As the title indicates, the talk consists of two vignettes: on hypothesis testing and early stopping for
boosting algorithms.

The first part focuses on a certain class of composite testing problems with null and alternative specified by cones; such geometric testing problems arise in various applications (e.g., treatment effects, radar detection, and shape-constrained testing). Despite the widespread use of the generalized likelihood ratio test (GLRT), its properties have yet to be fully understood. When is it optimal, and when can it be improved upon? How does its performance depend on the cones? I provide some answers to these and other questions, all based on a tight characterization of the GLRT's performance.

In the second part, I will discuss how to understand the behavior of early stopping with boosting for non -parametric regression. While non-parametric models offer great flexibility, they can lead to overfitting and thus poor generalization performance. For this reason, procedures for fitting these models must involve some form of regularization. Although early-stopping of iterative algorithms is a widely-used form of regularization in statistics and optimization, it is less well-understood than its analogue based on penalized regularization. In this talk, I will establish some precise connections between these two, and give an explicit and optimal stopping criteria for boosting algorithms run in a reproducing kernel Hilbert space.

This talk is based on joint works with Adityanand Guntuboyina, Martin Wainwright and Fanny Yang.