A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers

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 0, a line of recent work has studied models with various types of structure (e.g., sparse vectors; block-structured matrices; low-rank matrices; Markov assumptions). In such settings, a general approach to estimation is to solve a regularized convex program (known as a regularized M-estimator) which combines a loss function (measuring how well the model fits the data) with some regularization function that encourages the assumed structure. The goal of this paper is to provide a unified framework for establishing consistency and convergence rates for such regularized Mestimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive several existing results, and also to obtain several new results on consistency and converge...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Sahand Negahban, Pradeep Ravikumar, Martin J. Wainwright, Bin Yu
• Keywords: Convergence Rates, CORR 2010, Education, Regularization Functions, Regularized M-estimators