Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908409485328384 |
|---|---|
| author | Dereziński, Michał LeJeune, Daniel Needell, Deanna Rebrova, Elizaveta |
| author_facet | Dereziński, Michał LeJeune, Daniel Needell, Deanna Rebrova, Elizaveta |
| contents | 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 $κ_\ell$ 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 $\|A\bar{x} - b\| \le ε\|b\|$, in time $\bar O(κ_\ell\cdot n^2\log 1/ε)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems Dereziński, Michał LeJeune, Daniel Needell, Deanna Rebrova, Elizaveta Data Structures and Algorithms Machine Learning Numerical Analysis Optimization and Control 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 $κ_\ell$ 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 $\|A\bar{x} - b\| \le ε\|b\|$, in time $\bar O(κ_\ell\cdot n^2\log 1/ε)$, 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. |
| title | Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems |
| topic | Data Structures and Algorithms Machine Learning Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2405.05818 |