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

TL;DR

This paper investigates the effects of classical and Hessian sketching methods on matrix ridge regression, revealing their impacts on optimization, statistical bias-variance trade-offs, and how model averaging can mitigate increased risks.

Contribution

It provides a comprehensive analysis of sketching impacts on MRR, including bounds on bias and variance, and demonstrates the effectiveness of model averaging in reducing risks.

Findings

Classical sketch recovers nearly optimal solutions for MRR.

Hessian sketch's approximation error depends on response mass and optimal objective.

Model averaging significantly reduces the risk gap between true and sketched MRR solutions.

Abstract

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.