A Unified Framework for High-Dimensional Analysis of M-Estimators with Decomposable Regularizers

TL;DR

This paper introduces a unified theoretical framework for analyzing high-dimensional M-estimators with decomposable regularizers, providing new insights into their consistency and convergence rates across various structured models.

Contribution

It establishes a general theorem that unifies and extends existing results, highlighting the roles of restricted strong convexity and decomposability in high-dimensional estimation.

Findings

The framework applies to sparse vectors, matrices, and low-rank models.

It derives optimal convergence rates in multiple norms.

The analysis simplifies understanding of regularized M-estimators' behavior.

Abstract

High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides a unified framework for establishing consistency and convergence rates for such regularized M-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive some existing results, and also to obtain a number of new results on consistency and convergence rates, in both $\ell_2$-error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized M-estimators have fast convergence rates and which are optimal in many well-studied cases.