Shape constraints such as monotonicity, convexity, and log-concavity are naturally motivated in many applications, and can offer attractive alternatives to more traditional smoothness constraints in nonparametric estimation. In this talk we present some recent results on shape constrained estimation in high and low dimensions. First, we show how shape constrained additive models can be used to select variables in a sparse convex regression function. In contrast, additive models generally fail for variable selection under smoothness constraints. Next, we introduce graph-structured…

While simple supervised learning problems like binary classification and regression are fairly well understood, increasingly, many applications involve more complex learning problems: more complex label and prediction spaces, more complex loss structures, or both. The first part of the talk will discuss recent advances in our understanding of such problems, including the notion of convex calibration dimension of a loss function, unified approaches for designing convex calibrated surrogates for arbitrary losses, and connections between supervised learning and property elicitation. The…

Data in the form of pairwise comparisons between a collection of n items arises in many settings, including voting schemes, tournament play, and online search rankings. We study a flexible non-parametric model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity (SST). The SST class includes a large family of classical parametric models as special cases, among them the Bradley-Terry-Luce and Thurstone models, but is substantially richer. We provide…

We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a non-asymptotic point of view. In particular, we define estimators with a sub-Gaussian behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators. These results are in http://arxiv.org/abs/1509.05845, to appear in Ann Stat. (Joint work with L. Devroye, M. Lerasle, and G. Lugosi.)

Most supervised machine learning (ML) methods are explicitly designed to solve prediction problems very well. Achieving this goal does not imply that these methods automatically deliver good estimators of causal parameters. Examples of such parameters include individual regression coefficients, average treatment effects, average lifts, and demand or supply elasticities. In fact, estimates of such causal parameters obtained via naively plugging ML estimators into estimating equations for such parameters can behave very poorly, for example, by formally having inferior rates of…