  We explore in some detail the notion of algorithmic stability as a viable
framework for analyzing the generalization error of learning algorithms. We
introduce the new notion of training stability of a learning algorithm and show
that, in a general setting, it is sufficient for good bounds on generalization
error. In the PAC setting, training stability is both necessary and sufficient
for learnability.\ The approach based on training stability makes no reference
to VC dimension or VC entropy. There is no need to prove uniform convergence,
and generalization error is bounded directly via an extended McDiarmid
inequality. As a result it potentially allows us to deal with a broader class
of learning algorithms than Empirical Risk Minimization. \ We also explore the
relationships among VC dimension, generalization error, and various notions of
stability. Several examples of learning algorithms are considered.
