By Kamalika Chaudhuri, CLAUDIO GENTILE, Sandra Zilles

ISBN-10: 331924485X

ISBN-13: 9783319244853

ISBN-10: 3319244868

ISBN-13: 9783319244860

This publication constitutes the complaints of the twenty sixth overseas convention on Algorithmic studying idea, ALT 2015, held in Banff, AB, Canada, in October 2015, and co-located with the 18th foreign convention on Discovery technology, DS 2015. The 23 complete papers awarded during this quantity have been rigorously reviewed and chosen from forty four submissions. furthermore the publication comprises 2 complete papers summarizing the invited talks and a pair of abstracts of invited talks. The papers are geared up in topical sections named: inductive inference; studying from queries, instructing complexity; computational studying concept and algorithms; statistical studying conception and pattern complexity; on-line studying, stochastic optimization; and Kolmogorov complexity, algorithmic details theory.

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Extra resources for Algorithmic Learning Theory: 26th International Conference, ALT 2015, Banff, AB, Canada, October 4-6, 2015, Proceedings

Example text

Anandkumar et al. the case of document modeling; here, each document corresponds to a mixture over topics (as opposed to just a single topic). The distribution over such topic mixtures is a Dirichlet distribution Dir(α) with parameter vector α ∈ Rk++ with strictly positive entries; its density over the probability simplex Δk−1 := {v ∈ k Rk : vi ∈ [0, 1]∀i ∈ [k], i=1 vi = 1} is given by Γ (α0 ) pα (h) = k i=1 k Γ (αi ) i=1 i −1 hα , i h ∈ Δk−1 where α0 := α1 + α2 + · · · + αk . As before, the k topics are specified by probability vectors μ1 , μ2 , .

Ed is the standard coordinate basis for Rd . One advantage of this encoding of words is that the (cross) moments of these random vectors correspond to joint probabilities over words. For instance, observe that E[x1 ⊗ x2 ] = Pr[x1 = ei , x2 = ej ] ei ⊗ ej 1≤i,j≤d Pr[1st word = i, 2nd word = j] ei ⊗ ej , = 1≤i,j≤d so the (i, j)-the entry of the matrix E[x1 ⊗ x2 ] is Pr[1st word = i, 2nd word = j]. More generally, the (i1 , i2 , . . , i )-th entry in the tensor E[x1 ⊗ x2 ⊗ · · · ⊗ x ] is Pr[1st word = i1 , 2nd word = i2 , .

The details can be found in Supplementary D. 2 Computational Complexity for Nuclear-norm Minimization The optimization for nuclear-norm formulation is much more complex. Recently [10] proposed an active subspace method to solve Problem (10). The computational bottleneck is the approxSVD step and the inner problem step, both of which involve calculating a similar equation as shown on the left hand side of Eq (8). However, the rank of U or V is not fixed in each iteration as that of ALS, and in the worst case, it can be as large as min{d1 , d2 }.

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Algorithmic Learning Theory: 26th International Conference, ALT 2015, Banff, AB, Canada, October 4-6, 2015, Proceedings by Kamalika Chaudhuri, CLAUDIO GENTILE, Sandra Zilles

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