Universal ε-approximators for integrals

Michael Langberg, Leonard J. Schulman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 ℝd with an arbitrary norm o. That is, any set u1, ..., uk ∈ ℝd determines an f by f(x) = (min i o(x - ui))α. (Here α ≥ 0.) Then for every measure μ on ℝd 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)).

Original languageEnglish
Title of host publicationProceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherAssociation for Computing Machinery (ACM)
Number of pages10
ISBN (Print)9780898717013
StatePublished - 2010
Event21st Annual ACM-SIAM Symposium on Discrete Algorithms - Austin, TX, United States
Duration: 17 Jan 201019 Jan 2010

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference21st Annual ACM-SIAM Symposium on Discrete Algorithms
Country/TerritoryUnited States
CityAustin, TX


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