On the analysis of average time complexity of Estimation of Distribution Algorithms

Estimation of Distribution Algorithm (EDA) is a well-known stochastic optimization technique. The average time complexity is a crucial criterion that measures the performance of the stochastic algorithms. In the past few years, various kinds of EDAs have been proposed, but the related theoretical study on the time complexity of these algorithms is relatively few. This paper analyzed the time complexity of two early versions of EDA, the Univariate Marginal Distribution Algorithm (UMDA) and the Incremental UMDA (IUMDA). We generalize the concept of convergence to convergence time, and manage to estimate the upper bound of the mean First Hitting Times (FHTs) of UMDA (IUMDA) on a well-known pseudo-modular function, which is frequently studied in the field of genetic algorithms. Our analysis shows that UMDA (IUMDA) has O(n) behaviors on the pseudo-modular function. In addition, we analyze the mean FHT of IUMDA on a hard problem. Our result shows that IUMDA may spend exponential generations to find the global optimum. This is the first time that the mean first hitting times of UMDA (IUMDA) are theoretically studied.