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Confinement of Unimodal Probability Distributions and an FKG-Gaussian Correlation Inequality

Mark Sellke, Harvard University
E18-304

Abstract: While unimodal probability distributions are well understood in dimension 1, the same cannot be said in high dimension without imposing stronger conditions such as log-concavity. I will explain a new approach to proving confinement (e.g. variance upper bounds) for high-dimensional unimodal distributions which are not log-concave, based on an extension of Royen's celebrated Gaussian correlation inequality. We will see how it yields new localization results for Ginzberg-Landau random surfaces, a well-studied family of continuous-variable graphical models, with very general…

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Estimation of Functionals of High-Dimensional and Infinite-Dimensional Parameters of Statistical Models

Vladimir Koltchinskii, Georgia Institute of Technology
2-449

The mini-course will meet on Monday, April 1 and Wednesday, April 3rd from 1:30-3:00pm This mini-course deals with a circle of problems related to estimation of real valued functionals of high-dimensional and infinite-dimensional parameters of statistical models. In such problems, it is of interest to estimate one-dimensional features of a high-dimensional parameter represented by nonlinear functionals of certain degree of smoothness defined on the parameter space. The functionals of interest could be often estimated with faster convergence rates than the…

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Optimal nonparametric capture-recapture methods for estimating population size

Edward Kennedy, Carnegie Mellon University
E18-304

Abstract: Estimation of population size using incomplete lists has a long history across many biological and social sciences. For example, human rights groups often construct partial lists of victims of armed conflicts, to estimate the total number of victims. Earlier statistical methods for this setup often use parametric assumptions, or rely on suboptimal plug-in-type nonparametric estimators; but both approaches can lead to substantial bias, the former via model misspecification and the latter via smoothing. Under an identifying assumption that two lists…

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Lattices and the Hardness of Statistical Problems

Vinod Vaikuntanathan (MIT)
E18-304

Abstract: I will describe recent results that (a) show nearly optimal hardness of learning Gaussian mixtures, and (b) give evidence of average-case hardness of sparse linear regression w.r.t. all efficient algorithms, assuming the worst-case hardness of lattice problems. The talk is based on the following papers with Aparna Gupte and Neekon Vafa. https://arxiv.org/pdf/2204.02550.pdf https://arxiv.org/pdf/2402.14645.pdf Bio: Vinod Vaikuntanathan is a professor of computer science at MIT and the chief cryptographer at Duality Technologies. His research is in the foundations of cryptography…

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Emergent outlier subspaces in high-dimensional stochastic gradient descent

Reza Gheissari, Northwestern University
E18-304

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…

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Consensus-based optimization and sampling

Franca Hoffmann, California Institute of Technology
E18-304

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)…

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Matrix displacement convexity and intrinsic dimensionality

Yair Shenfeld, Brown University
E18-304

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…

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Adversarial combinatorial bandits for imperfect-information sequential games

Gabriele Farina, MIT
E18-304

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…

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Model-agnostic covariate-assisted inference on partially identified causal effects

Lihua Lei, Stanford University
E18-304

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…

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