Download Algorithmic Learning Theory: 19th International Conference, by Imre Csiszár (auth.), Yoav Freund, László Györfi, György PDF

By Imre Csiszár (auth.), Yoav Freund, László Györfi, György Turán, Thomas Zeugmann (eds.)

This e-book constitutes the refereed complaints of the nineteenth foreign convention on Algorithmic studying idea, ALT 2008, held in Budapest, Hungary, in October 2008, co-located with the eleventh overseas convention on Discovery technological know-how, DS 2008.

The 31 revised complete papers offered including the abstracts of five invited talks have been conscientiously reviewed and chosen from forty six submissions. The papers are devoted to the theoretical foundations of computer studying; they tackle issues akin to statistical studying; likelihood and stochastic approaches; boosting and specialists; lively and question studying; and inductive inference.

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Extra info for Algorithmic Learning Theory: 19th International Conference, ALT 2008, Budapest, Hungary, October 13-16, 2008. Proceedings

Example text

We now list some obvious properties of piecewise constant scoring function. Proposition 5. Consider some piecewise constant scoring function sN ∈ SN . (i) The ROC curve of sN is piecewise linear with N linear parts. (ii) The ROC curve of sN does not depend on the particular values of the sequence (aj )j≥1 appearing in its D-representation but only on their ordering. Our purpose in this section is to design an iterative procedure which outputs a piecewise constant scoring function sN ∈ SN whose ROC curve is as close as possible to the optimal ROC∗ .

2d − 1 with the convention that ∆∗d,0 = 0 and ∆∗d,2d = 1 for all d ≥ 0. ∗ ∗ ∗ ∗ ∗ ∗ by: Rd,k = Cd,k ∪ Rd,k−1 with Rd,0 = Cd,0 . We also define the sets Rd,k Remark 3 (A tree-structured recursive interpolation scheme). A nice feature of the recursive approximation procedure is its binary-tree structure. Owing to their crucial practical advantages regarding implementation and interpretation, tree-structured decision rules have been proved useful for a wide range of statistical tasks and are in particular among the most popular methods for regression and classification (we refer to Chapter 20 in [DGL96] for an excellent account of tree decision rules in the context of classification).

For a fixed j, we look for the optimal splitting which would increase the AUC by adding a knot (α(t), β(t)) such that α(t) is between αj and αj+1 . We take the notation sN +1,t (x) = sN (x) + I{η(x) > t}, with t ∈ (Q∗ (αj+1 ), Q∗ (αj )). (j) The AUC can then be written, for some constant cj , as: 1 1 (j) AN +1 (t) = AUC(sN+1,t ) = cj + (αj+1 − αj )β(t) − α(t)(βj+1 − βj ), 2 2 which is maximized at t∗ such that: dβ(t∗ ) = βj+1 − βj αj+1 − αj dα(t∗ ) . Set α∗j = α(t∗ ) and get, thanks to Proposition 3, the following relationship: Q∗ (α∗j ) 1−p βj+1 − βj · = .

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