Asymptotics of Learning on Dependent and Structured Random Objects

Classical statistical inference relies on numerous tools from probability theory to study the properties of estimators. However, these same tools are often inadequate to study modern machine problems that frequently involve structured data (e.g networks) or complicated dependence structures (e.g dependent random matrices). In this talk, we extend universal limit theorems beyond the classical setting. Firstly, we consider distributionally "structured" and dependent random objects i.e random objects whose distribution are invariant under the action of an amenable group. We show, under a mild moment and mixing conditions, a series of universal second and third-order limit theorems: central-limit theorems, concentration inequalities, Wigner semi-circular law, and Berry-Esseen bounds. The utility of these will be illustrated by a series of examples in machine learning, network, and information theory. Secondly, by building on these results, we establish the asymptotic distribution of the cross-validated risk with the number of folds allowed to grow at an arbitrary rate. Using this, we study the statistical speed-up of cross-validation compared to a train-test split procedure, which reveals surprising results even when used on simple estimators.