Optimizing Neural Networks for the Generation of Block Designs

This work describes the evaluation of several search algorithms , based on optimizing neural networks, as applied to a family of problems : the generation of block-designs. Given a set (v; b; u) of parameters (v rows, b columns and u ones), a block design is any v b-binary con guration that has the following properties: u ones, r ones per row, k ones per column, and correlation between pairs of rows. The values [u; r; k; ] are called here the descriptors of the design and, since they have to be integers, they impose admissibility constraints on the independent parameters. Admissiblity, though, does not imply existence. An optimizing algorithm can be decomposed into a cost function, that conforms the search landscape, and a search strategy that de nes the way to explore it. This work proposes a set of cost functions, based on the number of pairs as a measure of the distribution of each of the properties of a design. The resulting structure, then, is straightforwardly mapped onto an optimizing neural network with fourth-order connections. Three search strategies are evaluated: Two classical ones (down-hill search and simulated annealing), and a new one, cooperative search, which samples search space in parallel with several instances of the network that move together as a cluster , in the sense of decreasing average energy. The gradual reduction of the radius (maximum hamming distance to the center of the cluster), leads the search to deeper and deeper mean energy basins. Given a particular experimental case, the expected cost to the rst solution is the variable selected as the experimental result , because it takes into account the resources being used. Problem, function and strategy are the main factors of the experimental evaluation. But considering the sub-factors involved (such as function weights and strategy parameters), a full factorial analysis is intractable. Experimentation, therefore, has to proceed in three stages: A training stage, for weight and parameter tuning, a comparison stage, for factor performance analysis, and a third stage where the best algorithm is applied to problems of increasing size. Results from the comparison stage (over the 25 smallest problems) show that interaction between the three factors is high. Simulated annealing (24 solved problems) is the most e cient search strategy. Cooperative search outperforms down-hill search in the number of solved problems (22 against 13), but on the problems solved by both, it proves to be less e cient. Out of 9 di erent function structures, the simplest cost function works best. Results from the third stage show that, although some problems are intrinsically hard (or easy), the number of unsolved cases increases with problem size. In all, the generation of block designs has proven to be a good benchmark for neural optimization algorithms. Finally, a new family of combinatorial con gurations is proposed, called maximally balanced designs , that admit two correlative values in the last three design descriptors, therefore avoiding admissibility constraints. Some of their properties are analyzed, and experimentally veri ed. Over 38 test cases, 26 problems have been solved. Results for maximally balanced designs, thus, are similar to those for block desings.