Entropic optimal transport (entropic OT) is a regularized variant of the optimal transport problem, widely used in practice for its computational benefits. A key statistical question for both entropic and un-regularized OT is the extent to which low-dimensional structure, of the type conjectured by the well-known manifold hypothesis, affects the statistical rates of convergence. In this talk, we will present statistical results for entropic OT which clarify the statistical role of…

Born in the late 20th century, PAC-Bayes has recently re-emerged as a powerful framework for learning with guarantees. Its bounds offer a principled way to understand the generalization ability of randomized learning algorithms, even guiding the design of new ones. This introduction dives into the foundations of PAC-Bayes, explores its recent advancements, and tries to offer some insights into promising future research directions.