Solving Dense Linear Systems Faster Than via Preconditioning

TL;DR

This paper introduces a stochastic optimization algorithm that efficiently solves dense linear systems by leveraging spectral properties, outperforming traditional methods especially for matrices with flat spectra, and extends to sparse matrices and regression.

Contribution

The paper presents a novel randomized block coordinate descent algorithm with improved runtime for solving dense linear systems, utilizing determinantal point processes and matrix sketching techniques.

Findings

Achieves 7(n^2 + nk^{\u03A9-1})\u2212log(1/B5}) runtime for solving linear systems.

Outperforms direct solving and preconditioned iterative methods for matrices with flat spectra.

Extends methodology to sparse positive semidefinite matrices and least squares regression.

Abstract

We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq \epsilon\|b\|$ in time: $$\tilde O((n^2+nk^{\omega-1})\log1/\epsilon),$$ where $k$ is the number of singular values of $A$ larger than $O(1)$ times its smallest positive singular value, $\omega < 2.372$ is the matrix multiplication exponent, and $\tilde O$ hides a poly-logarithmic in $n$ factor. When $k=O(n^{1-\theta})$ (namely, $A$ has a flat-tailed spectrum, e.g., due to noisy data or regularization), this improves on both the cost of solving the system directly, as well as on the cost of preconditioning an iterative method such as conjugate gradient. In particular, our algorithm has an $\tilde O(n^2)$ runtime when $k=O(n^{0.729})$. We further adapt this result to sparse positive semidefinite matrices and least squares regression. Our main algorithm can be viewed as a randomized block coordinate descent method, where the key challenge is simultaneously ensuring good convergence and fast per-iteration time. In our analysis, we use theory of majorization for elementary symmetric polynomials to establish a sharp convergence guarantee when coordinate blocks are sampled using a determinantal point process. We then use a Markov chain coupling argument to show that similar convergence can be attained with a cheaper sampling scheme, and accelerate the block coordinate descent update via matrix sketching.