, previously unattempted in probabilistic optimization literature, of linking the theory of continuous evolutionary algorithms to that of differential optimization techniques such as particle swarm optimization [40] and differential evolution [41].…”

“…From a broader perspective, this paper can be seen, together with the works in [27,28], as an attempt of revisiting the classical theory of continuous evolutionary algorithms. Even if practitioners in the field will continue to use the normal multi-variate as mutation distribution, we claim that the theory can benefit from the uniform distribution inside the sphere.…”

“…The study of EAs with uniform distribution in the sphere recently culminated with two systematic studies, one for RIDGE [27], the other for SPHERE [28], comparable to classical theory of evolution strategies [6,29]. Under a carefully constructed normalization of mutation parameters (equalizing the expectations of normal and uniform multi-variates as n → ∞), those studies demonstrate the same behavior for the respective EA variants.…”