Iterative Hessian sketch: Fast and accurate solution approximation for constrained least-squares

TL;DR

This paper introduces the iterative Hessian sketch, a novel randomized method that efficiently approximates solutions to constrained least-squares problems with high accuracy, outperforming traditional sketching techniques.

Contribution

The paper presents the iterative Hessian sketch, providing theoretical guarantees and demonstrating its effectiveness for constrained least-squares approximation with fewer iterations and lower projection dimensions.

Findings

Iterative Hessian sketch achieves accurate solution approximations.

Traditional sketches are sub-optimal for solution approximation.

Method is effective for constrained and unconstrained least-squares.

Abstract

We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance measure between the approximate minimizer and the true minimizer (solution approximation). Focusing on the latter criterion, our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least-squares problem; as a surprising consequence, the most widely used least-squares sketch is sub-optimal for solution approximation. We then present a new method known as the iterative Hessian sketch, and show that it can be used to obtain approximations to the original least-squares problem using a projection dimension proportional to the statistical complexity of the least-squares minimizer, and a logarithmic number of iterations. We illustrate our general theory with simulations for both unconstrained and constrained versions of least-squares, including $\ell_1$-regularization and nuclear norm constraints. We also numerically demonstrate the practicality of our approach in a real face expression classification experiment.