Abstract

We construct data dependent upper bounds on the risk in function learning problems. The bounds are based on the local norms of the Rademacher process indexed by the underlying function class and they do not require prior knowledge about the distribution of training examples or any specific properties of the function class. Using Talagrand's type concentration inequalities for empirical and Rademacher processes, we show that the bounds hold with high probability that decreases exponentially fast when the sample size grows. In typical situations that are frequently encountered in the theory of function learning, the bounds give nearly optimal rate of convergence of the risk to zero. 1. Local Rademacher norms and bounds on the risk: main results Let (S; A) be a measurable space and let F be a class of A-measurable functions from S into [0; 1]: Denote P(S) the set of all probability measures on (S; A): Let f 0 2 F be an unknown target function. Given a probability measure P 2 P(S) (also unknown), let (X 1 ; : : : ; Xn ) be an i.i.d. sample in (S; A) with common distribution P (defined on a probability space(\Omega ; \Sigma; P)). In computer learning theory, the problem of estimating f 0 ; based on the labeled sample (X 1 ; Y 1 ); : : : ; (Xn ; Yn ); where Y j := f 0 (X j ); j = 1; : : : ; n; is referred to as function learning problem. The so called concept learning is a special case of function learning. In this case, F := fI C : C 2 Cg; where C ae A is called a class of concepts (see Vapnik (1998), Vidyasagar (1996), Devroye, Gyorfi and Lugosi (1996) for the account on statistical learning theory). The goal of function learning is to find an estimate

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