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 couple of valuable subject matters in computational studying concept for researchers and scholars in synthetic intelligence, neural networks, theoretical computing device technological know-how, and statistics.Computational studying thought is a brand new and swiftly increasing quarter of analysis that examines formal types of induction with the pursuits of gaining knowledge of the typical tools underlying effective studying algorithms and making a choice on the computational impediments to learning.Each subject within the booklet has been selected to clarify a normal precept, that is explored in an exact formal atmosphere. instinct has been emphasised within the presentation to make the fabric available to the nontheoretician whereas nonetheless offering specific arguments for the expert. This stability is the results of new proofs of verified theorems, and new shows of the traditional proofs.The subject matters lined comprise the incentive, definitions, and basic effects, either optimistic and detrimental, for the commonly studied L. G. Valiant version of potentially nearly right studying; Occam's Razor, which formalizes a courting among studying and knowledge compression; the Vapnik-Chervonenkis measurement; the equivalence of vulnerable and robust studying; effective studying within the presence of noise by means of the strategy of statistical queries; relationships among studying and cryptography, and the ensuing computational boundaries on effective studying; reducibility among studying difficulties; and algorithms for studying finite automata from lively experimentation.

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2. Let f(·) b e an integer-valued funct ion , and assume that there does not exist a randomized algori thm taking as in p ut a graph G and a p a­ rameter 0 < 6 :5 1 that runs in time polyno mial in 1/6 and the size of G, and that with probability at least 1 - 6 ou tputs "no" if G is not k­ colorable and outputs an f( k )- coloring of G o t herwise . Then show that for some k � 3, k-term DNF formulae are not efficiently using f(k)-term DNF formulae. 3. Consider the following two-oracle variant of the PAC model : when c E C is the target concept, there are separate and arbitrary distributions vt over only the positive examples of c and V; over only the negative examples of c.

The paper of Haussler, Kearns, Littlestone and Warmut h [49] contains many theorems giving equivalences and relationships between some of the different models in the literature. 3, 1 . 5 are contained in this paper. 1 al. 2 is from Pitt and Valiant [71] . Copyrighted Material 2 Occam's Razor The PAC model introduced in Chapter 1 defined learning directly in terms of the predictive power of the hypothesis output by the learning algorithm. It was possible to apply this measure of success to a learning algorithm because we made the assumption that the instances are drawn indep end ently from a fixed probability distri bution V, and then measured predictive power with respect to this same distribution .

Thus, an efficient learning algorithm for this p roblem is required to run in time polynomial in n, l/f and 1/6. 3 If RP '# NP, the representation class of 9-term Copyrighted Material DNF Probably Approximately Correct Learning 19 formulae is not efficiently PA C learnable. Proof: The high-level idea of the proof is to reduce an NP-complete la nguage A ( to be specified shortly ) to the problem of PAC learning 3- term DNF formulae. More precisely, the reduction will efficiently map any string �, for which we wish to determine membership in A, to a set So of labeled examples.

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