Investigating the distribution assumptions in the Pac learning model

Abstract

In this paper, we question two assumptions of the pac learning model related to the distribution of examples: that the learning algorithm must learn for an arbitrary distribution (the distribution-independence assumption), and that the distribution of training examples is identical to the distribution of examples that the learning system sees in use (the distribution-invariance assumption). We argue that the distribution-independence assumption is too stringent, and we propose a learning model in which the distributions are required to satisfy a parametrized “reasonableness” criterion. This model allows us to extend results for learning functions under uniform distributions to non-uniform distributions. As an example, a bound is given for the time to learn an arbitrary halfspace in this model using the perceptron algorithm. We also argue that the distribution-invariance assumption in the pac model is unrealistic and we give bounds on the sample complexity when the distributions of training and testing examples differ.