The weekly SILO Seminar Series is made possible through the generous support of the 3M Company and its Advanced Technology Group


with additional support from the Analytics Group of the Northwestern Mutual Life Insurance Company

Northwestern Mutual

Fast global convergence of gradient methods for high-dimensional statistical recovery

Sahand Negahban, Postdoc, MIT

Date and Time: Oct 24, 2012 (12:30 PM)
Location: Orchard room (3280) at the Wisconsin Institute for Discovery Building


Many statistical M-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension d to grow with (and possibly exceed) the sample size n. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie much of classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that projected gradient descent has a globally geometric rate of convergence up to the \emph{statistical precision} of the model, meaning the typical distance between the true unknown parameter $\theta^*$ and an optimal solution $\hat{\theta}$. This result is substantially sharper than previous convergence results, which yielded sublinear convergence, or linear convergence only up to the noise level. Our analysis applies to a wide range of M-estimators and statistical models, including sparse linear regression using Lasso ($\ell_1$-regularized regression); group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition.

This is based on joint work with Alekh Agarwal (MSR) and Martin Wainwright (UC Berkeley)