Abstract

Estimation of Distribution Algorithms (EDA) have been proposed as an extension of genetic algorithms. Our algorithm FDA assumes that the function to be optimized is additively decomposed (ADF). The interaction graph GADF is used to create exact or approximate factorizations of the Boltzmann distribution. Using Gibbs sampling instead of probabilistic logic sampling is investigated. We also discuss the algorithm LFDA which learns a Bayesian network from data. For both algorithms estimates of the necessary sample size N to find the optimum are derived. The bounds are based on statistical learning theory and PAC learning. If the assumptions of a factorization theorem are fulfilled, the upper bound of the sample size N of FDA is of order O(n ln n) where n is the size of the problem. The computational complexity per generation is O(N ∗ n). For LFDA a bound cannot be proven because the network learned might be far from optimal. In many applications the optimal network is not necessary for converge to the global optima. For the 2D Ising model only 60 % of the edges of GADF need to be contained in the learned graph. Bounds can be obtained for two new learning methods. The first one learns factor graphs instead of Bayesian networks, the second one detects the structure of the function by computing its Walsh or Fourier coefficients. The computational complexity to compute the Walsh coefficients is O(n 2 ln n). The networks computed by FDA and LFDA are analyzed for a set of benchmark functions.

Citations