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.

我们从 PAC-Bayesian 的角度提出了数据相关的均匀泛化界，通过将训练算法输出的数据相关假设集应用于随机集的严格方法，我们证明了数据相关的界，适用于多种情境，并将此方法应用于基于分形维度的泛化界和连续 Langevin 动力学以及随机梯度 Langevin 动力学的轨迹上，这些结果为噪声算法的泛化特性提供了新的信息。