We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

How worried should you be about an AI apocalypse? Fears that artificial intelligence could rise up to wipe out humanity are understandable given our steady diet of sci-fi stories depicting just that, but what is the real risk? Isaac Asimov's three laws of robotics are not a practical guide Super-intelligent artificial intelligence rising up and wiping out humanity has been a common trope in science fiction for decades. Now, we live in a world where real AI seems to be advancing faster than ever. Does that mean you should start worrying about an AI apocalypse?

This is the Picard rank one analogue to Proposition 5.3 in the main text. The number in each dimension is given in Table 1.

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science.

Then the phenomenon has been thought as inevitable. However, the phenomenon seldom occurs in the context of recent deep learning. There is a gap between theory and reality.

In this case, we view the geometric domain as the discrete 1d gridΩ = [1,...,d], and consider geometric transformationsG as subsets of the symmetric group of permutations ofd