Various authors have proposed probabilistic extensions of Valiant's PAC (Probably Approximately Correct) learning model in which the target to be learned is a (conditional) probability distribution. In this paper, we improve upon the best known upper bounds on the sample complexity of the parameter estimation part of the learning problem for distributions and stochastic rules over a finite domain with respect to the Kullback-Leibler divergence (KL-devergence). In particular, we improve the upper bound of order O(1/ε2) due to Abe, Takeuchi, and Warmuth to a bound of order O(1/ε). In obtaining our results, we made use of the properties of a specific estimator (slightly modified maximum likelihood estimator) with respect to the KL-divergence, while previously known upper bounds were obtained using the uniform convergence technique.

We consider the problem of efficient learning of probabilistic concepts (p-concepts) and more generally stochastic rules in the sense defined by Kearns and Schapire and by Yamanishi. Their models extend the PAC-learning model of Valiant to the learning scenario in which the target concept or function is stochastic rather than deterministic as in Valiant's original model. In this paper, we consider the learnability of stochastic rules with respect to the classic 'Kullback-Leibler divergence' (KL divergence) as well as the quadratic distance as the distance measure between the rules. First, we show that the notion of polynomial time learnability of p-concepts and stochastic rules with fixed range size using the KL divergence is in fact equivalent to the same notion using the quadratic distance, and hence any of the distances considered in [6] and [18]: the quadratic, variation, and Hellinger distances. As a corollary, it follows that a wide range of classes of p-concepts which were shown to be polynomially learnable with respect to the quadratic distance in [6] are also learnable with respect to the KL divergence. The sample and time complexity of algorithms that would be obtained by the above general equivalence, however, are far from optimal. We present a polynomial learning algorithm with reasonable sample and time complexity for the important class of convex linear combinations of stochastic rules. We also develop a simple and versatile technique for obtaining sample complexity bounds for learning classes of stochastic rules with respect to the KL-divergence and quadratic distance, and apply them to produce bounds for the classes of probabilistic finite state acceptors (automata), probabilistic decision lists, and convex linear combinations.