Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems

• URL: http://arxiv.org/abs/2405.05818v2
• Date: Tue, 17 Jun 2025 04:24:53 GMT
• Title: Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems
• Authors: Michał Dereziński, Daniel LeJeune, Deanna Needell, Elizaveta Rebrova,
• Abstract summary: Despite being a key bottleneck in many machine learning tasks, the cost of solving large linear systems has proven challenging to quantify.<n>We consider a fine-grained notion of complexity for solving linear systems, which is motivated by applications where the data exhibits low-dimensional structure.<n>We give a algorithm based on the Sketch-and-Project paradigm, that solves the linear system $Ax = b$, that is, finds $barx$ such that $|Abarx - b| le epsilon |b|$, in time
• Score: 9.30306458153248
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Despite being a key bottleneck in many machine learning tasks, the cost of solving large linear systems has proven challenging to quantify due to problem-dependent quantities such as condition numbers. To tackle this, we consider a fine-grained notion of complexity for solving linear systems, which is motivated by applications where the data exhibits low-dimensional structure, including spiked covariance models and kernel machines, and when the linear system is explicitly regularized, such as ridge regression. Concretely, let $\kappa_\ell$ be the ratio between the $\ell$th largest and the smallest singular value of $n\times n$ matrix $A$. We give a stochastic algorithm based on the Sketch-and-Project paradigm, that solves the linear system $Ax = b$, that is, finds $\bar{x}$ such that $\|A\bar{x} - b\| \le \epsilon \|b\|$, in time $\bar O(\kappa_\ell\cdot n^2\log 1/\epsilon)$, for any $\ell = O(n^{0.729})$. This is a direct improvement over preconditioned conjugate gradient, and it provides a stronger separation between stochastic linear solvers and algorithms accessing $A$ only through matrix-vector products. Our main technical contribution is the new analysis of the first and second moments of the random projection matrix that arises in Sketch-and-Project.

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