Download Agents and Artificial Intelligence: Third International by Boi Faltings (auth.), Joaquim Filipe, Ana Fred (eds.) PDF

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 awarded including invited paper have been rigorously reviewed and chosen from 367 submissions. The papers are geared up in topical sections on man made intelligence and on agents.

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

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2I (23) To conclude the proof, we need to relate somehow Remp (ωI ,k ) from each fold to Remp (ωI ). We need the relation in the direction Remp (ωI ,k ) ≤ · · · , so that we can plug the right-hand-side of it into (23) and keep it true. e. Remp (ωI ) ≥ Remp (ωI ,k ), because it is usually easier to fit fewer data points using models of equally rich complexities. But we don’t know with what probability that occurs. Contrarily, on may easily find a specific data subset for which Remp (ωI ) ≤ Remp (ωI ,k ).

For more information see [22,23,24]. We introduce the following notation. I and I stand for the size of training and testing sets respectively. n−1 I, I = n 1 I = I. n Without loss of generality for theorems and proofs, let I be dividable by n, so that I and I are integers. In a single fold, let {z1 , z2 , . . , zI }, {z1 , z2 , . . , zI } represent respectively the training set and the testing set, taken as a split of the whole permuted data set {z1 , z2 , . . , zI }. Similarly, empirical risks calculated as follows: Remp (ω) = 1 I ∑ Q(zi , ω), I i=1 Remp (ω) = 1 I (9) I ∑ Q(zi , ω), (10) i=1 represent respectively the training error and the testing error, calculated for any function ω.

The proof is analogous to the former proof, but we need to write most of the probabilistic inequalities in the different direction. With probability at least 1 − η, the following bound on true risk holds true: Remp (ωI ) ≤ R(ωI ) + ln N − ln η . 2I (30) By joining (30) and (21) we obtain, with probability at least 1 − 2η the system of inequalities: Remp (ωI ) − ln N − ln η ≤ R(ωI ) ≤ Remp (ωI ) 2I − ln η . (31) 2I + After n independent folds we obtain, with probability at least (1 − 2η)n: 1 n ∑ Remp(ωI ,k ) − n k=1 ln N − ln η − 2I − ln η 2I ≤ 1 n ∑ Remp (ωI ,k ) .

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