A statistical learning perspective of genetic programming

Abstract. This paper proposes a theoretical analysis of Genetic Pro-gramming (GP) from the perspective of statistical learning theory, a well grounded mathematical toolbox for machine learning. By computing the Vapnik-Chervonenkis dimension of the family of programs that can be inferred by a specific setting of GP, it is proved that a parsimonious fitness ensures universal consistency. This means that the empirical er-ror minimization allows convergence to the best possible error when the number of test cases goes to infinity. However, it is also proved that the standard method consisting in putting a hard limit on the program size still results in programs of infinitely increasing size in function of their accuracy. It is also shown that cross-validation or hold-out for choosing the complexity level that optimizes the error rate in generalization also leads to bloat. So a more complicated modification of the fitness is pro-posed in order to avoid unnecessary bloat while nevertheless preserving universal consistency.