Accurate estimates of the generalization ability for symmetric set of predictors and randomized learning algorithms

The main issue of the combinatorial approach to overfitting is to obtain computationally efficient formulas for overfitting probabilities. A group-theoretical approach is proposed to simplify derivation of such formulas when the set of predictors has a certain group of symmetries. Examples of the sets are given. The general estimate of overfitting probability is proved for the randomized learning algorithm. It is applied to four model sets of predictors—a layer of the Boolean cube, the Boolean cube, the unimodal chain, and a bundle of monotonic chains.