Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

Oliveto, Pietro S.; Witt, Carsten

doi:10.1007/s00453-010-9387-z

Cited by 118 publications

(56 citation statements)

References 20 publications

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“…Even more important, it quickly became one of the most powerful tools for both proving upper and lower bounds on the expected optimization times of evolutionary algorithms. For example, see [HY04,GW03,GL06,HJKN08,NOW09,OW].…”

Section: Introductionmentioning

confidence: 99%

Multiplicative Drift Analysis

2012

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In this work, we introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms.We give a multiplicative version of the classical drift theorem. This allows easier analyses in those settings where the optimization progress is roughly proportional to the current distance to the optimum.To display the strength of this tool, we regard the classical problem how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time O(n log n), where n is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most (1 + o(1))1.39en ln(n), again using multiplicative drift analysis. We also prove a corresponding lower bound of (1 − o(1))en ln(n) which actually holds for all functions with a unique global optimum.We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours. * Carola Winzen is a recipient of the Google Europe Fellowship in Randomized Algorithms, and this work is supported in part by this Google Fellowship.

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Section: Introductionmentioning

confidence: 99%

Multiplicative Drift Analysis

2012

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“…Theorem 5 (Negative Drift Theorem [OW11]). Let X t , t ≥ 0 be real-valued random variables describing a stochastic process over some state space, with filtration F t := (X 0 , .…”

Section: Negative Driftmentioning

confidence: 99%

OneMax in Black-Box Models with Several Restrictions

Doerr¹,

Lengler²

2016

Algorithmica

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Black-box complexity studies lower bounds for the efficiency of general-purpose black-box optimization algorithms such as evolutionary algorithms and other search heuristics. Different models exist, each one being designed to analyze a different aspect of typical heuristics such as the memory size or the variation operators in use. While most of the previous works focus on one particular such aspect, we consider in this work how the combination of several algorithmic restrictions influence the blackbox complexity. Our testbed are so-called OneMax functions, a classical set of test functions that is intimately related to classic coin-weighing problems and to the board game Mastermind.We analyze in particular the combined memory-restricted ranking-based blackbox complexity of OneMax for different memory sizes. While its isolated memoryrestricted as well as its ranking-based black-box complexity for bit strings of length n is only of order n/ log n, the combined model does not allow for algorithms being faster than linear in n, as can be seen by standard information-theoretic considerations. We show that this linear bound is indeed asymptotically tight. Similar results are obtained for other memory-and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of OneMax in the recently introduced elitist model, in which only the best-so-far solution can be kept in the memory. Finally, we also provide improved lower bounds for the complexity of OneMax in the regarded models.Our result enlivens the quest for natural evolutionary algorithms optimizing OneMax in o(n log n) iterations.

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“…P N i=1 Q i j=1 x j have been analysed in many articles. We refer to [18] for the mathematical analysis of the runtime of these algorithms using drift analysis. Roughly speaking, the runtime of RLS or (1+1)-EA on OneM ax N is ⇥(N ln(N )) and the runtime of RLS or (1+1)-EA on LO N is ⇥(N 2 ).…”

Section: State Of the Artmentioning

confidence: 99%

Analysis of runtime of optimization algorithms for noisy functions over discrete codomains

Akimoto

Astete-Morales

Teytaud

2015

Theoretical Computer Science

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We consider in this work the application of optimization algorithms to problems over discrete codomains corrupted by additive unbiased noise.We propose a modification of the algorithms by repeating the fitness evaluation of the noisy function su ciently so that, with a fix probability, the function evaluation on the noisy case is identical to the true value.If the runtime of the algorithms on the noise-free case is known, the number of resampling is chosen accordingly. If not, the number of resampling is chosen regarding to the number of fitness evaluations, in an anytime manner.We conclude that if the additive noise is Gaussian, then the runtime on the noisy case, for an adapted algorithm using resamplings, is similar to the runtime on the noise-free case: we incur only an extra logarithmic factor. If the noise is non-Gaussian but with finite variance, then the total runtime of the noisy case is quadratic in function of the runtime on the noise-free case.

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Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

Cited by 118 publications

References 20 publications

Multiplicative Drift Analysis

Multiplicative Drift Analysis

OneMax in Black-Box Models with Several Restrictions

Analysis of runtime of optimization algorithms for noisy functions over discrete codomains

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