The use of tail inequalities on the probable computational time of randomized search heuristics

Zhou, Dong; Luo, Dan; Lu, Ruqian; Han, Zhangang

doi:10.1016/j.tcs.2012.01.010

Cited by 18 publications

(17 citation statements)

References 31 publications

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“…e 2 nµ 2 = (1 + o(1)) e 2 n 3 µ 2 M 2 and p min = min{ M enµ , 1 eµ } = M enµ , noting that M = o(n) and M ≤ n. Apart from a small improvement in the constants, stemming from slightly more careful estimates, the result on the expected runtime is the same as the one in [Wit06], which is E[T ] ≤ 3en max{µ ln(en), n}, see Theorem 1 of [Wit06] and recall that we count the number of iterations, that is, we ignore the µ fitness evaluations of the initial individuals. The tail bound, as discussed earlier, is stronger than the one in [ZLLH12] due to the stronger Chernoff bounds for sums of geometric random variables which are available now.…”

Section: Performance Of the (µ + 1) Ea On Leadingonesmentioning

confidence: 88%

Section: Related Workmentioning

confidence: 99%

“…What comes closest to this work is the paper [ZLLH12], which also tries to establish runtime analysis beyond expectations in a formalized manner. The notion proposed in [ZLLH12], called probable computational time L(δ), is the smallest time T such that the algorithm under investigation within the first T fitness evaluations finds an optimal solution with probability at least 1 − δ. In some sense, this notion can be seen as an inverse of the classic tail bound language, which makes statements of the type that the probability that the algorithm needs more than T iterations, is at most some function in T , which is often has an exponential tail.…”

Section: Related Workmentioning

confidence: 99%

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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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Section: Performance Of the (µ + 1) Ea On Leadingonesmentioning

confidence: 88%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Better Runtime Guarantees via Stochastic Domination

Doerr

2018

Evolutionary Computation in Combinatorial Optimization

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“…The fitness-level method has also been applied to other elitist optimization methods, including elitist ant colony optimizers [12,27] and a binary particle swarm optimizer [37]. It gives rise to powerful tail inequalities [40] and it can be used to prove lower bounds as well, when combined with additional knowledge on transition probabilities [35]. Finally, Lehre [22] recently showed that the fitness-level method can be extended towards non-elitist EAs with additional mild conditions on transition probabilities and the population size.…”

Section: Theorem 2 (Fitness-level Method) For Two Setsmentioning

confidence: 99%

General Upper Bounds on the Runtime of Parallel Evolutionary Algorithms*

Lässig

Sudholt

2014

Evolutionary Computation

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We present a general method for analyzing the runtime of parallel evolutionary algorithms with spatially structured populations. Based on the fitness-level method, it yields upper bounds on the expected parallel runtime. This allows for a rigorous estimate of the speedup gained by parallelization. Tailored results are given for common migration topologies: ring graphs, torus graphs, hypercubes, and the complete graph. Example applications for pseudo-Boolean optimization show that our method is easy to apply and that it gives powerful results. In our examples the performance guarantees improve with the density of the topology. Surprisingly, even sparse topologies such as ring graphs lead to a significant speedup for many functions while not increasing the total number of function evaluations by more than a constant factor. We also identify which number of processors lead to the best guaranteed speedups, thus giving hints on how to parameterize parallel evolutionary algorithms.

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“…There were slight variations on this considering the number of iterations/generations or fitness evaluations. Jansen and Zarges [7] and Zhou et al [18] pointed out that there is a gap between the empirical results and the theoretical results obtained on the optimization time. Theoretical research most often yields asymptotic results on finding the global optimum while practitioners concern more about achieving some good result within a reasonable time budget.…”

Section: Introductionmentioning

confidence: 99%

A fixed budget analysis of randomized search heuristics for the traveling salesperson problem

Nallaperuma

Neumann

Sudholt

2014

Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation

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Randomized Search heuristics are frequently applied to NP-hard combinatorial optimization problems. The runtime analysis of randomized search heuristics has contributed tremendously to their theoretical understanding. Recently, randomized search heuristics have been examined regarding their achievable progress within a fixed time budget. We follow this approach and present a first fixed budget runtime analysis for a NP-hard combinatorial optimization problem. We consider the well-known Traveling Salesperson problem (TSP) and analyze the fitness increase that randomized search heuristics are able to achieve within a given fixed budget.

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The use of tail inequalities on the probable computational time of randomized search heuristics

Cited by 18 publications

References 31 publications

Better Runtime Guarantees via Stochastic Domination

Better Runtime Guarantees via Stochastic Domination

General Upper Bounds on the Runtime of Parallel Evolutionary Algorithms*

A fixed budget analysis of randomized search heuristics for the traveling salesperson problem

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