Multi-objective methods for tree size control

Variable length methods for evolutionary computation can lead to a progressive and mainly unnecessary growth of individuals, known as bloat. First, we propose to measure performance in genetic programming as a function of the number of nodes, rather than trees, that have been evaluated. Evolutionary Multi-Objective Optimization (EMOO) constitutes a principled way to optimize both size and fitness and may provide parameterless size control. Reportedly, its use can also lead to minimization of size at the expense of fitness. We replicate this problem, and an empirical analysis suggests that multi-objective size control particularly requires diversity maintenance. Experiments support this explanation. The multi-objective approach is compared to genetic programming without size control on the 11-multiplexer, 6-parity, and a symbolic regression problem. On all three test problems, the method greatly reduces bloat and significantly improves fitness as a function of computational expense. Using the FOCUS algorithm, multi-objective size control is combined with active pursuit of diversity, and hypothesized minimum-size solutions to 3-, 4- and 5-parity are found. The solutions thus found are furthermore easily interpretable. When combined with diversity maintenance, EMOO can provide an adequate and parameterless approach to size control in variable length evolution.