Abstract:  It has been empirically observed that the spectrum of neural network Hessians after training have a bulk concentrated near zero, and a few outlier eigenvalues. Moreover, the eigenspaces associated to these outliers have been associated to a low-dimensional subspace in which most of the training occurs, and this implicit low-dimensional structure has been used as a heuristic for the success of high-dimensional classification. We will describe recent rigorous results in this direction for the Hessian spectrum over the course…

Abstract: Particle methods provide a powerful paradigm for solving complex global optimization problems leading to highly parallelizable algorithms. Despite widespread and growing adoption, theory underpinning their behavior has been mainly based on meta-heuristics. In application settings involving black-box procedures, or where gradients are too costly to obtain, one relies on derivative-free approaches instead. This talk will focus on two recent techniques, consensus-based optimization and consensus-based sampling. We explain how these methods can be used for the following two goals: (i)…

Abstract: The space of probability measures endowed with the optimal transport metric has a rich structure with applications in probability, analysis, and geometry. The notion of (displacement) convexity in this space was discovered by McCann, and forms the backbone of this theory.  I will introduce a new, and stronger, notion of displacement convexity which operates on the matrix level. The motivation behind this definition is to capture the intrinsic dimensionality of probability measures which could have very different behaviors along…

Abstract: This talk will focus on learning policies for tree-form decision problems (extensive-form games) from adversarial feedback. In principle, one could convert learning in any extensive-form game (EFG) into learning in an equivalent normal-form game (NFG), that is, a multi-armed bandit problem with one arm per tree-form policy. However, doing so comes at the cost of an exponential blowup of the strategy space. So, progress on NFGs and EFGs has historically followed separate tracks, with the EFG community often having…

Abstract: Many causal estimands are only partially identifiable since they depend on the unobservable joint distribution between potential outcomes. Stratification on pretreatment covariates can yield sharper partial identification bounds; however, unless the covariates are discrete with relatively small support, this approach typically requires consistent estimation of the conditional distributions of the potential outcomes given the covariates. Thus, existing approaches may fail under model misspecification or if consistency assumptions are violated. In this study, we propose a unified and model-agnostic inferential approach…

Abstract: The interchange process $\sigma_T$ is a random permutation valued stochastic process on a graph evolving in time by transpositions on its edges at rate 1. On $Z^d$, when $T$ is small all the cycles of the permutation $\sigma_T$ are finite almost surely but it is conjectured that infinite cycles appear in dimensions 3 and higher for large times.  In this talk I will focus on the finite volume case where we establish that macroscopic cycles with Poisson-Dirichlet statistics appear for large times in…

Abstract: This talk discusses uncertainty quantification and inference using ensemble methods. Recent theoretical developments inspired by random forests have cast bagging-type methods as U-statistics when bootstrap samples are replaced by subsamples, resulting in a central limit theorem and hence the potential for inference. However, to carry this out requires estimating a variance for which all proposed estimators exhibit substantial upward bias. In this talk, we convert subsamples without replacement to subsamples with replacement resulting in V-statistics for which we prove…

Abstract: One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. Recent work has identified a class of plugin estimators of Brenier maps which achieve the minimax L^2 risk, and are simple to compute. In this talk, we show that such estimators obey pointwise central limit theorems. This provides a first step toward the question of performing statistical…

Abstract: Sampling from a target measure when only partial information is available (e.g. unnormalized density as in Bayesian inference, or true samples as in generative modeling) is a fundamental problem in computational statistics and machine learning. The sampling problem can be cast as an optimization one over the space of probability distributions of a well-chosen discrepancy,  e.g. a divergence or distance to the target. In this talk, I will discuss several properties of sampling algorithms for some choices of discrepancies (standard ones,…

Abstract: Across science and policy, decision-makers often need to draw conclusions about the best candidate among competing alternatives. For instance, researchers may seek to infer the effectiveness of the most successful treatment or determine which demographic group benefits most from a specific treatment. Similarly, in machine learning, practitioners are often interested in the population performance of the model that empirically performs best. However, cherry-picking the best candidate leads to the winner’s curse: the observed performance for the winner is biased…