Probabilistic Tools for the Analysis of Randomized Optimization Heuristics

This chapter collects several probabilistic tools that have proven to be useful in the analysis of randomized search heuristics. This includes classic material such as the Markov, Chebyshev, and Chernoff inequalities, but also lesser-known topics such as stochastic domination and coupling, and Chernoff bounds for geometrically distributed random variables and for negatively correlated random variables. Most of the results presented here have appeared previously, but some only in recent conference publications. While the focus is on presenting tools for the analysis of randomized search heuristics, many of these may be useful as well for the analysis of classic randomized algorithms or discrete random structures.

[1]  Per Kristian Lehre,et al.  Theoretical analysis of rank-based mutation - combining exploration and exploitation , 2009, 2009 IEEE Congress on Evolutionary Computation.

[2]  Benjamin Doerr,et al.  The unrestricted black-box complexity of jump functions , 2017, GECCO.

[3]  Marvin Künnemann,et al.  Optimizing linear functions with the (1+λ) evolutionary algorithm - Different asymptotic runtimes for different instances , 2015, Theor. Comput. Sci..

[4]  Carsten Witt,et al.  Runtime Analysis of the ( μ +1) EA on Simple Pseudo-Boolean Functions , 2006 .

[5]  Dong Zhou,et al.  The use of tail inequalities on the probable computational time of randomized search heuristics , 2012, Theor. Comput. Sci..

[6]  Per Kristian Lehre,et al.  Faster black-box algorithms through higher arity operators , 2010, FOGA '11.

[7]  Duc-Cuong Dang,et al.  Self-adaptation of Mutation Rates in Non-elitist Populations , 2016, PPSN.

[8]  Frank Neumann,et al.  Optimal Fixed and Adaptive Mutation Rates for the LeadingOnes Problem , 2010, PPSN.

[9]  Frank Neumann,et al.  Runtime Analysis of a Simple Ant Colony Optimization Algorithm , 2006, ISAAC.

[10]  Dirk Sudholt,et al.  When do evolutionary algorithms optimize separable functions in parallel? , 2013, FOGA XII '13.

[11]  Ingo Wegener,et al.  Randomized local search, evolutionary algorithms, and the minimum spanning tree problem , 2004, Theor. Comput. Sci..

[12]  Ingo Wegener,et al.  The analysis of evolutionary algorithms on sorting and shortest paths problems , 2004, J. Math. Model. Algorithms.

[13]  Benjamin Doerr,et al.  A Tight Bound for the (1+1)-EA on the Single Source Shortest Path Problem , 2007 .

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

[15]  Frank Neumann,et al.  Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods Runtime Analysis of a Simple Ant Colony Optimization Algorithm Runtime Analysis of a Simple Ant Colony Optimization Algorithm , 2022 .

[16]  Thomas Jansen,et al.  A method to derive fixed budget results from expected optimisation times , 2013, GECCO '13.

[17]  Kenneth A. De Jong,et al.  Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods on the Choice of the Offspring Population Size in Evolutionary Algorithms on the Choice of the Offspring Population Size in Evolutionary Algorithms , 2004 .

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

[19]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[20]  Hsien-Kuei Hwang,et al.  Probabilistic Analysis of the (1+1)-Evolutionary Algorithm , 2014, Evolutionary Computation.

[21]  Carsten Witt,et al.  Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax , 2017, FOGA '17.

[22]  Benjamin Doerr,et al.  The Impact of Random Initialization on the Runtime of Randomized Search Heuristics , 2015, Algorithmica.

[23]  Benjamin Doerr,et al.  Tight Analysis of the (1+1)-EA for the Single Source Shortest Path Problem , 2011, Evolutionary Computation.

[24]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[25]  Benjamin Doerr,et al.  A lower bound for the discrepancy of a random point set , 2012, J. Complex..

[26]  Benjamin Doerr,et al.  Analyzing Randomized Search Heuristics: Tools from Probability Theory , 2011, Theory of Randomized Search Heuristics.

