By Nader H. Bshouty, Gilles Stoltz, Nicolas Vayatis, Thomas Zeugmann
This e-book constitutes the refereed lawsuits of the twenty third foreign convention on Algorithmic studying concept, ALT 2012, held in Lyon, France, in October 2012. The convention was once co-located and held in parallel with the fifteenth overseas convention on Discovery technological know-how, DS 2012. The 23 complete papers and five invited talks awarded have been conscientiously reviewed and chosen from forty seven submissions. The papers are equipped in topical sections on inductive inference, instructing and PAC studying, statistical studying thought and category, kin among types and knowledge, bandit difficulties, on-line prediction of person sequences, and different types of on-line learning.
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Extra info for Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings
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34) Now, from Birg´e and Massart (2001, Theorem 5), we know that the minimax risk over A (p, d, M ) satisfies inf sup s˜ s∈A(p,d,M) E s − s˜ 2 ≥ κd min M 2 , ε2 1 + ln ≥κ RM −1 2 p d q min M 2 , ε2 1 + ln p d (35) Some Rates of Convergence for the Selected Lasso Estimator 31 where κ > 0 denotes some absolute constant. Moreover, we have ⎛ ⎡ ⎤⎞ 2−q −1 2r RM p ⎦⎠ ≥ ε2 ⎝1 + ln ⎣ ε2 1 + ln q d 2 (RM −1 ) ≥ ε2 ln RM −1 = ε2 ln Rε−1 ς ς εM −1 ς ς = M 2 + ε2 ln εM −1 ς = M 2 − ε2 ln ς ln Rε−1 2 . (36) But the assumption Rε−1 ≥ max(e2 , ς 2 ) implies that (36) is greater than M 2 /2.
Massart and C. Meynet Due to computer progress and development of state of the art technologies such as DNA microarrays, we are faced with high-dimensional data where the number of variables can be much larger than the sample size. To solve this problem, the sparsity scenario has been widely studied. It consists in assuming that there exists a sparse representation of the function s in the dictionary D, that is to say that most coefficients α ˆ j can be taken to zero. Then, one could consider an 0 -penalty in (5) in order to penalize the number of non-zero coefficients αj and favor a sparse estimation of s.
Let [S] = {f [n] | f ∈ S}. Thus, [R] = Seq. For notation simplification, [f ] = [{f }]. A class S is said to be dense if [S] = [R]. A class S is everywhere sparse iff for all τ ∈ Seq, there exists a τ τ such that τ ∈ [S]. A total function f is an accumulation point of S iff there exist pairwise distinct functions g0 , g1 , . . in S such that, for all n ∈ N, f [n] gn . }. We let M , N and P range over learners and let C range over classes of learners. Let M0 , M1 , . . denote an acceptable numbering of all the learners.