Runtime analysis of the (1+1) evolutionary algorithm on strings over finite alphabets

We analyze the run-time of the (1 + 1) Evolutionary Algorithm optimizing an arbitrary linear function f : {0, 1, . . . , r} n → R. If the mutation probability of the algorithm is p = c/n, then (1 + o(1))(e c /c)rn log n + O(r 3 n log log n) is an upper bound for the expected time needed to find the optimum. We also give a lower bound of (1 + o(1))(1/c)rn log n. Hence for constant c and all r slightly smaller than (log n) 1/3 , our bounds deviate by only a constant factor, which is e(1 + o(1)) for the standard mutation probability of 1/n. The proof of the upper bound uses multiplicative adaptive drift analysis as developed in a series of recent papers. We cannot close the gap for larger values of r, but find indications that multiplicative drift is not the optimal analysis tool for this case.

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“…Doerr, Johanssen, and Schmidt in [5] were the first to regard the problem for linear functions of the form…”

Section: Linear Functions Over Larger Domainsmentioning

confidence: 99%

“…The work [5] showed that the classic drift methods used to analyze evolutionary algorithms (EA) optimizing linear functions f : {0, 1} n → R cannot be extended to the natural generalization of the problem to linear functions f : {0, . .…”

Section: Introductionmentioning

confidence: 99%

Run-time analysis of the (1+1) evolutionary algorithm optimizing linear functions over a finite alphabet

Doerr

Pöhl

2012

Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation

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“…We extend the (1+1) EA by letting mutation change any position independently with probability 1/n; any changed position is randomly increased or decreased by one (with probability 1/2 each). Note that similar extensions of the OneMax function (without dynamic changes) have been studied by Doerr, Johannsen, and Schmidt (2011) and Doerr and Pohl (2012); they considered arbitrary linear functions over {0, . .…”

Section: Introductionmentioning

confidence: 99%

(1+1) EA on Generalized Dynamic OneMax

Kötzing

Lissovoi

Witt

2015

Proceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms XIII

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Evolutionary algorithms (EAs) perform well in settings involving uncertainty, including settings with stochastic or dynamic fitness functions. In this paper, we analyze the (1+1) EA on dynamically changing OneMax, as introduced by Droste (2003). We re-prove the known results on first hitting times using the modern tool of drift analysis. We extend these results to search spaces which allow for more than two values per dimension.Furthermore, we make an anytime analysis as suggested by Jansen and Zarges (2014), analyzing how closely the (1+1) EA can track the dynamically moving optimum over time. We get tight bounds both for the case of bit strings, as well as for the case of more than two values per position. Surprisingly, in the latter setting, the expected quality of the search point maintained by the (1+1) EA does not depend on the number of values per dimension.

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“…, n}, that is, in the factors [0..r − 1], then the natural analogue of the standard-bit mutation operator is to select each component i ∈ [1..n] independently and mutate the selected components by changing the current value to a random other value in [0..r − 1]. This operator was used in [STW04,Gun05] as well as in the theoretical works [DJS11,DP12].…”

Section: Mutation Operators For Multi-valued Search Spacesmentioning

confidence: 99%

The Right Mutation Strength for Multi-Valued Decision Variables

Doerr

Koetzing

2016

Proceedings of the Genetic and Evolutionary Computation Conference 2016

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The most common representation in evolutionary computation are bit strings. This is ideal to model binary decision variables, but less useful for variables taking more values. With very little theoretical work existing on how to use evolutionary algorithms for such optimization problems, we study the run time of simple evolutionary algorithms on some OneMax-like functions defined over Ω = {0, 1, . . . , r − 1}n . More precisely, we regard a variety of problem classes requesting the component-wise minimization of the distance to an unknown target vector z ∈ Ω.For such problems we see a crucial difference in how we extend the standard-bit mutation operator to these multi-valued domains. While it is natural to select each position of the solution vector to be changed independently with probability 1/n, there are various ways to then change such a position. If we change each selected position to a random value different from the original one, we obtain an expected run time of Θ(nr log n). If we change each selected position by either +1 or −1 (random choice), the optimization time reduces to Θ(nr + n log n). If we use a random mutation strength i ∈ {0, 1, . . . , r − 1} n with probability inversely proportional to i and change the selected position by either +i or −i (random choice), then the optimization time becomes Θ(n log(r)(log(n) + log(r))), bringing down the dependence on r from linear to polylogarithmic.One of our results depends on a new variant of the lower bounding multiplicative drift theorem.

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Runtime analysis of the (1+1) evolutionary algorithm on strings over finite alphabets

Cited by 10 publications

References 23 publications

Run-time analysis of the (1+1) evolutionary algorithm optimizing linear functions over a finite alphabet

Run-time analysis of the (1+1) evolutionary algorithm optimizing linear functions over a finite alphabet

(1+1) EA on Generalized Dynamic OneMax

The Right Mutation Strength for Multi-Valued Decision Variables

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