[27]  Marc Schoenauer,et al.  Rigorous Hitting Times for Binary Mutations , 1999, Evolutionary Computation.

[28]  Stefan Droste,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Analysis of the (1+1) EA for a Noisy OneMax , 2004 .

[29]  Pietro Simone Oliveto,et al.  Analysis of the $(1+1)$-EA for Finding Approximate Solutions to Vertex Cover Problems , 2009, IEEE Transactions on Evolutionary Computation.

[30]  S. Janson.  Tail bounds for sums of geometric and exponential variables , 2017, 1709.08157.

[31]  Thomas Jansen,et al.  On benefits and drawbacks of aging strategies for randomized search heuristics , 2011, Theor. Comput. Sci..

[32]  Bernhard Klar,et al.  BOUNDS ON TAIL PROBABILITIES OF DISCRETE DISTRIBUTIONS , 2000, Probability in the Engineering and Informational Sciences.

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

[34]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[35]  Benjamin Doerr,et al.  Multiplicative Drift Analysis , 2010, GECCO '10.

[36]  Thomas Jansen,et al.  Mutation Rate Matters Even When Optimizing Monotonic Functions , 2013, Evolutionary Computation.

[37]  Yu. V. Prokhorov.  Convergence of Random Processes and Limit Theorems in Probability Theory , 1956 .

[38]  Pietro Simone Oliveto,et al.  Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation , 2008, Algorithmica.

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

[40]  Anton V. Eremeev,et al.  Comparing evolutionary algorithms to the (1+1)-EA , 2008, Theor. Comput. Sci..

[41]  Mahmoud Fouz,et al.  Sharp bounds by probability-generating functions and variable drift , 2011, GECCO '11.

[42]  E. Slud.  Distribution Inequalities for the Binomial Law , 1977 .

[43]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[44]  Timo Kötzing.  Concentration of First Hitting Times Under Additive Drift , 2015, Algorithmica.

[45]  Benjamin Doerr,et al.  Money for Nothing: Speeding Up Evolutionary Algorithms Through Better Initialization , 2015, GECCO.

[46]  Thomas Jansen,et al.  On the analysis of a dynamic evolutionary algorithm , 2006, J. Discrete Algorithms.

[47]  Ingo Wegener,et al.  On the Optimization of Monotone Polynomials by Simple Randomized Search Heuristics , 2005, Combinatorics, Probability and Computing.

[48]  Benjamin Doerr,et al.  Fast genetic algorithms , 2017, GECCO.

[49]  Mehryar Mohri,et al.  Tight Lower Bound on the Probability of a Binomial Exceeding its Expectation , 2013, ArXiv.

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

[51]  Carsten Witt,et al.  Fitness levels with tail bounds for the analysis of randomized search heuristics , 2014, Inf. Process. Lett..

[52]  Benjamin Doerr,et al.  Directed trees: A powerful representation for sorting and ordering problems , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[53]  Pietro Simone Oliveto,et al.  On the runtime analysis of generalised selection hyper-heuristics for pseudo-boolean optimisation , 2017, GECCO.

[54]  Benjamin Doerr,et al.  Runtime analysis of the (1 + (λ, λ)) genetic algorithm on random satisfiable 3-CNF formulas , 2017, GECCO.

[55]  Benjamin Doerr,et al.  Crossover can provably be useful in evolutionary computation , 2008, GECCO '08.

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

[57]  C. Scheideler.  Probabilistic Methods for Coordination Problems , 2000 .

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

[59]  Jan Ramon,et al.  A lower bound on the probability that a binomial random variable is exceeding its mean , 2016 .

[60]  Dirk Sudholt,et al.  The impact of parametrization in memetic evolutionary algorithms , 2009, Theor. Comput. Sci..

[61]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[62]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

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

[64]  Benjamin Doerr,et al.  Better Runtime Guarantees via Stochastic Domination , 2018, EvoCOP.

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

[66]  Benjamin Doerr,et al.  The (1+λ) evolutionary algorithm with self-adjusting mutation rate , 2017, GECCO.

