A Characterization of Strong Learnability in the Statistical Query Model
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In this paper, we consider Kearns' Statistical Query Model of learning. It is well known that the number of statistical queries, needed for ``weakly learning'' an unknown target concept (i.e.~for gaining significant advantage over random guessing) is polynomially related to the so-called Statistical Query dimension of the concept class. In this paper, we provide a similar characterization for ``strong learning'' where the learners final hypothesis is required to approximate the unknown target concept up to a small rate of misclassification. The quantity that characterizes strong learnability in the Statistical Query model is a surprisingly close relative of (though not identical to) the Statistical Query dimension. For the purpose of proving the main result, we provide other characterizations of strong learnability which are given in terms of covering numbers and related notions. These results might find some interest in their own right. All characterizations are purely information-theoretical and ignore computational issues.
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