Fast Convex Quadratic Optimization Solvers with Adaptive Sketching-based Preconditioners

TL;DR

This paper introduces adaptive sketching-based iterative methods for regularized least-squares problems that automatically adjust sketch size, ensuring linear convergence without prior knowledge of data complexity, and outperform existing solvers.

Contribution

The paper presents adaptive versions of the iterative Hessian sketch and preconditioned conjugate gradient methods that do not require estimating the effective dimension beforehand.

Findings

Adaptive methods guarantee linear convergence.

Methods outperform standard solvers on synthetic and real datasets.

Sketch size adapts to data complexity, improving efficiency.

Abstract

We consider least-squares problems with quadratic regularization and propose novel sketching-based iterative methods with an adaptive sketch size. The sketch size can be as small as the effective dimension of the data matrix to guarantee linear convergence. However, a major difficulty in choosing the sketch size in terms of the effective dimension lies in the fact that the latter is usually unknown in practice. Current sketching-based solvers for regularized least-squares fall short on addressing this issue. Our main contribution is to propose adaptive versions of standard sketching-based iterative solvers, namely, the iterative Hessian sketch and the preconditioned conjugate gradient method, that do not require a priori estimation of the effective dimension. We propose an adaptive mechanism to control the sketch size according to the progress made in each step of the iterative solver. If enough progress is not made, the sketch size increases to improve the convergence rate. We prove that the adaptive sketch size scales at most in terms of the effective dimension, and that our adaptive methods are guaranteed to converge linearly. Consequently, our adaptive methods improve the state-of-the-art complexity for solving dense, ill-conditioned least-squares problems. Importantly, we illustrate numerically on several synthetic and real datasets that our method is extremely efficient and is often significantly faster than standard least-squares solvers such as a direct factorization based solver, the conjugate gradient method and its preconditioned variants.