We present the BHUNT scheme for automatically discovering algebraic constraints between pairs of columns in relational data. The constraints may be "fuzzy" in that they hold for most, but not all, of the records, and the columns may be in the same table or different tables. Such constraints are of interest in the context of both data mining and query optimization, and the BHUNT methodology can potentially be adapted to discover fuzzy functional dependencies and other useful relationships. BHUNT first identifies candidate sets of column value pairs that are likely to satisfy an algebraic constraint. This discovery process exploits both system catalog information and data samples, and employs pruning heuristics to control processing costs.

We present a new perspective for investigating the Probably Approximate Correct (PAC) learnability of classes of concepts. We focus on special sets of points for characterizing the concepts within their class. This gives rise to a general notion of boundary of a concept, which holds even in discrete spaces, and to a special probability measuring technique. This technique is applied (i) to narrow the gap between the minimum and maximum sample sizes necessary to learn under a more stringent learnability definition , and (ii) to get self-explanatory indices of the complexity of the learning task. These indices can be roughly estimated during the learning process and appear very useful in the treatment of nonsymbolic procedures, e.g. in the context of neural networks.

ABSTRACT. We provide some theoretical results on sample complexity of PAC learning when the hypotheses are given by subsymbolical devices such as neural networks. In this framework we give new foundations to the notion of degrees of freedom of a statistic and relate it to the complexity of a concept class. Thus, for a given concept class and a given sample size, we discuss the efficiency of subsymbolical learning algorithms in terms of degrees of freedom of the computed statistic. In this setting we appraise the sample complexity overhead coming from relying on approximate hypotheses and display an increase in the degrees of freedom yield by embedding available formal knowledge into the algorithm. For known sample distribution, these quantities are related to the learning approximation goal and a special production prize is shown. Finally, we prove that testing the approximation capability of a neural network generally demands smaller sample size than training it.

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This paper studies the asymptotic properties of a class of partitioning regression estimators of the conditional expectation function and its derivatives. Mean-square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions. The uniform convergence rate explicitly accounts for the effect of moment assumptions, which is useful in semiparametric inference, and is derived both in probability and almost surely. A uniform (weak and strong) Bahadur representation is then developed for linear functionals of the partitioning estimator. Using this representation, asymptotic normality and consistency of a suitable standard-error estimator is established for linear functionals of the partitioning estimators, under simple sufficient conditions.

Abstract The error probability of the partitioning classification rule is shown to converge to the Bayes error faster than 1= p n under certain conditions. The resubstitution and the deleted error estimates for the partitioning classification rule from a sample (X1; Y1); : : : ; (Xn; Yn) are studied. The random part of the resubstitution estimate is shown to be small for arbitrary partition and for any distribution of (X; Y). If we assume that X has a density f and the partitions consist of rectangles, then the difference between the expected value of the estimate and the Bayes error restricted to the partition is less than a constant times 1= p n. The main result of the paper is that, under the same conditions, for both estimates the difference between the estimate and the real error probability of the classification rule is asymptotically normal with 0 mean and variance L\Lambda =2, where L\Lambda is the Bayes error. 1 Introduction Let X be the d-dimensional feature vector with distribution _, and let Y be the binary valued label. Denote the posterior probabilities by Pi(x) = PfY = ijX = xg; i = 0; 1: In pattern recognition the value of the label Y is to be predicted upon observing the feature vector X. The prediction rule or classifier g is a function Rd! f0; 1g, whose performance is measured by the probability of error