Achieving Tight \(O(4^k)\) Runtime Bounds on Jumpk by Proving that Genetic Algorithms Evolve Near-Maximal Population Diversity

IF 0.7 4区 计算机科学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING
Andre Opris, Johannes Lengler, Dirk Sudholt
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引用次数: 0

Abstract

The \(\textsc {Jump} _k\) benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (Algorithmica 2002) proved an upper bound of \(O(\textrm{poly}(n) + 4^k/p_c)\) for the (\(\mu \)+1) Genetic Algorithm ((\(\mu \)+1) GA), but only for unrealistically small crossover probabilities \(p_c\). To this date, it remains an open problem to prove similar upper bounds for realistic \(p_c\); the best known runtime bound, in terms of function evaluations, for \(p_c = \Omega (1)\) is \(O((n/\chi )^{k-1})\), \(\chi \) a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the (\(\mu \)+1) GA on \(\textsc {Jump} _k\). The (\(\mu \)+1)-\({\lambda _c}\)-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover \({\lambda _c}\) times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the (\(\mu \)+1)-\({\lambda _c}\)-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of \(O(\mu n \log (\mu ) + 4^k)\) function evaluations for a range of k under the mild assumptions \(p_c = O(1/k)\) and \(\mu \in \Omega (kn)\). For all constant k, the restriction is satisfied for some \(p_c = \Omega (1)\) and it implies that the expected runtime for all constant k and an appropriate \(\mu = \Theta (kn)\) is bounded by \(O(n^2 \log n)\), irrespective of k. For larger k, the expected time of the (\(\mu \)+1)-\({\lambda _c}\)-GA is \(\Theta (4^k)\), which is tight for a large class of unbiased black-box algorithms and faster than the original (\(\mu \)+1) GA by a factor of \(\Omega (1/p_c)\). We also show that our analysis can be extended to other unitation functions such as \(\textsc {Jump} _{k, \delta }\) and Hurdle.

通过证明遗传算法进化出接近最大种群多样性来实现跳跃的紧密\(O(4^k)\)运行时间界限
\(\textsc {Jump} _k\)基准是第一个证明交叉比仅突变的进化算法更快的问题。Jansen和Wegener (Algorithmica 2002)证明了(\(\mu \) +1)遗传算法((\(\mu \) +1) GA)的上界\(O(\textrm{poly}(n) + 4^k/p_c)\),但仅适用于不切实际的小交叉概率\(p_c\)。到目前为止,证明类似的上界对于现实\(p_c\)仍然是一个悬而未决的问题;就函数求值而言,\(p_c = \Omega (1)\)最著名的运行时边界是\(O((n/\chi )^{k-1})\), \(\chi \)是一个正常数。我们提供了一种新颖的方法,并分析了种群多样性的演变,以\(\textsc {Jump} _k\)上(\(\mu \) +1)遗传变异的成对汉明距离和来衡量。(\(\mu \) +1)- \({\lambda _c}\) -遗传在每一代中产生一个后代,要么对亲本中的一个施加突变,要么对同样的两个亲本施加\({\lambda _c}\)次交叉(接着是突变),以增加在有交叉的几代中产生被接受的后代的概率。结果表明,(\(\mu \) +1)- \({\lambda _c}\) -遗传算法的种群多样性收敛到接近完美的均衡状态。在温和的假设\(p_c = O(1/k)\)和\(\mu \in \Omega (kn)\)下,对于k的范围,这产生了一个改进的\(O(\mu n \log (\mu ) + 4^k)\)函数计算的时间界限。对于所有常数k,对于某些\(p_c = \Omega (1)\)满足限制,这意味着所有常数k和适当的\(\mu = \Theta (kn)\)的期望运行时间都以\(O(n^2 \log n)\)为界,而与k无关。对于较大的k, (\(\mu \) +1)- \({\lambda _c}\) -GA的期望时间为\(\Theta (4^k)\),这对于大型无偏黑箱算法来说是紧密的,并且比原始(\(\mu \) +1) GA快了一个\(\Omega (1/p_c)\)因子。我们还表明,我们的分析可以扩展到其他统一函数,如\(\textsc {Jump} _{k, \delta }\)和障碍。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Algorithmica
Algorithmica 工程技术-计算机:软件工程
CiteScore
2.80
自引率
9.10%
发文量
158
审稿时长
12 months
期刊介绍: Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential. Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming. In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.
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