Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems
- URL: http://arxiv.org/abs/2405.05818v1
- Date: Thu, 9 May 2024 14:56:49 GMT
- Title: Fine-grained Analysis and Faster Algorithms for Iteratively Solving Linear Systems
- Authors: Michał Dereziński, Daniel LeJeune, Deanna Needell, Elizaveta Rebrova,
- Abstract summary: 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.
- Score: 9.30306458153248
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: While effective in practice, iterative methods for solving large systems of linear equations can be significantly affected by problem-dependent condition number quantities. This makes characterizing their time complexity challenging, particularly when we wish to make comparisons between deterministic and stochastic methods, that may or may not rely on preconditioning and/or fast matrix multiplication. In this work, we consider a fine-grained notion of complexity for iterative linear solvers which we call 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. Concretely, we prove the following main algorithmic result: Given an $n\times n$ matrix $A$ and a vector $b$, we can find $\tilde{x}$ such that $\|A\tilde{x}-b\|\leq\epsilon\|b\|$ in time $\tilde{O}(\kappa_\ell\cdot n^2\log 1/\epsilon)$ for any $\ell = O(n^{\frac1{\omega-1}})=O(n^{0.729})$, where $\omega \approx 2.372$ is the current fast matrix multiplication exponent. This guarantee is achieved by Sketch-and-Project with Nesterov's acceleration. 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 Gradient method, suggesting a stronger separation between deterministic and stochastic iterative solvers; and relating the complexity of iterative solvers to the ongoing algorithmic advances in fast matrix multiplication, since the bound on $\ell$ improves with $\omega$. Our main technical contributions are new sharp characterizations for the first and second moments of the random projection matrix that commonly arises in sketching algorithms, building on a combination of techniques from combinatorial sampling via determinantal point processes and Gaussian universality results from random matrix theory.
Related papers
- Solving Dense Linear Systems Faster Than via Preconditioning [1.8854491183340518]
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.
arXiv Detail & Related papers (2023-12-14T12:53:34Z) - Fast Minimization of Expected Logarithmic Loss via Stochastic Dual
Averaging [8.990961435218544]
We propose a first-order algorithm named $B$-sample dual averaging with the logarithmic barrier.
For the Poisson inverse problem, our algorithm attains an $varepsilon$ solution in $smashtildeO(d3/varepsilon2)$ time.
When computing the maximum-likelihood estimate for quantum state tomography, our algorithm yields an $varepsilon$-optimal solution in $smashtildeO(d3/varepsilon2)$ time.
arXiv Detail & Related papers (2023-11-05T03:33:44Z) - Efficiently Learning One-Hidden-Layer ReLU Networks via Schur
Polynomials [50.90125395570797]
We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss.
Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
arXiv Detail & Related papers (2023-07-24T14:37:22Z) - Fast and Practical Quantum-Inspired Classical Algorithms for Solving
Linear Systems [11.929584800629673]
We propose fast and practical quantum-inspired classical algorithms for solving linear systems.
Our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems.
arXiv Detail & Related papers (2023-07-13T08:46:19Z) - Linear Query Approximation Algorithms for Non-monotone Submodular
Maximization under Knapsack Constraint [16.02833173359407]
This work introduces two constant factor approximation algorithms with linear query complexity for non-monotone submodular over a ground set of size $n$ subject to a knapsack constraint.
$mathsfDLA$ is a deterministic algorithm that provides an approximation factor of $6+epsilon$ while $mathsfRLA$ is a randomized algorithm with an approximation factor of $4+epsilon$.
arXiv Detail & Related papers (2023-05-17T15:27:33Z) - 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) - Linear Bandit Algorithms with Sublinear Time Complexity [67.21046514005029]
We propose to accelerate existing linear bandit algorithms to achieve per-step time complexity sublinear in the number of arms $K$.
We show that our proposed algorithms can achieve $O(K1-alpha(T))$ per-step complexity for some $alpha(T) > 0$ and $widetilde O(stT)$ regret, where $T$ is the time horizon.
arXiv Detail & Related papers (2021-03-03T22:42:15Z) - Quantum algorithms for spectral sums [50.045011844765185]
We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices.
We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
arXiv Detail & Related papers (2020-11-12T16:29:45Z) - Linear-Sample Learning of Low-Rank Distributions [56.59844655107251]
We show that learning $ktimes k$, rank-$r$, matrices to normalized $L_1$ distance requires $Omega(frackrepsilon2)$ samples.
We propose an algorithm that uses $cal O(frackrepsilon2log2fracepsilon)$ samples, a number linear in the high dimension, and nearly linear in the matrices, typically low, rank proofs.
arXiv Detail & Related papers (2020-09-30T19:10:32Z) - Learning nonlinear dynamical systems from a single trajectory [102.60042167341956]
We introduce algorithms for learning nonlinear dynamical systems of the form $x_t+1=sigma(Thetastarx_t)+varepsilon_t$.
We give an algorithm that recovers the weight matrix $Thetastar$ from a single trajectory with optimal sample complexity and linear running time.
arXiv Detail & Related papers (2020-04-30T10:42:48Z) - Solving the Robust Matrix Completion Problem via a System of Nonlinear
Equations [28.83358353043287]
We consider the problem of robust matrix completion, which aims to recover a low rank matrix $L_*$ and a sparse matrix $S_*$ from incomplete observations of their sum $M=L_*+S_*inmathbbRmtimes n$.
The algorithm is highly parallelizable and suitable for large scale problems.
Numerical simulations show that the simple method works as expected and is comparable with state-of-the-art methods.
arXiv Detail & Related papers (2020-03-24T17:28:15Z)
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.