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.

翻译：我们通过PAC-贝叶斯视角提出数据依赖的统一泛化界。首先，我们以严格的方式将PAC-贝叶斯框架应用于"随机集"，假设训练算法在观测训练数据后输出数据依赖的假设集。该方法使我们能够证明适用于多种场景的数据依赖界。为突出该方法的优势，我们考虑两个主要应用。首先，我们提出基于分形维数的泛化界的PAC-贝叶斯表述，所得结果被证明更为紧致，并通过单一简洁证明技术统一了现有结果。其次，我们对连续朗之万动力学和随机梯度朗之万动力学的轨迹证明了统一界，这些结果为含噪算法的泛化特性提供了新信息。