Many statistical estimators are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizer. We analyze the convergence rates of first-order gradient methods for solving such problems 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 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 a simple first-order method has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical Euclidean distance between the true unknown parameter $\theta^*$ and the optimal solution $\widehat{\theta}$. This globally linear rate is substantially faster than previous analyses of global convergence for specific methods that yielded only sublinear rates. Our analysis applies to a wide range of $M$-estimators and statistical models, including sparse linear regression using Lasso ($\ell_1$-regularized regression), generalized linear models, block-sparse models, and low-rank matrix recovery using nuclear norm regularization.
Joint work with Alekh Agarwal and Sahand Negahban, UC Berkeley
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