Concentration of First Hitting Times Under Additive Drift

Kötzing, Timo

doi:10.1007/s00453-015-0048-0

Cited by 52 publications

(37 citation statements)

References 59 publications

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“…The next theorem gives a lower bound on hitting times of random walks even if we start close to the goal, provided that the drift towards the goal is weak. We remark that the statement on the expectation is similar to other lower bounds for additive drift [9], but the existing tail bounds are tailored to the regime of strong drift, and are thus not tight in our case. We prove it by martingale theory.…”

Section: Theorem 33 (Variable Driftsupporting

confidence: 51%

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Bounding bloat in genetic programming

Doerr

Kötzing

Lagodzinski

et al. 2017

Proceedings of the Genetic and Evolutionary Computation Conference

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While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat (unnecessary growth of solutions) slowing down optimization. Theoretical analyses could so far not bound bloat and required explicit assumptions on the magnitude of bloat.In this paper we analyze bloat in mutation-based genetic programming for the two test functions Order and Majority. We overcome previous assumptions on the magnitude of bloat and give matching or close-to-matching upper and lower bounds for the expected optimization time.In particular, we show that the (1+1) GP takes (i) ΘpT init`n log nq iterations with bloat control on Order as well as Majority; and (ii) OpT init log T init`n plog nq 3 q and ΩpT init`n log nq (and ΩpT init log T init q for n " 1) iterations without bloat control on Majority. 1

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Section: Theorem 33 (Variable Driftsupporting

confidence: 51%

“…For bloat estimation we need a lower bound drift theorem in the regime of weak additive drift. A related theorem (Theorem 3.5) follows from Theorem 10 and 12 in [9]. Theorem 3.5 is not directly applicable to our situation, since it gives only tight bounds in the regime of strong drift.…”

Section: Theorem 33 (Variable Driftmentioning

confidence: 99%

Bounding bloat in genetic programming

Doerr

Kötzing

Lagodzinski

et al. 2017

Proceedings of the Genetic and Evolutionary Computation Conference

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“…By a refinement to the negative drift theorem of Oliveto and Witt [25,24] (cf Theorem 3 of [18]), since (1) . So, for any polynomial s = poly(n), with probability superpolynomially close to one, Y s has not yet reached a state larger than (1/2 − a)K, and so p i,t > a for all 1 ≤ t ≤ s. As this holds for arbitrary i, applying a union bound retains a superpolynomially small probability that any of the n frequencies have gone below a by s = poly(n) steps.…”

Section: Compact Gamentioning

confidence: 97%

The Benefit of Recombination in Noisy Evolutionary Search

Friedrich

Kötzing

Krejca

et al. 2015

Lecture Notes in Computer Science

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The benefit of sexual recombination is one of the most fundamental questions both in population genetics and evolutionary computation. It is widely believed that recombination helps solving difficult optimization problems. We present the first result, which rigorously proves that it is beneficial to use sexual recombination in an uncertain environment with a noisy fitness function. For this, we model sexual recombination with a simple estimation of distribution algorithm called the Compact Genetic Algorithm (cGA), which we compare with the classical µ + 1 EA. For a simple noisy fitness function with additive Gaussian posterior noise N (0, σ 2 ), we prove that the mutation-only µ + 1 EA typically cannot handle noise in polynomial time for σ 2 large enough while the cGA runs in polynomial time as long as the population size is not too small. This shows that in this uncertain environment sexual recombination is provably beneficial. We observe the same behavior in a small empirical study.

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“…By Lemma 5 and 6, we see that with probability at least p = 1 − (1/e) − o(1), one iteration of the main loop of Algorithm 1 produces a solution y with Om(y) ≥ Om(x) + ∆, where ∆ is some number satisfying ∆ = Ω(log(λ)/ log log(λ)). This seems to call for an application of additive drift (Theorem 4), but in particular for the derivation of the large deviation claim, the following handmade solution seems to be easier (despite several tail bounds for additive drift existing, see, e.g., [46] and the references therein).…”

Section: Proof (Of Theorem 7)mentioning

confidence: 99%

Optimal Static and Self-Adjusting Parameter Choices for the $$(1+(\lambda ,\lambda ))$$ ( 1 + ( λ , λ ) ) Genetic Algorithm

2017

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The (1 + (λ, λ)) genetic algorithm (GA) proposed in [Doerr, Doerr, and Ebel. From black-box complexity to designing new genetic algorithms. Theoretical Computer Science (2015)] is one of the few examples for which a super-constant speed-up of the expected optimization time through the use of crossover could be rigorously demonstrated. It was proven that the expected optimization time of this algorithm on OneMax is O(max{n log(n)/λ, λn}) for any offspring population size λ ∈ {1, . . . , n} (and the other parameters suitably dependent on λ) and it was shown experimentally that a self-adjusting choice of λ leads to a better, most likely linear, runtime.In this work, we study more precisely how the optimization time depends on the parameter choices, leading to the following results on how to optimally choose the population size, the mutation probability, and the crossover bias both in a static and a dynamic fashion.For the mutation probability and the crossover bias depending on λ as in [DDE15], we improve the previous runtime bound to O(max{n log(n)/λ, nλ log log(λ)/ log(λ)}). This expression is minimized by a value of λ slightly larger than what the previous result suggested and gives an expected optimization time of O n log(n) log log log(n)/ log log(n) .We show that no static choice in the three-dimensional parameter space of offspring population, mutation probability, and crossover bias gives an asymptotically better runtime.Results presented in this work are based on [12][13][14]. B. DoerŕEcole Polytechnique, LIX -UMR 7161,

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Concentration of First Hitting Times Under Additive Drift

Cited by 52 publications

References 59 publications

Bounding bloat in genetic programming

Bounding bloat in genetic programming

The Benefit of Recombination in Noisy Evolutionary Search

Optimal Static and Self-Adjusting Parameter Choices for the $$(1+(\lambda ,\lambda ))$$ ( 1 + ( λ , λ ) ) Genetic Algorithm

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