Stochastic Runtime Analysis of the Cross-Entropy Algorithm

• Published in: IEEE transactions on evolutionary computation 2017-08, Vol.21 (4), p.616-628
• Main Authors: Zijun Wu, Kolonko, Michael, Mohring, Rolf H.
• Format: Article
• Language: English
• Subjects: 1-ANT algorithm; Algorithm design and analysis; Algorithms; analysis of randomized algorithms; Ant colony optimization; Complexity theory; cross-entropy (CE) algorithm; Entropy (Information theory); Evolutionary computation; Functions (mathematics); Iterative methods; Mathematical analysis; Mathematical model; Optimization; Probability theory; Randomness; Runtime; Sampling; stochastic complexity; Stochastic processes; stochastic runtime analysis of evolutionary algorithms (EAs)
• ISSN: 1089-778X; 1941-0026
• DOI: 10.1109/TEVC.2017.2667713

This paper analyzes the stochastic runtime of the cross-entropy (CE) algorithm for the well-studied standard problems ONEMAX and LEADINGONES. We prove that the total number of solutions the algorithm needs to evaluate before reaching the optimal solution (i.e., its runtime) is bounded by a polynomial Q(n) in the problem size n with a probability growing exponentially to 1 with n if the parameters of the algorithm are adapted to n in a reasonable way. Our polynomial bound Q(n) for ONEMAX outperforms the well-known runtime bound of the 1-ANT algorithm, a particular ant colony optimization algorithm. Our adaptation of the parameters of the CE algorithm balances the number of iterations needed and the size of the samples drawn in each iteration, resulting in an increased efficiency. For the LEADINGONES problem, we improve the runtime of the algorithm by bounding the sampling probabilities away from 0 and 1. The resulting runtime outperforms the known stochastic runtime for a univariate marginal distribution algorithm, and is very close to the known expected runtime of variants of max-min ant systems. Bounding the sampling probabilities allows the CE algorithm to explore the search space even for test functions with a very rugged landscape as the LEADINGONES function.