Abstract: The training of neural networks optimizes complex non-convex objective functions, yet in practice simple algorithms achieve great performances. Recent works suggest that over-parametrization could be a key ingredient in explaining this discrepancy. However, current theories could not fully explain the role of over-parameterization. In particular, they either work in a regime where neurons don't move much, or require large number of neurons. In this paper we develop a local convergence theory for mildly over-parameterized two-layer neural net. We show…

Abstract: The goal of list learning is to understand how to learn basic statistics of a dataset when it has been corrupted by an overwhelming fraction of outliers. More formally, one is given a set of points $S$, of which an $\alpha$-fraction $T$ are promised to be well-behaved. The goal is then to output an $O(1 / \alpha)$ sized list of candidate means, so that one of these candidates is close to the true mean of the points in $T$.…

Abstract: Introduced by Kiefer and Wolfowitz 1956, the nonparametric maximum likelihood estimator (NPMLE) is a widely used methodology for learning mixture models and empirical Bayes estimation. Sidestepping the non-convexity in mixture likelihood, the NPMLE estimates the mixing distribution by maximizing the total likelihood over the space of probability measures, which can be viewed as an extreme form of over parameterization. In this work we discover a surprising property of the NPMLE solution. Consider, for example, a Gaussian mixture model on…

Abstract: Two of the fundamental problems in non-parametric statistical inference are goodness-of-fit and two-sample testing. These two problems have been extensively studied and several multivariate tests have been proposed over the last thirty years, many of which are based on geometric graphs. These include, among several others, the celebrated Friedman-Rafsky two-sample test based on the minimal spanning tree and the K-nearest neighbor graphs, and the Bickel-Breiman spacings tests for goodness-of-fit. These tests are asymptotically distribution-free, universally consistent, and computationally efficient…