| Invited Talk

Name of the Speaker: Ms. Shubhangi Ghosh
Name of the Organizer: Prof. Krishna Jagannathan
Venue: ESB-234 (Malaviya Hall)
Date/Time: 29th July 2024 (Monday), 3:00 PM
Title: Minimax risk of sparse linear regression and higher-order asymptotics

Abstract :

The minimax framework has been one of the cornerstones of theoretical statistics, and has contributed to the popularity of many well-known estimators, such as the regularized M-estimators and regularized linear regression estimators for high-dimensional problems. In this paper, we demonstrate that numerous theoretical results within the classical minimax framework are inadequate in explaining empirical observations. In some instances, these minimax outcomes offer insights that contradict empirical findings. For example, although LASSO has been proven to be minimax optimal for the sparse linear regression problem, numerous empirical studies have shown its suboptimal performance across various signal-to-noise (SNR) levels. In this study, we aim to introduce an enhanced version of the minimax framework that not only elucidates these disparities but also offers more precise insights into the optimality of different estimators.

Our novel approach has two two distinctive components: (1) it integrates the signal-to-noise ratio into the construction of the parameter space. (2) It obtains accurate approximation of the minimax risk through asymptotic arguments. The theoretical findings derived from this refined framework provide new insights and practical guidance. For instance, in the context of estimating sparse signals under the linear regression model, our approach demonstrates that in the low SNR, ridge regression surpasses all other estimators, even when the regression coefficients are sparse.

Bio: Presently doing her PhD at the Department of Statistics, Columbia University.