  We study the generalization error of randomized learning algorithms --
focusing on stochastic gradient descent (SGD) -- using a novel combination of
PAC-Bayes and algorithmic stability. Importantly, our generalization bounds
hold for all posterior distributions on an algorithm's random hyperparameters,
including distributions that depend on the training data. This inspires an
adaptive sampling algorithm for SGD that optimizes the posterior at runtime. We
analyze this algorithm in the context of our generalization bounds and evaluate
it on a benchmark dataset. Our experiments demonstrate that adaptive sampling
can reduce empirical risk faster than uniform sampling while also improving
out-of-sample accuracy.
