  Most generalization bounds in learning theory are based on some measure of
the complexity of the hypothesis class used, independently of any algorithm. In
contrast, the notion of algorithmic stability can be used to derive tight
generalization bounds that are tailored to specific learning algorithms by
exploiting their particular properties. However, as in much of learning theory,
existing stability analyses and bounds apply only in the scenario where the
samples are independently and identically distributed. In many machine learning
applications, however, this assumption does not hold. The observations received
by the learning algorithm often have some inherent temporal dependence.
  This paper studies the scenario where the observations are drawn from a
stationary phi-mixing or beta-mixing sequence, a widely adopted assumption in
the study of non-i.i.d. processes that implies a dependence between
observations weakening over time. We prove novel and distinct stability-based
generalization bounds for stationary phi-mixing and beta-mixing sequences.
These bounds strictly generalize the bounds given in the i.i.d. case and apply
to all stable learning algorithms, thereby extending the use of
stability-bounds to non-i.i.d. scenarios.
  We also illustrate the application of our phi-mixing generalization bounds to
general classes of learning algorithms, including Support Vector Regression,
Kernel Ridge Regression, and Support Vector Machines, and many other kernel
regularization-based and relative entropy-based regularization algorithms.
These novel bounds can thus be viewed as the first theoretical basis for the
use of these algorithms in non-i.i.d. scenarios.
