Chi-Square Matrix: An Approach for Building-Block Identification

Abstract

This paper presents a line of research in genetic algorithms (GAs), called building-block identification. The building blocks (BBs) are common structures inferred from a set of solutions. In simple GA, crossover operator plays an important role in mixing BBs. However, the crossover probably disrupts the BBs because the cut point is chosen at random. Therefore the BBs need to be identified explicitly so that the solutions are efficiently mixed. Let S be a set of binary solutions and the solution s = b ₁ ... b _ℓ,   b _i ∈ {0, 1}. We construct a symmetric matrix of which the element in row i and column j, denoted by m _ij , is the chi-square of variables b _i and b _j . The larger the m _ij is, the higher the dependency is between bit i and bit j. If m _ij is high, bit i and bit j should be passed together to prevent BB disruption. Our approach is validated for additively decomposable functions (ADFs) and hierarchically decomposable functions (HDFs). In terms of scalability, our approach shows a polynomial relationship between the number of function evaluations required to reach the optimum and the problem size. A comparison between the chi-square matrix and the hierarchical Bayesian optimization algorithm (hBOA) shows that the matrix computation is 10 times faster and uses 10 times less memory than constructing the Bayesian network.