[67]  Ingo Wegener,et al.  Theoretical Aspects of Evolutionary Algorithms , 2001, ICALP.

[68]  Benjamin Doerr,et al.  An Elementary Analysis of the Probability That a Binomial Random Variable Exceeds Its Expectation , 2017, Statistics & Probability Letters.

[69]  Vasek Chvátal,et al.  The tail of the hypergeometric distribution , 1979, Discret. Math..

[70]  Benjamin Doerr,et al.  Lower bounds for the runtime of a global multi-objective evolutionary algorithm , 2013, 2013 IEEE Congress on Evolutionary Computation.

[71]  Benjamin Doerr,et al.  Optimal Static and Self-Adjusting Parameter Choices for the (1+(λ,λ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( , 2017, Algorithmica.

[72]  Benjamin Doerr,et al.  Adjacency list matchings: an ideal genotype for cycle covers , 2007, GECCO '07.

[73]  Carsten Witt,et al.  The Interplay of Population Size and Mutation Probability in the ($$1+\lambda $$1+λ) EA on OneMax , 2016, Algorithmica.

[74]  Dirk Sudholt,et al.  A Simple Ant Colony Optimizer for Stochastic Shortest Path Problems , 2012, Algorithmica.

[75]  Pietro Simone Oliveto,et al.  Erratum: Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation , 2008, PPSN.

[76]  Benjamin Doerr,et al.  k-Bit Mutation with Self-Adjusting k Outperforms Standard Bit Mutation , 2016, PPSN.

[77]  Aravind Srinivasan,et al.  Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds , 1997, SIAM J. Comput..

[78]  Thomas Jansen,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Evolutionary Algorithms-How to Cope With Plateaus of Constant Fitness and When to Reject Strings of the Same Fitness , 2001 .

[79]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[80]  Carsten Witt,et al.  Population size versus runtime of a simple evolutionary algorithm , 2008, Theor. Comput. Sci..

[81]  Xin Yao,et al.  Drift analysis and average time complexity of evolutionary algorithms , 2001, Artif. Intell..

[82]  Thomas Jansen,et al.  On the analysis of the (1+1) evolutionary algorithm , 2002, Theor. Comput. Sci..

[83]  H. Robbins.  A Remark on Stirling’s Formula , 1955 .

[84]  Stefan Droste,et al.  Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods Analysis of the (1+1) Ea for a Dynamically Bitwise Changing Onemax Analysis of the (1+1) Ea for a Dynamically Bitwise Changing Onemax , 2003 .

[85]  Mahmoud Fouz,et al.  Quasirandom evolutionary algorithms , 2010, GECCO '10.

[86]  Duc-Cuong Dang,et al.  Level-Based Analysis of Genetic Algorithms and Other Search Processes , 2014, bioRxiv.

[87]  Leslie Ann Goldberg,et al.  Adaptive Drift Analysis , 2011, Algorithmica.

[88]  Xin Tong,et al.  Neyman-Pearson Classification, Convexity and Stochastic Constraints , 2011, J. Mach. Learn. Res..

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

[90]  Benjamin Doerr,et al.  Ranking-Based Black-Box Complexity , 2011, Algorithmica.

[91]  Uriel Feige,et al.  On sums of independent random variables with unbounded variance, and estimating the average degree in a graph , 2004, STOC '04.

[92]  Frank Neumann,et al.  Computing single source shortest paths using single-objective fitness , 2009, FOGA '09.

[93]  Per Kristian Lehre,et al.  Unbiased Black-Box Complexity of Parallel Search , 2014, PPSN.

[94]  S. V. Nagaev.  Lower Bounds on Large Deviation Probabilities for Sums of Independent Random Variables , 2002 .

[95]  Benjamin Doerr,et al.  Edge-based representation beats vertex-based representation in shortest path problems , 2010, GECCO '10.

[96]  Benjamin Doerr,et al.  Runtime Analysis for Self-adaptive Mutation Rates , 2018, Algorithmica.

[97]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .