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 rigorously reviewed and chosen from fifty five submissions. The papers are geared up in topical sections on statistical studying thought, on-line studying, inductive inference, PAC studying, boosting, and different studying paradigms.

**Read Online 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**

**Example text**

2ε(p − 1), 2εp}. n Since t − ε > 3t/4, then by approximation one can ﬁnd √ n √ an subset A1 ⊂ T for n which A1 ⊂ A+εB∞ and N (A1 , nB2 , t−ε) ≥ N (A, √nB2 , t). Therefore, there √ exists a subset A2 ⊂ A1 of cardinality |A2 | ≥ N (A, nB2n , t), which is 3t n4 separated with respect to the · 2 -norm. Note that every two distinct points n x, y ∈ A2 satisfy that i=1 |x(i) − y(i)|2 ≥ (9t2 /16)n ≥ t2 n/2 and that |x(i) − 2 y(i)| ≤ 4 for all i. Hence |x(i) − y(i)| ≥ t/2 on at least t2 n/16 coordinates i.

Clearly, to prove the assertion of the theorem for the set A ⊂ T n , it is suﬃcient to prove it for the set PI A ⊂ T I (with |I| instead of n), whose cardinality already satisﬁes log |PI A| = log |A| ≥ cε(k − 1)/2 ≥ cε|I|/4. Therefore, we can assume that |A| = exp(αn) with α > cε for some absolute constant c. 3 in [1] (see also [3]). A set is called a cube if it is of the form Dσ = i∈σ {ai , bi }, where σ is a subset of {1, . . , n} and ai , bi ∈ T . We will be interested only in large cubes, which are the cubes in which ai and bi are separated for all i ∈ σ.

In a similar way to theorem 7 one can obtain the following: Theorem 8. There are absolute constants C and c for which the following holds. , xn } to n be a sample and let Gn = n−1/2 supf ∈F | i=1 gi f (xi )|. Then, Gn ≤ C where σF2 (sn ) = n−1 on sn . n i=1 σF (sn ) √ cG/ n fatct (F ) log(2/t) dt, (13) f 2 (xi ) and µn is the empirical measure supported To complement this result, let us mention another one of Talagrand’s results, which enables one to estimate the expectation of σF2 when sn is selected randomly according to an underlying probability measure.

### 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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