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Algorithmic Learning Theory: 23rd International Conference, by Nader H. Bshouty, Gilles Stoltz, Nicolas Vayatis, Thomas PDF

By Nader H. Bshouty, Gilles Stoltz, Nicolas Vayatis, Thomas Zeugmann

ISBN-10: 3642341055

ISBN-13: 9783642341052

ISBN-10: 3642341063

ISBN-13: 9783642341069

This booklet constitutes the refereed lawsuits of the twenty third overseas convention on Algorithmic studying thought, ALT 2012, held in Lyon, France, in October 2012. The convention used to be co-located and held in parallel with the fifteenth overseas convention on Discovery technology, DS 2012. The 23 complete papers and five invited talks offered have been conscientiously reviewed and chosen from forty seven submissions. The papers are geared up in topical sections on inductive inference, instructing and PAC studying, statistical studying conception and type, family members among versions and information, bandit difficulties, on-line prediction of person sequences, and different types of on-line learning.

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Additional info for Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings

Example text

14) j=J Upper Bound of the Quadratic Risk Proposition 1. Assume that the dictionary D is an orthonormal basis of the Hilbert space H. Let 1 < q < 2, r > 0, R > 0 such that Rε−1 ≥ e, and assume r that s ∈ wLq (R) ∩ B2,∞ (R). Consider the selected Lasso estimator sˆpˆ defined by (10) with parameters λp and pen(p) given by (11). Then, there exists Cq,r > 0 depending only on q and r such that the quadratic risk of sˆpˆ satisfies E s − sˆpˆ 2 ≤ Cq,r Rq ε ln (Rε−1 ) 2−q . (15) Some Rates of Convergence for the Selected Lasso Estimator 23 Proof.

By applying Lemma 2 and Lemma 3 with aj = α∗j for all j ∈ {1, . . , p} and δ = λ/2, p ∞ and by using the fact that j=1 1{|α∗j |>t} ≤ j=1 1{|α∗j |>t} ≤ Rq t−q for all t > 0 if s ∈ wLq (R), we get that (ii) is bounded by p α∗j 2 1{|α∗j |≤λ/2} ≤ 2q−1 q 2−q R λ , 2−q (27) |α∗j |1{|α∗j |>λ/2} ≤ q 2q−1 q 1−q R λ . q−1 (28) j=1 while (iii) is bounded by p j=1 Gathering (25) and (28) on the one hand and (24), (26), (27) and (28) on the other hand, we get that there exists Cq > 0 depending only on q and Cr > 0 depending only on r such that sp,λ L1 (Dp ) ≤ Cq Rq λ1−q and s − sp,λ 2 ≤ Cr R2 p−2r + Cq Rq λ2−q .

20) Proof. Page 31. Upper Bound of the Quadratic Risk Proposition 3. Assume that supj∈N φj ≤ 1. Let 1 < q < 2, r > 0, R > 0 such that Rε−1 ≥ e and assume that s ∈ Bq,r (R). Consider the selected Lasso estimator sˆpˆ defined by (10) with parameters λp and pen(p) given by (11). Then, there exists Cq,r > 0 depending only on q and r such that the quadratic risk of sˆpˆ satisfies E s − sˆpˆ 2 ≤ Cq,r Rq ε ln (Rε−1 ) 2−q . (21) Proof. Page 31. Proposition 3 is to be compared with Proposition 1 established in the orthonormal case.

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Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings by Nader H. Bshouty, Gilles Stoltz, Nicolas Vayatis, Thomas Zeugmann


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