We study the question of determining whether an unknown function has a particular property or is ffl-far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being k-colorable or having a ae-clique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edge-queries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...

Consider the following type of problem: There is an unknown function, f : R n ! R m , there is also a black-box that on query x (2 R n ) returns f(x). Is there an algorithm that, using probes to the black-box, can figure out analytic information about f? (For an example: "Is f a polynomial? ", "Is f a second order differentiable at x = (0; 0; : : : ; 0)?" etc.). Clearly, for examples as these, if we bound the number of probes an algorithm has to settle for, no algorithm can carry the task. On the other hand, if one allows an infinite iteration of a `probe compute and guess' process, then, (quite surprisingly) for many such questions, there are algorithms that are guaranteed to be correct in all but finitely many of their guesses. We call such questions Decidable In the Limit, (DIL). We analyze the class of DIL problems and provide a necessary and sufficient condition for the membership of a decision problem in this class. We offer an algorithm for any DIL problem, and apply it to several types of learning tasks. We introduce a an extension of the usual Inductive Inference learning model - Inductive Inference with a Cheating Teacher. In this model the teacher may choose to present to the learner, not only a language belonging to the agreed - upon family of languages, but also an arbitrary language outside this family. In such a case we require that the learner will be able to eventually detect the faulty choice made by the teacher. We show that such strong type of learning is possible, and there exist learning algorithms that will fail only on arbitrarily small sets of faulty languages. Furthermore, if an a-priori probability distribution P , according to which f is being chosen, is available to the algorithm, then it can be strengthened into a finite A prelimi...

... In this article we present the general outlook and discuss general families of elaborations for use in practice; the exponential connection elaboration plays a key role. We then describe model elaborations for use in diagnosing: departures from normality, goodness of fit in generalized linear models, and variable selection in regression and outlier detection. We illustrate our approach with two applications.

Given an i.i.d. sample drawn from a density f on the real line, one has to test whether f is in a given class of densities. We investigate testing procedures constructed on the basis of minimizing the L 1 -distance between the kernel density estimate and any density in the hypothesized class. General nonasymptotic bounds are derived for the power of the test. It is shown that the concentratedness of the data-dependent smoothing factor plays a key role in the performance of the test, as well as the "size" of the hypothesized class of densities.