Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax

The Univariate Marginal Distribution Algorithm (UMDA), a popular estimation of distribution algorithm, is studied from a run time perspective. On the classical OneMax benchmark function, a lower bound of Ω(μ√n + n log n), where μ is the population size, on its expected run time is proved. This is the first direct lower bound on the run time of the UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation of distribution algorithms in general.

[1]  Pietro Simone Oliveto,et al.  Improved time complexity analysis of the Simple Genetic Algorithm , 2015, Theor. Comput. Sci..

[2]  Xin Yao,et al.  Analysis of Computational Time of Simple Estimation of Distribution Algorithms , 2010, IEEE Transactions on Evolutionary Computation.

[3]  S. M. Samuels.  On the Number of Successes in Independent Trials , 1965 .

[4]  Per Kristian Lehre,et al.  Concentrated Hitting Times of Randomized Search Heuristics with Variable Drift , 2014, ISAAC.

[5]  Jonathan E. Rowe,et al.  Theoretical analysis of local search strategies to optimize network communication subject to preserving the total number of links , 2009, Int. J. Intell. Comput. Cybern..

[6]  Andrew M. Sutton,et al.  The Benefit of Recombination in Noisy Evolutionary Search , 2015, GECCO.

[7]  J. A. Lozano,et al.  Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms (Studies in Fuzziness and Soft Computing) , 2006 .

[8]  M. T. Alodat,et al.  the Normal Distribution Function , 2008 .

[9]  Dirk Sudholt,et al.  The choice of the offspring population size in the (1, λ) evolutionary algorithm , 2014, Theor. Comput. Sci..

[10]  B. Hajek.  Hitting-time and occupation-time bounds implied by drift analysis with applications , 1982, Advances in Applied Probability.

[11]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[12]  Stefan Droste,et al.  A rigorous analysis of the compact genetic algorithm for linear functions , 2006, Natural Computing.

[13]  Xin Yao,et al.  Rigorous time complexity analysis of Univariate Marginal Distribution Algorithm with margins , 2009, 2009 IEEE Congress on Evolutionary Computation.

[14]  Andrew M. Sutton,et al.  The Benefit of Recombination in Noisy Evolutionary Search , 2015, ISAAC.

[15]  KaleJyoti,et al.  Genetic Algorithms and Evolutionary Computation , 2010 .

[16]  Thomas Jansen,et al.  UNIVERSITY OF DORTMUND REIHE COMPUTATIONAL INTELLIGENCE COLLABORATIVE RESEARCH CENTER 531 Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Upper and Lower Bounds for Randomized Search Heuristics in Black-Box Optimization , 2004 .

[17]  Dirk Sudholt,et al.  Update Strength in EDAs and ACO: How to Avoid Genetic Drift , 2016, GECCO.

[18]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[19]  Per Kristian Lehre,et al.  When is an estimation of distribution algorithm better than an evolutionary algorithm? , 2009, 2009 IEEE Congress on Evolutionary Computation.

[20]  J. Darroch.  On the Distribution of the Number of Successes in Independent Trials , 1964 .

[21]  Daniel Johannsen,et al.  Random combinatorial structures and randomized search heuristics , 2010 .

[22]  Jens Jägersküpper,et al.  When the Plus Strategy Outperforms the Comma Strategyand When Not , 2007, 2007 IEEE Symposium on Foundations of Computational Intelligence.

[23]  B. Harshbarger.  An Introduction to Probability Theory and its Applications, Volume I , 1958 .

[24]  Dirk Sudholt,et al.  A few ants are enough: ACO with iteration-best update , 2010, GECCO '10.

[25]  Duc-Cuong Dang,et al.  Simplified Runtime Analysis of Estimation of Distribution Algorithms , 2015, GECCO.

[26]  Andrew Rosalsky,et al.  ON THE NUMBER OF SUCCESSES IN INDEPENDENT TRIALS , 2016 .

[27]  Dirk Sudholt,et al.  A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms , 2011, IEEE Transactions on Evolutionary Computation.

[28]  M. Habib.  Probabilistic methods for algorithmic discrete mathematics , 1998 .

[29]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[30]  Per Kristian Lehre,et al.  Improved runtime bounds for the univariate marginal distribution algorithm via anti-concentration , 2017, GECCO.

[31]  Xin Yao,et al.  On the analysis of average time complexity of estimation of distribution algorithms , 2007, 2007 IEEE Congress on Evolutionary Computation.

[32]  Carsten Witt,et al.  Tight Bounds on the Optimization Time of a Randomized Search Heuristic on Linear Functions† , 2013, Combinatorics, Probability and Computing.

[33]  Tobias Friedrich,et al.  EDAs cannot be Balanced and Stable , 2016, GECCO.

[34]  Martin Pelikan,et al.  An introduction and survey of estimation of distribution algorithms , 2011, Swarm Evol. Comput..