Parallel black-box complexity with tail bounds

We propose a new black-box complexity model for search algorithms evaluating λ search points in parallel. The parallel unary unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unary unbiased black-box algorithm needs to optimise a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. We present complexity results for LeadingOnes and OneMax. Our main result is a general performance limit: we prove that on every function every λ-parallel unary unbiased algorithm needs at least a certain number of evaluations (a function of problem size and λ) to find any desired target set of up to exponential size, with an overwhelming probability. This yields lower bounds for the typical optimisation time on unimodal and multimodal problems, for the time to find any local optimum, and for the time to even get close to any optimum. The power and versatility of this approach is shown for a wide range of illustrative problems from combinatorial optimisation. Our performance limits can guide parameter choice and algorithm design; we demonstrate the latter by presenting an optimal λ-parallel algorithm for OneMax that uses parallelism most effectively

Level-Based Analysis of Genetic Algorithms and Other Search Processes

The fitness-level technique is a simple and old way to derive upper bounds for the expected runtime of simple elitist evolutionary algorithms (EAs). Recently, the technique has been adapted to deduce the runtime of algorithms with non-elitist populations and unary variation operators [2,8]. In this paper, we show that the restriction to unary variation operators can be removed. This gives rise to a much more general analytical tool which is applicable to a wide range of search processes. As introductory examples, we provide simple runtime analyses of many variants of the Genetic Algorithm on well-known benchmark functions, such as OneMax, LeadingOnes, and the sorting problem

Escaping Local Optima Using Crossover with Emergent Diversity

Population diversity is essential for avoiding premature convergence in Genetic Algorithms and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous run time analysis to gain insight into population dynamics and Genetic Algorithm performance for the (μ+1) Genetic Algorithm and the Jump test function. We show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to significant improvements of the expected optimisation time compared to mutation-only algorithms like the (1+1) Evolutionary Algorithm. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to even larger speedups. Experiments were conducted to complement our theoretical findings and further highlight the benefits of crossover on the function class

Emergence of Diversity and its Benefits for Crossover in Genetic Algorithms

Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for a standard (µ+1) GA and the Jumpk test function. By studying the stochastic process underlying the size of the largest collection of identical genotypes we show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order Ω(n/ log n) compared to mutationonly algorithms like the (1+1) EA

Editorial to the special issue on "Theoretical foundations of evolutionary computation"

Abstract not availablePer Kristian Lehre, Frank Neumann, Jonathan E. Rowe, Xin Ya

General Lower Bounds for the Running Time of Evolutionary Algorithms

Abstract. We present a new method for proving lower bounds in evolutionary computation based on fitness-level arguments and an additional condition on transition probabilities between fitness levels. The method yields exact or near-exact lower bounds for LO, OneMax, and all functions with a unique optimum. All lower bounds hold for every evolutionary algorithm that only uses standard mutation as variation operator.

Negative Drift in Populations

Abstract. An important step in gaining a better understanding of the stochastic dynamics of evolving populations, is the development of appropriate analytical tools. We present a new drift theorem for populations that allows properties of their long-term behaviour, e. g. the runtime of evolutionary algorithms, to be derived from simple conditions on the onestep behaviour of their variation operators and selection mechanisms.