Solving Dense Linear Systems Faster Than via Preconditioning
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910475948654592 |
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| author | Dereziński, Michał Yang, Jiaming |
| author_facet | Dereziński, Michał Yang, Jiaming |
| contents | We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq ε\|b\|$ in time: $$\tilde O((n^2+nk^{ω-1})\log1/ε),$$ where $k$ is the number of singular values of $A$ larger than $O(1)$ times its smallest positive singular value, $ω< 2.372$ is the matrix multiplication exponent, and $\tilde O$ hides a poly-logarithmic in $n$ factor. When $k=O(n^{1-θ})$ (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08893 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solving Dense Linear Systems Faster Than via Preconditioning Dereziński, Michał Yang, Jiaming Data Structures and Algorithms Machine Learning Numerical Analysis Optimization and Control We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq ε\|b\|$ in time: $$\tilde O((n^2+nk^{ω-1})\log1/ε),$$ where $k$ is the number of singular values of $A$ larger than $O(1)$ times its smallest positive singular value, $ω< 2.372$ is the matrix multiplication exponent, and $\tilde O$ hides a poly-logarithmic in $n$ factor. When $k=O(n^{1-θ})$ (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. |
| title | Solving Dense Linear Systems Faster Than via Preconditioning |
| topic | Data Structures and Algorithms Machine Learning Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2312.08893 |