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An Iterative, Sketching-based Framework for Ridge Regression

Ridge regression is a variant of regularized least squares regression that 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 highquality approximations to the optimal solution vector. Our analysis builds upon two simple structural results that boil down to randomized matrix multiplication, a fundamental and wellunderstood 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.