  In statistical learning theory, generalization error is used to quantify the
degree to which a supervised machine learning algorithm may overfit to training
data. Recent work [Xu and Raginsky (2017)] has established a bound on the
generalization error of empirical risk minimization based on the mutual
information $I(S;W)$ between the algorithm input $S$ and the algorithm output
$W$, when the loss function is sub-Gaussian. We leverage these results to
derive generalization error bounds for a broad class of iterative algorithms
that are characterized by bounded, noisy updates with Markovian structure. Our
bounds are very general and are applicable to numerous settings of interest,
including stochastic gradient Langevin dynamics (SGLD) and variants of the
stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm. Furthermore, our
error bounds hold for any output function computed over the path of iterates,
including the last iterate of the algorithm or the average of subsets of
iterates, and also allow for non-uniform sampling of data in successive updates
of the algorithm.
