A statistical learning theory approach of bloat

Code bloat, the excessive increase of code size, is an important issue in Genetic Programming (GP). This paper proposes a theoretical analysis of code bloat in the framework of symbolic regression in GP, from the viewpoint of Statistical Learning Theory, a well grounded mathematical toolbox for Machine Learning. Two kinds of bloat must be distinguished in that context, depending whether the target function lies in the search space or not. Then, important mathematical results are proved using classical results from Statistical Learning. Namely, the Vapnik-Chervonenkis dimension of programs is computed, and further results from Statistical Learning allow to prove that a parsimonious fitness ensures Universal Consistency (the solution minimizing the empirical error does converge to the best possible error when the number of examples goes to infinity). However, it is proved that the standard method consisting in choosing a maximal program size depending on the number of examples might still result in programs of infinitely increasing size with their accuracy; a more complicated modification of the fitness is proposed that theoretically avoids unnecessary bloat while nevertheless preserving the Universal Consistency.