Timo Kötzing
ABSTRACT
Recent advances in drift analysis have given us better and better tools for understanding random processes, including the run time of randomized search heuristics. In the setting of multiplicative drift we do not only have excellent bounds on the expected run time, but also more general results showing the concentration}of the run time. In this paper we investigate the setting of additive drift under the assumption of strong concentration of the "step size" of the process. Under sufficiently strong drift towards the goal we show a strong concentration of the hitting time. In contrast to this, we show that in the presence of small drift a Gambler's-Ruin-like behavior of the process overrides the influence of the drift. Finally, in the presence of sufficiently strong negative drift the hitting time is superpolynomial with high probability; this corresponds to the so-called negative drift theorem, for which we give new variants.
- Kazuoki Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19:357--367, 1967.Google Scholar
Cross Ref
- Benjamin Doerr and Leslie Ann Goldberg. Adaptive drift analysis. Algorithmica, 65:224--250, 2013.Google ScholarDigital Library
- Benjamin Doerr, Daniel Johannsen, and Carola Winzen. Multiplicative drift analysis. Algorithmica, 64:673--697, 2012.Google ScholarDigital Library
- Benjamin Doerr and Marvin Künnemann. How the (1+łambda) evolutionary algorithm optimizes linear functions. In Proc. of GECCO'13, pages 1589--1596, 2013. Google ScholarDigital Library
- Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, 2009. Google ScholarDigital Library
- Xiequan Fan, Ion Grama, and Quansheng Liu. Hoeffding's inequality for supermartingales. Stochastic Processes and their Applications, 122:3545--3559, 2012.Google Scholar
- Bruce Hajek. Hitting-time and occupation-time bounds implied by drift analysis with applications. Advances in Applied Probability, 13:502--525, 1982.Google ScholarCross Ref
- Jun He and Xin Yao. A study of drift analysis for estimating computation time of evolutionary algorithms. Natural Computing, 3:21--35, 2004. Google ScholarDigital Library
- Daniel Johannsen. Random Combinatorial Structures and Randomized Search Heuristics. PhD thesis, Universitat des Saarlandes, 2010. Available online at http://scidok.sulb.uni-saarland.de/volltexte/2011/3529/pdf/Dissertation_3166_Joha_Dani_2010.pdf.Google Scholar
- Per Kristian Lehre and Carsten Witt. General drift analysis with tail bounds. CoRR, abs/1307.2559, 2013.Google Scholar
- Boris Mitavskiy, Jonathan E. Rowe, and Chris Cannings. Preliminary theoretical analysis of a local search algorithm to optimize network communication subject to preserving the total number of links. In Proc. of CEC'08, pages 1484--1491, 2008.Google ScholarCross Ref
- Pietro Simone Oliveto and Carsten Witt. Simplified drift analysis for proving lower bounds in evolutionary computation. Algorithmica, 59:369--386, 2011. Google ScholarDigital Library
- Pietro Simone Oliveto and Carsten Witt. Erratum: Simplified drift analysis for proving lower bounds in evolutionary computation. arXiv, abs/1211.7184, 2012.Google Scholar
- Pietro Simone Oliveto and Carsten Witt. On the runtime analysis of the simple genetic algorithm. Theoretical Computer Science, 2013. In press.Google Scholar
- Jonathan E. Rowe and Dirk Sudholt. The choice of the offspring population size in the (1, łambda) ea. In Proc. of GECCO'12, pages 1349--1356, 2012. Google ScholarDigital Library
- Nicholas C. Wormald. The differential equation method for random graph processes and greedy algorithms. In Lectures on Approximation and Randomized Algorithms, pages 73--155, 1999.Google Scholar
Index Terms
- Concentration of first hitting times under additive drift
Recommendations
Concentration of First Hitting Times Under Additive Drift
Recent advances in drift analysis have given us better and better tools for understanding random processes, including the run time of randomized search heuristics. In the setting of multiplicative drift we do not only have excellent bounds on the ...
Unknown solution length problems with no asymptotically optimal run time
GECCO '17: Proceedings of the Genetic and Evolutionary Computation ConferenceWe revisit the problem of optimizing a fitness function of unknown dimension; that is, we face a function defined over bit-strings of large length N, but only n ≪ N of them have an influence on the fitness. Neither the position of these relevant bits ...
First-hitting times under drift
AbstractFor the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of ...
Comments