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…

Abstract: For many causal effect parameters of interest, doubly robust machine learning (DRML) estimators ψ̂ 1 are the state-of-the-art, incorporating the good prediction performance of machine learning; the decreased bias of doubly robust estimators; and the analytic tractability and bias reduction of sample splitting with cross fitting. Nonetheless, even in the absence of confounding by unmeasured factors, the nominal (1−α) Wald confidence interval ψ̂ 1±zα/2ˆ may still undercover even in large samples, because the bias of ψ̂ 1 may be of the same…

Abstract: We consider sequential prediction with expert advice when data are generated from distributions varying arbitrarily within an unknown constraint set. We quantify relaxations of the classical i.i.d. assumption in terms of these constraint sets, with i.i.d. sequences at one extreme and adversarial mechanisms at the other. The Hedge algorithm, long known to be minimax optimal in the adversarial regime, was recently shown to be minimax optimal for i.i.d. data. We show that Hedge with deterministic learning rates is suboptimal…

Abstract: Mainstream machine learning, despite its recent successes, has a serious drawback: while its state-of-the-art algorithms often produce excellent predictions, they do not provide measures of their accuracy and reliability that would be both practically useful and provably valid. Conformal prediction adapts rank tests, popular in nonparametric statistics, to testing the IID assumption (the observations being independent and identically distributed). This gives us practical measures, provably valid under the IID assumption, of the accuracy and reliability of predictions produced by…

Abstract: The magnitude of the weights of a neural network is a fundamental measure of complexity that plays a crucial role in the study of implicit and explicit regularization. For example, in recent work, gradient descent updates in overparameterized models asymptotically lead to solutions that implicitly minimize the ell_2 norm of the parameters of the model, resulting in an inductive bias that is highly architecture dependent. To investigate the properties of learned functions, it is natural to consider a function…