Abstract: Reinforcement learning (RL), which is frequently modeled as sequential learning and decision making in the face of uncertainty, is garnering growing interest in recent years due to its remarkable success in practice. In contemporary RL applications, it is increasingly more common to encounter environments with prohibitively large state and action space, thus imposing stringent requirements on the sample efficiency of the RL algorithms in use. Despite the empirical success, however, the theoretical underpinnings for many popular RL algorithms remain…

Abstract: One of the main reasons behind the success of high-dimensional statistics and modern machine learning in taming the curse of dimensionality is that many classes of high-dimensional distributions are surprisingly well-behaved and, when viewed correctly, exhibit a simple structure. This emergent simplicity is in the center of the theory of "high-dimensional phenomena", and is manifested in principles such as "Gaussian-like behavior" (objects of interest often inherit the properties of the Gaussian measure), "dimension-free behavior" (expressed in inequalities which do…

Abstract:  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 object i.e random objects whose distribution are invariant under the action of an amenable group. We…

Abstract:  Sorted L1 regularization has been incorporated into many methods for solving high-dimensional statistical estimation problems, including the SLOPE estimator in linear regression. In this talk, we study how this relatively new regularization technique improves variable selection by characterizing the optimal SLOPE trade-off between the false discovery proportion (FDP) and true positive proportion (TPP) or, equivalently, between measures of type I and type II error. Additionally, we show that on any problem instance, SLOPE with a certain regularization sequence outperforms…

Abstract: This talk will introduce a precise high-dimensional asymptotic theory for AdaBoost on separable data, taking both statistical and computational perspectives. We will consider the common modern setting where the number of features p and the sample size n are both large and comparable, and in particular, look at scenarios where the data is asymptotically separable. Under a class of statistical models, we will provide an (asymptotically) exact analysis of the max-min-L1-margin and the min-L1-norm interpolant. In turn, this will…

Abstract: Since the 1960s, particle physicists have developed a variety of data analysis strategies for the goal of comparing experimental measurements to theoretical predictions.  Despite their numerous successes, these techniques can seem esoteric and ad hoc, even to practitioners in the field.  In this talk, I explain how many particle physics analysis tools have a natural geometric interpretation in an emergent "space" of collider events induced by the Wasserstein metric.  This in turn suggests new analysis strategies to interpret generic…

Abstract: The existence of a transport map from the standard Gaussian leads to succinct​representations for, potentially complicated, measures.​ Inspired by result from optimal transport, we introduce the Brownian transport map that pushes forward the Wiener measure to a target measure in a finite-dimensional Euclidean space. Using tools from Ito's and Malliavin's calculus, we show that the map is Lipschitz in several cases of interest. Specifically, our results apply when the target measure satisfies one of the following: - More log-concave than the Gaussian, recovering…