Sparse Variations and Reducibility among Prediction Problems Authors :

We investigate the relationship between two di erent notions of reducibility among prediction (learning) problems within the distribution-free learning model of Valiant (PAC learning model [7, 2].) The notions of reducibility we consider are the analogues for prediction problems of the many-one reducibility and of the Turing reducibility. The former is the notion of prediction preserving reducibility developed by Pitt and Warmuth [6], and its generalization. Concerning these two notions of reducibility, we show that there exist a pair of prediction problems A and B, whose membership problems are polynomial time solvable, such that A is reducible to B with respect to the Turing reducibility, but not with respect to the prediction preserving reducibility. We show this result by making use of the notion of a class of polynomially sparse variants of a concept representation class. We rst show that any class A of polynomially sparse variants of another class B is reducible to B with respect to the `Turing reducibility.' We then prove the existence of a prediction problem R and a class R 0 of polynomially sparse variants of R, such that R 0 does not reduce to R with respect to the prediction preserving reducibility.