  How quickly can a given class of concepts be learned from examples? It is
common to measure the performance of a supervised machine learning algorithm by
plotting its "learning curve", that is, the decay of the error rate as a
function of the number of training examples. However, the classical theoretical
framework for understanding learnability, the PAC model of Vapnik-Chervonenkis
and Valiant, does not explain the behavior of learning curves: the
distribution-free PAC model of learning can only bound the upper envelope of
the learning curves over all possible data distributions. This does not match
the practice of machine learning, where the data source is typically fixed in
any given scenario, while the learner may choose the number of training
examples on the basis of factors such as computational resources and desired
accuracy.
  In this paper, we study an alternative learning model that better captures
such practical aspects of machine learning, but still gives rise to a complete
theory of the learnable in the spirit of the PAC model. More precisely, we
consider the problem of universal learning, which aims to understand the
performance of learning algorithms on every data distribution, but without
requiring uniformity over the distribution. The main result of this paper is a
remarkable trichotomy: there are only three possible rates of universal
learning. More precisely, we show that the learning curves of any given concept
class decay either at an exponential, linear, or arbitrarily slow rates.
Moreover, each of these cases is completely characterized by appropriate
combinatorial parameters, and we exhibit optimal learning algorithms that
achieve the best possible rate in each case.
  For concreteness, we consider in this paper only the realizable case, though
analogous results are expected to extend to more general learning scenarios.
