Randomized Least Squares Regression: Combining Model- and Algorithm-Induced Uncertainties

We analyze the uncertainties in the minimum norm solution of full-rank regression problems, arising from Gaussian linear models, computed by randomized (row-wise sampling and, more generally, sketching) algorithms. From a deterministic perspective, our structural perturbation bounds imply that least squares problems are less sensitive to multiplicative perturbations than to additive perturbations. From a probabilistic perspective, our expressions for the total expectation and variance with regard to both model- and algorithm-induced uncertainties, are exact, hold for general sketching matrices, and make no assumptions on the rank of the sketched matrix. The relative differences between the total bias and variance on the one hand, and the model bias and variance on the other hand, are governed by two factors: (i) the expected rank deficiency of the sketched matrix, and (ii) the expected difference between projectors associated with the original and the sketched problems. A simple example, based on uniform sampling with replacement, illustrates the statistical quantities.