An iterative, sketching-based framework for ridge regression

Ridge regression is a variant of regularized leasl squares regression Thai is particularly suitable in settings where the number of predictor variables greatly exceeds the number of observations. We present a simple, iterative, sketching-based algorithm for ridge regression that guarantees high- quality approximations to the optimal solution vector. Our analysis builds upon two simple structural results that boil down to randomized matrix multiplication, a fundamental and well- understood primitive of randomized linear algebra. An important contribution of our work is the analysis of the behavior of sub-sampled ridge regression problems when the ridge leverage scores are used: we prove that accurate approximations can be achieved by a sample whose size depends on the degrees of freedom of the ridge-regression problem rather than the dimensions of the design matrix. Our empirical evaluations verify our theoretical results on both real and synthetic data.