Abstract

We present new results on the well-studied problem of learning DNF expressions. We prove that an algorithm due to Kushilevitz and Mansour [13] can be used to weakly learn DNF formulas with membership queries with respect to the uniform distribution. This is the rst positive result known for learning general DNF in polynomial time in a nontrivial model. Our results should be contrasted with those of Kharitonov [12], who proved that AC 0 is not eciently learnable in this model based on cryptographic assumptions. We also present ecient learning algorithms in various models for the read-k and SAT-k subclasses of DNF. We then turn our attention to the recently introduced statistical query model of learning [9]. This model is a restricted version of the popular Probably Approximately Correct (PAC) model, and practically every PAC learning algorithm falls into the statistical query model [9]. We prove that DNF and decision trees are not even weakly learnable in polynomial time in this model. This result is information-theoretic and therefore does not rely on any unproven assumptions, and demonstrates that no straightforward modication of the existing algorithms for learning various restricted forms of DNF and decision trees will solve the general problem. These lower bounds are a corollary of a more general characterization of the complexity of statistical query learning in terms of the number of uncorrelated functions in the concept class. The underlying tool for all of our results is the Fourier analysis of the concept class to be learned.

Citations