By Shahar Mendelson, Robert C. Williamson (auth.), Jyrki Kivinen, Robert H. Sloan (eds.)

ISBN-10: 354043836X

ISBN-13: 9783540438366

ISBN-10: 3540454357

ISBN-13: 9783540454359

This booklet constitutes the refereed lawsuits of the fifteenth Annual convention on Computational studying idea, COLT 2002, held in Sydney, Australia, in July 2002.

The 26 revised complete papers offered have been conscientiously reviewed and chosen from fifty five submissions. The papers are equipped in topical sections on statistical studying concept, on-line studying, inductive inference, PAC studying, boosting, and different studying paradigms.

**Read or Download Computational Learning Theory: 15th Annual Conference on Computational Learning Theory, COLT 2002 Sydney, Australia, July 8–10, 2002 Proceedings PDF**

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**Additional resources for Computational Learning Theory: 15th Annual Conference on Computational Learning Theory, COLT 2002 Sydney, Australia, July 8–10, 2002 Proceedings**

**Sample text**

The ﬁrst step in the proof is to show that if one can establish such an exponential tail for a class of functions, then all the Lp norms are equivalent on the class. In fact, we show a little more: Lemma 44 Let G be a class of nonnegative functions which satisﬁes that there is some absolute constant c such that for every g ∈ G and every integer m, P r |g − Eg| ≥ mEg ≤ 2e−cm . Then, for every 0 < p < ∞ there is are constants cp and Cp which depends only on p and c, such that for every g ∈ G, 1 1 cp (Eg p ) p ≤ Eg ≤ Cp (Eg p ) p .

23 5. E. Gin´e, J. Zinn, Gaussian charachterization of uniform Donsker classes of functions, Annals of Probability, 19 (1991), 758–782. 23 6. D. Haussler, Sphere packing numbers for subsets of Boolean n-cube with bounded Vapnik-Chervonenkis dimension, Journal of Combinatorial Theory A 69 (1995), 217–232. 14, 24 7. W. Hoeﬀding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30. 19 8. M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer, 1991.

In [1] it was shown that F ⊂ B L∞ (Ω) satisﬁes the uniform law of large numbers if and only if fatε (F ) < ∞ for every ε > 0. Another important application of covering numbers estimates is the analysis of the uniform central limit property. Definition 2. Let F ⊂ B L∞ (Ω) , set P to be a probability measure on Ω and assume GP to be a gaussian process indexed by F , which has mean 0 and covariance EGP (f )GP (g) = f gdP − f dP gdP. A class F is called a universal Donsker class if for any probability measure P the law GP is tight in ∞ (F ) and νnP = n1/2 (Pn − P ) ∈ ∞ (F ) converges in law to GP in ∞ (F ).

### Computational Learning Theory: 15th Annual Conference on Computational Learning Theory, COLT 2002 Sydney, Australia, July 8–10, 2002 Proceedings by Shahar Mendelson, Robert C. Williamson (auth.), Jyrki Kivinen, Robert H. Sloan (eds.)

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