Sketched Ridge Regression: Optimization Perspective, Statistical Perspective, and Model Averaging
This work addresses the problem of efficiently solving large-scale ridge regression for practitioners in machine learning and statistics, offering a method to mitigate the statistical risks of sketching, though it is incremental by building on prior sketching research for simpler regression problems.
The paper tackles the statistical and optimization impacts of sketching methods (classical and Hessian sketch) on Matrix Ridge Regression (MRR), showing that classical sketch yields near-optimal solutions but increases variance, while Hessian sketch increases bias, leading to risks up to an order-of-magnitude higher than optimal MRR. It demonstrates that model averaging reduces this risk gap, enabling near-optimal solutions in distributed settings.
We address the statistical and optimization impacts of the classical sketch and Hessian sketch used to approximately solve the Matrix Ridge Regression (MRR) problem. Prior research has quantified the effects of classical sketch on the strictly simpler least squares regression (LSR) problem. We establish that classical sketch has a similar effect upon the optimization properties of MRR as it does on those of LSR: namely, it recovers nearly optimal solutions. By contrast, Hessian sketch does not have this guarantee, instead, the approximation error is governed by a subtle interplay between the "mass" in the responses and the optimal objective value. For both types of approximation, the regularization in the sketched MRR problem results in significantly different statistical properties from those of the sketched LSR problem. In particular, there is a bias-variance trade-off in sketched MRR that is not present in sketched LSR. We provide upper and lower bounds on the bias and variance of sketched MRR, these bounds show that classical sketch significantly increases the variance, while Hessian sketch significantly increases the bias. Empirically, sketched MRR solutions can have risks that are higher by an order-of-magnitude than those of the optimal MRR solutions. We establish theoretically and empirically that model averaging greatly decreases the gap between the risks of the true and sketched solutions to the MRR problem. Thus, in parallel or distributed settings, sketching combined with model averaging is a powerful technique that quickly obtains near-optimal solutions to the MRR problem while greatly mitigating the increased statistical risk incurred by sketching.