An Introduction to Computational Learning Theory by Michael J. Kearns

By Michael J. Kearns

Emphasizing problems with computational potency, Michael Kearns and Umesh Vazirani introduce a few imperative issues in computational studying concept for researchers and scholars in man made intelligence, neural networks, theoretical desktop technological know-how, and statistics.Computational studying conception is a brand new and quickly increasing sector of analysis that examines formal types of induction with the objectives of studying the typical tools underlying effective studying algorithms and determining the computational impediments to learning.Each subject within the ebook has been selected to clarify a common precept, that's explored in an actual formal surroundings. instinct has been emphasised within the presentation to make the cloth obtainable to the nontheoretician whereas nonetheless delivering targeted arguments for the professional. This stability is the results of new proofs of demonstrated theorems, and new displays of the traditional proofs.The subject matters lined comprise the inducement, definitions, and basic effects, either confident and unfavourable, for the generally studied L. G. Valiant version of potentially nearly right studying; Occam's Razor, which formalizes a dating among studying and knowledge compression; the Vapnik-Chervonenkis measurement; the equivalence of vulnerable and powerful studying; effective studying within the presence of noise through the strategy of statistical queries; relationships among studying and cryptography, and the ensuing computational obstacles on effective studying; reducibility among studying difficulties; and algorithms for studying finite automata from lively experimentation.

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For every triple of literals u, v, w over the original variable set Xb " " Xn t the new variable set contains a variable Yu ,v,w whose value is defined by Yu,v,w = u V v V w. Note that when u = v = w, then Yu,1J ,W = u, so all of the original variables are present in the new set. Also, note that the number of new variables Yu ,1J ,W is (2n) 3 = O(n3 ). ,v,w } ' Furthermore, it should be clear that any 3-CNF ' formula c over Xl , , Xn is equivalent to a simple conjunction c over the • • • • Copyrighted Material .

Then IUi+l1 � IUil- o;�l) = lUi l (1- oP:(S» ) . So by in duc ti on on i: lu,l:<; (1- opi(sS m. Choosing i � opt(S) log m suffic es to drive this upper bound below 1. Thus all the elements of U are covered after the algorithm has chosen opt(S) logm sets. Copyrighted Material Chapter 2 40 We now return to the problem of PAC learn ing conjunctions with few relevant variables. 2 to o btain the required sa mple size for PAC learning. Thus, given a sample S of m e xamples of a target conjunction, the new Occam algori th m starts by applying our o riginal conjunctions algorithm - which uses only the positive examples - to S in order to produce a hypothes is conjunction h.

The new algorithm will then use the negative examples in S to e xclude several additional literals from h in a manner described below, to compute a new hypothesis conjunction hi containing at most size ( c) log m of the literals appearing in h. 2. Recal l th at e xcluding literals from h does no t affect consistency with the positive examples in S, since the set of positive examples of h only as we delete literals. However, the new algo rithm has to carefully choose which literals of h it excludes in order to ensure that the hypothes is grows is still consistent with all the negative examples in S.

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