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Valid Generalisation from Approximate Interpolation

Published online by Cambridge University Press:  12 September 2008

Martin Anthony ,
Peter Bartlett ,
Yuval Ishai  and
John Shawe-Taylor
Martin Anthony
Affiliation:
Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK Email: anthony@vax.lse.ac.uk
Peter Bartlett
Affiliation:
Department of Systems Engineering, Research School of Information Sciences and Engineering, Australian National University, Canberra 0200, Australia Email: Peter.Bartlett@anu.edu.au
Yuval Ishai
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
John Shawe-Taylor
Affiliation:
Computer Science Department, Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK Email: john@dcs.rhbnc.ac.uk
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Abstract

Let and be sets of functions from domain X to ℝ. We say that validly generalises from approximate interpolation if and only if for each η > 0 and ∈, δ ∈ (0,1) there is m0(η, ∈, δ) such that for any function t ∈ and any probability distribution on X, if m > m0 then with m-probability at least 1 – δ, a sample X = (x1, X2,…,xm) ∈ Xm satisfies

We find conditions that are necessary and sufficient for to validly generalise from approximate interpolation, and we obtain bounds on the sample length m0{η,∈,δ) in terms of various parameters describing the expressive power of .

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 3 , September 1996 , pp. 191 - 214
Copyright
Copyright © Cambridge University Press 1996

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