Solving Dense Linear Systems Faster Than via Preconditioning

• URL: http://arxiv.org/abs/2312.08893v2
• Date: Thu, 6 Jun 2024 21:33:16 GMT
• Title: Solving Dense Linear Systems Faster Than via Preconditioning
• Authors: Michał Dereziński, Jiaming Yang,
• Abstract summary: We show that our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. In particular, our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. Our main algorithm can be viewed as a randomized block coordinate descent method.
• Score: 1.8854491183340518
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Approaching Optimality for Solving Dense Linear Systems with Low-Rank Structure [16.324043075920564]
  We provide new high-accuracy randomized algorithms for solving linear systems and regression problems.<n>Our algorithms nearly-match a natural complexity limit under dense inputs for these problems.<n>We show how to obtain these running times even under the weaker assumption that all but $k$ of the singular values have a bounded generalized mean.
  arXiv  Detail & Related papers  (2025-07-15T20:48:30Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Improved Sample Complexity for Private Nonsmooth Nonconvex Optimization [28.497079108813924]
  We provide a single-pass $(epsilon,delta)$-DP algorithm that returns an $(alpha,beta)$-stationary point as long as the dataset is of size $widetildeOmega(sqrtd/alphabeta3+d/epsilonalphabeta2)$.<n>We then provide a multi-pass time algorithm which further improves the sample complexity to $widetildeOmegaleft(d/beta2+d3/4/epsilonalpha
  arXiv  Detail & Related papers  (2024-10-08T10:15:49Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Variance-Reduced Fast Krasnoselkii-Mann Methods for Finite-Sum Root-Finding Problems [8.0153031008486]
  We propose a new class of fast Krasnoselkii--Mann methods with variance reduction to solve a finite-sum co-coercive equation $Gx = 0$.<n>Our algorithm is single-loop and leverages a new family of unbiased variance-reduced estimators specifically designed for a wider class of root-finding algorithms.<n> numerical experiments validate our algorithms and demonstrate their promising performance compared to state-of-the-art methods.
  arXiv  Detail & Related papers  (2024-06-04T15:23:29Z)
• Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning [10.690769339903941]
  We present a new class of preconditioned iterative methods for solving linear systems of the form $Ax = b$. Our methods are based on constructing a low-rank Nystr"om approximation to $A$ using sparse random sketching. We prove that the convergence of our methods depends on a natural average condition number of $A$, which improves as the rank of the Nystr"om approximation.
  arXiv  Detail & Related papers  (2024-05-09T15:53:43Z)
• Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems [9.30306458153248]
  We consider the spectral tail condition number, $kappa_ell$, defined as the ratio between the $ell$th largest and the smallest singular value of the matrix representing the system. Some of the implications of our result, and of the use of $kappa_ell$, include direct improvement over a fine-grained analysis of the Conjugate method.
  arXiv  Detail & Related papers  (2024-05-09T14:56:49Z)
• Hardness of Low Rank Approximation of Entrywise Transformed Matrix Products [9.661328973620531]
  Inspired by fast algorithms in natural language processing, we study low rank approximation in the entrywise transformed setting. We give novel reductions from the Strong Exponential Time Hypothesis (SETH) that rely on lower bounding the leverage scores of flat sparse vectors. Since our low rank algorithms rely on matrix-vectors, our lower bounds extend to show that computing $f(UV)W$, for even a small matrix $W$, requires $Omega(n2-o(1))$ time.
  arXiv  Detail & Related papers  (2023-11-03T14:56:24Z)
• Structured Semidefinite Programming for Recovering Structured Preconditioners [41.28701750733703]
  We give an algorithm which, given positive definite $mathbfK in mathbbRd times d$ with $mathrmnnz(mathbfK)$ nonzero entries, computes an $epsilon$-optimal diagonal preconditioner in time. We attain our results via new algorithms for a class of semidefinite programs we call matrix-dictionary approximation SDPs.
  arXiv  Detail & Related papers  (2023-10-27T16:54:29Z)
• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Sketching Algorithms and Lower Bounds for Ridge Regression [65.0720777731368]
  We give a sketching-based iterative algorithm that computes $1+varepsilon$ approximate solutions for the ridge regression problem. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression.
  arXiv  Detail & Related papers  (2022-04-13T22:18:47Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• Clustering Mixture Models in Almost-Linear Time via List-Decodable Mean Estimation [58.24280149662003]
  We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. We develop new algorithms for list-decodable mean estimation, achieving nearly-optimal statistical guarantees.
  arXiv  Detail & Related papers  (2021-06-16T03:34:14Z)
• Sparse sketches with small inversion bias [79.77110958547695]
  Inversion bias arises when averaging estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(epsilon,delta)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, the estimator $(epsilon,delta)$-unbiased for $(Atop A)-1$ with a sketch of size $m=O(d+sqrt d/
  arXiv  Detail & Related papers  (2020-11-21T01:33:15Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.