  We propose data-dependent uniform generalization bounds by approaching the
problem from a PAC-Bayesian perspective. We first apply the PAC-Bayesian
framework on `random sets' in a rigorous way, where the training algorithm is
assumed to output a data-dependent hypothesis set after observing the training
data. This approach allows us to prove data-dependent bounds, which can be
applicable in numerous contexts. To highlight the power of our approach, we
consider two main applications. First, we propose a PAC-Bayesian formulation of
the recently developed fractal-dimension-based generalization bounds. The
derived results are shown to be tighter and they unify the existing results
around one simple proof technique. Second, we prove uniform bounds over the
trajectories of continuous Langevin dynamics and stochastic gradient Langevin
dynamics. These results provide novel information about the generalization
properties of noisy algorithms.
