Abstract
Escaping local optima is one of the major obstacles to function optimisation. Using the metaphor of a fitness landscape, local optima correspond to hills separated by fitness valleys that have to be overcome. We define a class of fitness valleys of tunable difficulty by considering their length, representing the Hamming path between the two optima and their depth, the drop in fitness. For this function class we present a runtime comparison between stochastic search algorithms using different search strategies. The (1+1) EA is a simple and well-studied evolutionary algorithm that has to jump across the valley to a point of higher fitness because it does not accept worsening moves (elitism). In contrast, the Metropolis algorithm and the Strong Selection Weak Mutation (SSWM) algorithm, a famous process in population genetics, are both able to cross the fitness valley by accepting worsening moves. We show that the runtime of the (1+1) EA depends critically on the length of the valley while the runtimes of the non-elitist algorithms depend crucially on the depth of the valley. Moreover, we show that both SSWM and Metropolis can also efficiently optimise a rugged function consisting of consecutive valleys.
Highlights
Black box algorithms are general purpose optimisation tools typically used when no good problem specific algorithm is known for the problem at hand
We demonstrate the generality of the presented mathematical tool by using it to prove that the same asymptotic results achieved by Strong Selection Weak Mutation (SSWM) hold for the well-known Metropolis algorithm that, differently from SSWM, always accepts improving moves
We presented an analysis of randomised search heuristics for crossing fitness valleys where no mutational bias exists and the probability for moving forwards or backwards on the path depends only on the fitness difference between neighbouring search points
Summary
Black box algorithms are general purpose optimisation tools typically used when no good problem specific algorithm is known for the problem at hand. It has been shown that the interplay between the two variation operators, mutation and crossover, may efficiently give rise to the necessary burst of diversity without the need of any artificial diversity mechanism [3] Another combination that has been proven to be effective for elitist algorithms to overcome local optima is to alternate mutations with variable depth search [35]. A very different approach is to attempt to escape by accepting solutions of lower fitness in the hope of eventually leaving the basin of attraction of the local optimum This approach is the main driving force behind non-elitist algorithms. This generalisation allows the results to hold for a broader family of functions
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