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We propose a fast algorithm for computing the entire ridge regressionregularization path in nearly linear time. Our method constructs a basis onwhich the solution of ridge regression can be computed instantly for any valueof the regularization parameter. Consequently, linear models can be tuned viacross-validation or other risk estimation strategies with substantially betterefficiency. The algorithm is based on iteratively sketching the Krylov subspacewith a binomial decomposition over the regularization path. We provide aconvergence analysis with various sketching matrices and show that it improvesthe state-of-the-art computational complexity. We also provide a technique toadaptively estimate the sketching dimension. This algorithm works for both theover-determined and under-determined problems. We also provide an extension formatrix-valued ridge regression. The numerical results on real medium and largescale ridge regression tasks illustrate the effectiveness of the proposedmethod compared to standard baselines which require super-linear computationaltime.

Sketching the Krylov Subspace: Faster Computation of the Entire Ridge Regularization Path