Algorithmic Learning Theory: 17th International Conference, by José L. Balcázar, Philip M. Long, Frank Stephan

By José L. Balcázar, Philip M. Long, Frank Stephan

This booklet constitutes the refereed court cases of the seventeenth overseas convention on Algorithmic studying conception, ALT 2006, held in Barcelona, Spain in October 2006, colocated with the ninth foreign convention on Discovery technological know-how, DS 2006.

The 24 revised complete papers provided including the abstracts of 5 invited papers have been conscientiously reviewed and chosen from fifty three submissions. The papers are devoted to the theoretical foundations of computing device learning.

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Additional resources for Algorithmic Learning Theory: 17th International Conference, ALT 2006, Barcelona, Spain, October 7-10, 2006, Proceedings (Lecture Notes in Computer Science)

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Additional tools: weak hypotheses and boosting. Let f : [b]n → {−1, 1} and D be a probability distribution over [b]n . A function g : [b]n → R is said to be a weak hypothesis for f with advantage γ under D if ED [f g] ≥ γ. The first boosting algorithm was described by Schapire [20] in 1990; since then boosting has been intensively studied (see [9] for an overview). The basic idea is that by combining a sequence of weak hypotheses h1 , h2 , . . (the i-th of which has advantage γ with respect to a carefully chosen distribution Di ) it is possible to obtain a high accuracy final hypothesis h which satisfies Pr[h(x) = f (x)] ≥ 1− .

Suppose that Algorithm B is given: – 0 < , δ < 1, and membership query access MEM(f ) to f : [b]n → {−1, 1}; – access to an algorithm WL which has the following property: given a value δ and access to MEM(f ) and to EX(f, D) (the latter is an example oracle which generates random examples from [b]n drawn with respect to distribution D), it constructs a weak hypothesis for f with advantage γ under D with probability at least 1 − δ in time polynomial in n, log b, log(1/δ ). Then Algorithm B behaves as follows: – It runs for S = O(log(1/ )/γ 2) stages and runs in total time polynomial in n, log b, −1 , γ −1 , log(δ −1 ).

A. Servedio Lemma 9. Let f : [b]n → {−1, 1} be expressed as an s-Majority of Parity of basic b-literals. Then for each index 1 ≤ i ≤ n, there are at most 2s i-sensitive values with respect to f . Proof. A literal on variable xi induces two i-sensitive values. The lemma follows directly from our assumption (see Section 2) that for each variable xi , each of the s Parity gates has at most one incoming literal which depends on xi . Algorithm 1. An improved algorithm for learning Majority of Parity of basic b-literals.

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