Regularized Linear Regression: A Precise Analysis of the Estimation Error
published: Aug. 20, 2015, recorded: July 2015, views: 130
Report a problem or upload filesIf you have found a problem with this lecture or would like to send us extra material, articles, exercises, etc., please use our ticket system to describe your request and upload the data.
Enter your e-mail into the 'Cc' field, and we will keep you updated with your request's status.
Non-smooth regularized convex optimization procedures have emerged as a powerful tool to recover structured signals (sparse, low-rank, etc.) from (possibly compressed) noisy linear measurements. We focus on the problem of linear regression and consider a general class of optimization methods that minimize a loss function measuring the misfit of the model to the observations with an added “structured-inducing” regularization term. Celebrated instances include the LASSO, Group-LASSO, Least-Absolute Deviations method, etc.. We develop a quite general framework for how to determine precise prediction performance guaranties (e.g. mean-square-error) of such methods for the case of Gaussian measurement ensemble. The machinery builds upon Gordon's Gaussian min-max theorem under additional convexity assumptions that arise in many practical applications. This theorem associates with a primary optimization (PO) problem a simplified auxiliary optimization (AO) problem from which we can tightly infer properties of the original (PO), such as the optimal cost, the norm of the optimal solution, etc. Our theory applies to general loss functions and regularization and provides guidelines on how to optimally tune the regularizer coefficient when certain structural properties (such as sparsity level, rank, etc.) are known.
Download slides: colt2015_thrampoulidis_estimation_error_01.pdf (1.0 MB)
Link this pageWould you like to put a link to this lecture on your homepage?
Go ahead! Copy the HTML snippet !