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Shifting : one-inclusion mistake bounds and sample compression

Rubinstein, B., Bartlett, P.L., & Rubinstein, J. (2009) Shifting : one-inclusion mistake bounds and sample compression. Journal of Computer and System Sciences, 75(1), pp. 37-59.

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Abstract

We present new expected risk bounds for binary and multiclass prediction, and resolve several recent conjectures on sample compressibility due to Kuzmin and Warmuth. By exploiting the combinatorial structure of concept class F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy. The key step in their proof is a d = VC(F) bound on the graph density of a subgraph of the hypercube—oneinclusion graph. The first main result of this paper is a density bound of n [n−1 <=d-1]/[n <=d] < d, which positively resolves a conjecture of Kuzmin and Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved one-inclusion mistake bound. The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization.

Our second main result is an algebraic topological property of maximum classes of VC-dimension d as being d contractible simplicial complexes, extending the well-known characterization that d = 1 maximum classes are trees. We negatively resolve a minimum degree conjecture of Kuzmin and Warmuth—the second part to a conjectured proof of correctness for Peeling—that every class has one-inclusion minimum degree at most its VCdimension.

Our final main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This result improves on known PAC-based expected risk bounds by a factor of O(logn) and is shown to be optimal up to an O(logk) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout.

Impact and interest:

5 citations in Scopus
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3 citations in Web of Science®

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ID Code: 43977
Item Type: Journal Article
Additional URLs:
Keywords: One-inclusion mistake bounds , Shifting, Sample compression, Multiclass prediction, Worst-case expected risk
DOI: 10.1016/j.jcss.2008.07.005
ISSN: 0022-0000
Subjects: Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > COMPUTATION THEORY AND MATHEMATICS (080200)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Past > Schools > Mathematical Sciences
Copyright Owner: Copyright 2009 Academic Press/Elsevier
Deposited On: 18 Aug 2011 11:01
Last Modified: 01 Mar 2012 00:34

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