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We address the statistical and optimization impacts of the classical sketchand Hessian sketch used to approximately solve the Matrix Ridge Regression(MRR) problem. Prior research has quantified the effects of classical sketch onthe strictly simpler least squares regression (LSR) problem. We establish thatclassical sketch has a similar effect upon the optimization properties of MRRas it does on those of LSR: namely, it recovers nearly optimal solutions. Bycontrast, Hessian sketch does not have this guarantee, instead, theapproximation error is governed by a subtle interplay between the "mass" in theresponses and the optimal objective value.  For both types of approximation, the regularization in the sketched MRRproblem results in significantly different statistical properties from those ofthe sketched LSR problem. In particular, there is a bias-variance trade-off insketched MRR that is not present in sketched LSR. We provide upper and lowerbounds on the bias and variance of sketched MRR, these bounds show thatclassical sketch significantly increases the variance, while Hessian sketchsignificantly increases the bias. Empirically, sketched MRR solutions can haverisks that are higher by an order-of-magnitude than those of the optimal MRRsolutions.  We establish theoretically and empirically that model averaging greatlydecreases the gap between the risks of the true and sketched solutions to theMRR problem. Thus, in parallel or distributed settings, sketching combined withmodel averaging is a powerful technique that quickly obtains near-optimalsolutions to the MRR problem while greatly mitigating the increased statisticalrisk incurred by sketching.

Sketched Ridge Regression: Optimization Perspective, Statistical Perspective, and Model Averaging