Crossover can provably be useful in evolutionary computation

Doerr, Benjamin; Happ, Edda; Klein, Christian

doi:10.1016/j.tcs.2010.10.035

Cited by 97 publications

(58 citation statements)

References 15 publications

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“…The utility of crossover has also been proved for less artificial problems such as coloring problems inspired by the Ising model from physics [22], computing input-output sequences in finite state machines [23], shortest path problems [24], vertex cover [25] and multi-objective optimization problems [26]. The above works show that crossover allows to escape from local optima that have large basins of attraction for the mutation operator.…”

Section: Related Workmentioning

confidence: 97%

Standard Steady State Genetic Algorithms Can Hillclimb Faster Than Mutation-Only Evolutionary Algorithms

Corus

Oliveto

2018

IEEE Trans. Evol. Computat.

108

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Explaining to what extent the real power of genetic algorithms lies in the ability of crossover to recombine individuals into higher quality solutions is an important problem in evolutionary computation. In this paper we show how the interplay between mutation and crossover can make genetic algorithms hillclimb faster than their mutation-only counterparts. We devise a Markov Chain framework that allows to rigorously prove an upper bound on the runtime of standard steady state genetic algorithms to hillclimb the OneMax function. The bound establishes that the steady-state genetic algorithms are 25% faster than all standard bit mutation-only evolutionary algorithms with static mutation rate up to lower order terms for moderate population sizes. The analysis also suggests that larger populations may be faster than populations of size 2. We present a lower bound for a greedy (2+1) GA that matches the upper bound for populations larger than 2, rigorously proving that 2 individuals cannot outperform larger population sizes under greedy selection and greedy crossover up to lower order terms. In complementary experiments the best population size is greater than 2 and the greedy genetic algorithms are faster than standard ones, further suggesting that the derived lower bound also holds for the standard steady state (2+1) GA.

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Section: Related Workmentioning

confidence: 97%

Standard Steady State Genetic Algorithms Can Hillclimb Faster Than Mutation-Only Evolutionary Algorithms

Corus

Oliveto

2018

IEEE Trans. Evol. Computat.

108

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“…As an example, we use another combinatorial problem from [STW04], the singlesource shortest path problem, and obtain a simplified and more natural proof of the currently strongest runtime bound for this problem from [DHK11]. We note without proof that similar arguments could be used in the analysis of other problems where the evolutionary algorithm builds up the optimal solution incrementally from structurally smaller solutions, such as other path problems [The09,DJ10,DHK12] or dynamic programming [DEN + 11].…”

Section: Beyond the Fitness Level Theoremmentioning

confidence: 99%

Better Runtime Guarantees via Stochastic Domination

Doerr

2018

Evolutionary Computation in Combinatorial Optimization

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Apart from few exceptions, the mathematical runtime analysis of evolutionary algorithms is mostly concerned with expected runtimes. In this work, we argue that stochastic domination is a notion that should be used more frequently in this area. Stochastic domination allows to formulate much more informative performance guarantees, it allows to decouple the algorithm analysis into the true algorithmic part of detecting a domination statement and the probabilitytheoretical part of deriving the desired probabilistic guarantees from this statement, and it helps finding simpler and more natural proofs.As particular results, we prove a fitness level theorem which shows that the runtime is dominated by a sum of independent geometric random variables, we prove the first tail bounds for several classic runtime problems, and we give a short and natural proof for Witt's result that the runtime of any (µ, p) mutation-based algorithm on any function with unique optimum is subdominated by the runtime of a variant of the (1 + 1) EA on the OneMax function. As sideproducts, we determine the fastest unbiased (1+1) algorithm for the * Extended version of a paper that appeared at EvoCOP 2018 [Doe18a]. This version contains as new material a section on known precise runtime distributions, a Chernoff bound for sums of independent coupon collector runtimes, several new tail bounds for classic runtime results, and a section on counter-examples.LeadingOnes benchmark problem, both in the general case and when restricted to static mutation operators, and we prove a Chernoff-type tail bound for sums of independent coupon collector distributions.

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“…We make this elementary insight precise in the following lemma. This type of argument was used, among others, in works on shortest path problems [DJ10, DHK11,DHK12]. There one can show that, in each iteration, independent of the past, with at least a certain probability an extra edge of a desired path is found.…”

Section: Unconditional Sequential Dominationmentioning

confidence: 99%

Probabilistic Tools for the Analysis of Randomized Optimization Heuristics

Doerr¹

2019

Natural Computing Series

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118

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This chapter collects several probabilistic tools that proved to be useful in the analysis of randomized search heuristics. This includes classic material like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics like stochastic domination and coupling or Chernoff bounds for geometrically distributed random variables and for negatively correlated random variables. Most of the results presented here have appeared previously, some, however, only in recent conference publications. While the focus is on collecting tools for the analysis of randomized search heuristics, many of these may be useful as well in the analysis of classic randomized algorithms or discrete random structures.

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Crossover can provably be useful in evolutionary computation

Cited by 97 publications

References 15 publications

Standard Steady State Genetic Algorithms Can Hillclimb Faster Than Mutation-Only Evolutionary Algorithms

Standard Steady State Genetic Algorithms Can Hillclimb Faster Than Mutation-Only Evolutionary Algorithms

Better Runtime Guarantees via Stochastic Domination

Probabilistic Tools for the Analysis of Randomized Optimization Heuristics

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