Continuous Wasserstein PAC-Bayes Formulation Without Spatial Partitions

Formulate PAC-Bayes generalization bounds directly using continuous Wasserstein or optimal-transport distances on the space of probability measures \(\mathcal{M}_1(\mathcal{Z})\), without relying on finite spatial partitions.

Background

The paper’s data-space PAC-Bayes certificate evaluates the relative entropy between a Dirichlet-process posterior and prior after projecting them onto a finite measurable partition of the instance space. Finite partitions make the divergence computationally tractable through a closed-form expression involving Gamma and Digamma functions, but they discretize the underlying continuous geometry.

The authors note that taking the supremum over all measurable partitions recovers the exact Dirichlet-process divergence, whereas practical implementations use finite spatial grids or vector-quantization cells. Extending the framework to continuous Wasserstein or optimal-transport geometries would remove this spatial discretization and provide a more intrinsically geometric formulation of data-space PAC-Bayes bounds.