Learning distributions by their density-levels - a paradigm for learning without a teacher

We propose a mathematical model for learning the high density areas of an un known distribution from unlabeled random points drawn according to this distri bution While this type of a learning task has not been previously addressed in the Computational Learnability literature we believe that this it a rather basic problem that appears in many practical learning scenarios From a statistical theory standpoint our model may be viewed as a restricted in stance of the fundamental issue of inferring information about a probability distribution from the random samples it generates From a computational learning angle what we propose is a new framework of un supervised concept learning The examples provided to the learner in our model are not labeled and are not necessarily all positive or all negative The only information about their membership is indirectly disclosed to the student through the sampling distribution We investigate the basic features of the proposed model and provide lower and upper bounds on the sample complexity of such learning tasks Our main result is that the learnability of a class of distributions in this setting is equivalent to the niteness of the VC dimension of the class of the high density areas of these distributions One direction of the proof involves a reduction of the density level learnability to p concepts learnability while the su ciency condition is proved through the introduction of a generic learning algorithm