Universal ε-approximators for integrals

Let X be a space and F a family of 0,1-valued functions on X. Vapnik and Chervonenkis showed that if F is "simple" (finite VC dimension), then for every probability measure μ on X and ε > 0 there is a finite set S such that for all f ∈ F, ∑_x∈Sf(x)/|S| = [∫ f(x)dμ(x)]± ε. Think of S as a "universal ε-approximator" for integration in F. S can actually be obtained w.h.p. just by sampling a few points from μ. This is a mainstay of computational learning theory. It was later extended by other authors to families of bounded (e.g., [0, 1]-valued) real functions. In this work we establish similar "universal ε-approximators" for families of unbounded nonnegative real functions - in particular, for the families over which one optimizes when performing data classification. (In this case the ε-approximation should be multiplicative.) Specifically, let F be the family of "k-median functions" (or k-means, etc.) on ℝ^d with an arbitrary norm o. That is, any set u₁, ..., u_k ∈ ℝ^d determines an f by f(x) = (min _i o(x - u_i))^α. (Here α ≥ 0.) Then for every measure μ on ℝ^d there exists a set S of cardinality poly(k, d, 1/ε) and a measure ν supported on S such that for every f ∈ F, ∑_x∈s f(x)ν(x) ∈ (1 ± ε)·(∫ f(x)dμ(x)).