  In this paper, a mathematical theory of learning is proposed that has many
parallels with information theory. We consider Vapnik's General Setting of
Learning in which the learning process is defined to be the act of selecting a
hypothesis in response to a given training set. Such hypothesis can, for
example, be a decision boundary in classification, a set of centroids in
clustering, or a set of frequent item-sets in association rule mining.
Depending on the hypothesis space and how the final hypothesis is selected, we
show that a learning process can be assigned a numeric score, called learning
capacity, which is analogous to Shannon's channel capacity and satisfies
similar interesting properties as well such as the data-processing inequality
and the information-cannot-hurt inequality. In addition, learning capacity
provides the tightest possible bound on the difference between true risk and
empirical risk of the learning process for all loss functions that are
parametrized by the chosen hypothesis. It is also shown that the notion of
learning capacity equivalently quantifies how sensitive the choice of the final
hypothesis is to a small perturbation in the training set. Consequently,
algorithmic stability is both necessary and sufficient for generalization.
While the theory does not rely on concentration inequalities, we finally show
that analogs to classical results in learning theory using the Probably
Approximately Correct (PAC) model can be immediately deduced using this theory,
and conclude with information-theoretic bounds to learning capacity.
