Dr. Nicholas Pischke, University of Bath, UK

Quasi-Fejer monotonicity is a fundamental property enjoyed by many methods in stochastic approximation. Even further, it offers a streamlined and abstract theory with general convergence results that allows one to unify the study of the asymptotic behavior of various iterative processes. In this talk, I give an overview of recent work that tries to make these general results quantitative. Over general metric spaces, such monotone processes only converge under a compactness assumption. However, if the associated problem is sufficiently regular, characterized by a broad and abstract condition, then this compactness assumption can be removed, and one can actually derive explicit full rates of convergence for the process, leveraging martingale as well as measurable selection theory. I present the general approach for constructing such rates of convergence, and moreover discuss various methods for which these constructions yield new quantitative results on their asymptotic behavior. At the end, I give an overview of the kind of quantitative results that are possible without such a regularity assumption, and in particular of the new area of finitary martingale theory that such considerations lead to, and I frame these results as part of a recent broad approach towards deriving quantitative results in stochastic optimization through applications of methods from proof theory, the foundations of mathematics.