By Boi Faltings (auth.), Joaquim Filipe, Ana Fred (eds.)

This ebook constitutes the completely refereed post-conference complaints of the 3rd foreign convention on brokers and synthetic Intelligence, ICAART 2011, held in Rome, Italy, in January 2011. The 26 revised complete papers offered including invited paper have been rigorously reviewed and chosen from 367 submissions. The papers are prepared in topical sections on man made intelligence and on agents.

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Extra info for Agents and Artificial Intelligence: Third International Conference, ICAART 2011, Rome, Italy, January, 28-30, 2011. Revised Selected Papers

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2 Regression Estimation Learning Task (Infinite Set of Functions) Again, for shortness, we give only two theorems for bounds on V − C and C − V , the bound on |V −C| is their straightforward consequence (analogically as in previous sections). Probabilistic Connection between Cross-Validation and Vapnik Bounds 47 Theorem 9. Let {Q(z, ω)}ω∈Ω be an infinite set of real-valued bounded functions, 0 ≤ Q(ω, z) ≤ B, with finite Vapnik–Chervonenkis dimension h. Then, with probability 1 − α(η, n) or greater, the following inequality holds true: h(1 + 2Ih ) − ln η4 I n −1 B n−1 C −V ≤ √ n+ + n n−1 − ln η .

We do not focus on the leave-one-out cross-validation; we consider a more general n-fold non-stratified cross-validation (also: more convenient for our purposes); the leave-one-out case can be read out from our results as a special case. 1 Notation Related to Statistical Learning Theory We keep the notation similar to Vapnik’s [2,1]. – We denote the finite set of samples as: (x1 , y1 ), (x2 , y2 ), . . , (xI , yI ) , or more shortly by encapsulating pairs as {z1 , z2 , . . , zI }, where xi ∈ Rd are input points, yi are output values corresponding to them, and I is the set size.

24) On the level of sums of errors, not means, the total error for a larger sample will always surpass the total error for a smaller sample. This gives us I Remp (ωI ) ≤ IRemp (ωI ) and further: n Remp (ωI ). Remp (ωI ) ≤ (25) n−1 n Unfortunately it is of no use, because of the coefficient n−1 . Thinking of C − V in the theorem, we need a relation with coefficients 1 at both C and V . In [2, pp. 124] we find the following helpful assertion: Lemma 2. With probability at least 1 − 2η: Z Q(z, ωI )dF(z) − inf 1≤ j≤N Z Q(z, ω j )dF(z) R(ω0 ) ≤ ln N − ln η + 2I − ln η 2I (26) — the true risk for the selected function ωI is not farther from the minimal possible risk for this set of functions than ln N−ln η 2I + − ln η 2I .

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