Principled Design and Runtime Analysis of Abstract Convex Evolutionary Search

Geometric crossover is a formal class of crossovers which includes many well-known recombination operators across representations. In previous work, it was shown that all evolutionary algorithms with geometric crossover (but with no mutation) do the same form of convex search regardless of the underlying representation, the specific selection mechanism, offspring distribution, search space, and problem at hand. Furthermore, it was suggested that the generalised convex search could perform well on generalised forms of concave and approximately concave fitness landscapes, regardless of the underlying space and representation. In this article, we deepen this line of enquiry and study the runtime of generalised convex search on concave fitness landscapes. This is a first step towards linking a geometric theory of representations and runtime analysis in the attempt to (i) set the basis for a more general, unified approach for the runtime analysis of evolutionary algorithms across representations, and (ii) identify the essential matching features of evolutionary search behaviour and landscape topography that cause polynomial performance. We present a general runtime result that can be systematically instantiated to specific search spaces and representations, and present its specifications to three search spaces. As a corollary, we obtain that the convex search algorithm optimises LeadingOnes in O (n log n) fitness evaluations, which is faster than all unbiased unary black-box algorithms

Runtime analysis of convex evolutionary search algorithm with standard crossover

This is the final version. Available on open access from Elsevier via the DOI in this recordEvolutionary Algorithms (EAs) with no mutation can be generalized across representations as Convex Evolu- tionary Search algorithms (CSs). However, the crossover operator used by CSs does not faithfully generalize the standard two-parents crossover: it samples a convex hull instead of a segment. Segmentwise Evolutionary Search algorithms (SESs) are defined as a more faithful generalization, equipped with a crossover operator that samples the metric segment of two parents. In metric spaces where the union of all possible segments of a given set is always a convex set, a SES is a particular CS. Consequently, the representation-free analysis of the CS on quasi- concave landscapes can be extended to the SES in these particular metric spaces. When instantiated to binary strings of the Hamming space (resp. -ary strings of the Manhattan space), a polynomial expected runtime upper bound is obtained for quasi-concave landscapes with at most polynomially many level sets for well-chosen popu- lation sizes. In particular, the SES solves Leading Ones in at most 288 ln [4 (2 + 1)] expected fitness evaluations when the population size is equal to 144 ln [4 (2 + 1)]

Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point

Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multi-objective problems as the population of such an algorithm can be used to represent the trade-offs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multi-objective problems. We consider indicator-based algorithms whose goal is to maximize the hypervolume for a given problem by distributing {\mu} points on the Pareto front. To gain new theoretical insights into the behavior of hypervolume-based algorithms we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of bi-objective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolume-based approaches and examine Pareto fronts of different shapes by numerical calculations

Statistical and Computational Tradeoff in Genetic Algorithm-Based Estimation

When a Genetic Algorithm (GA), or a stochastic algorithm in general, is employed in a statistical problem, the obtained result is affected by both variability due to sampling, that refers to the fact that only a sample is observed, and variability due to the stochastic elements of the algorithm. This topic can be easily set in a framework of statistical and computational tradeoff question, crucial in recent problems, for which statisticians must carefully set statistical and computational part of the analysis, taking account of some resource or time constraints. In the present work we analyze estimation problems tackled by GAs, for which variability of estimates can be decomposed in the two sources of variability, considering some constraints in the form of cost functions, related to both data acquisition and runtime of the algorithm. Simulation studies will be presented to discuss the statistical and computational tradeoff question.Comment: 17 pages, 5 figure