Transport distances and novel uncertainty quantification results

In this talk, we will see how concentration inequalities for the sample mean, like those due to Bernstein and Hoeffding, are valid for any sample size but overly conservative, yielding unnecessarily wide confidence intervals. Motivated by applications to reinforcement learning we develop new results on transport distances and obtain new computable concentration inequalities with asymptotically optimal size, finite-sample validity, and sub-Gaussian decay. Beyond empirical averages, we will tie this to the concept of Gaussian universality that has been recently observed to hold for a large class of estimators. We will precisely determine the class of functions and, hence, estimators, for which it holds. Beyond the independent setting we will see how it holds for a class of dependent processes and see how this can allow us to gain insight into the behavior of data augmentation. Time permitting we will finish the talk by investigating what happens with structured data in the context of graph neural networks.