DMS-EPSRC: Certifying Accuracy of Randomized Algorithms in Numerical Linear Algebra

DMS-EPSRC：验证数值线性代数中随机算法的准确性

• 批准号: EP/Y030990/1
• 负责人: Yuji Nakatsukasa
• 金额: $ 38.14万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY030990%2F1
• 关键词: DMS; EPSRC; Certifying; Accuracy; Randomized

Among the most exciting developments over the last two decades is the rapid advances in randomized numerical linear algebra (RNLA). This has caused a paradigmatic shift in the computational sciences, and has enabled matrix computations of unprecedented scale. In addition to speed, RNLA often provides solutions with exceptional accuracy and robustness. Success stories in RNLA include low-rank approximation, least-squares problems, and trace estimation. In addition, the field has witnessed recent progress in linear systems, eigenvalue problems, and tensor approximation. Despite these success stories, there is often a wide gap between theory and practice. To illustrate, consider the task of "subspace sketching", which is a key idea in RNLA. There is theory to support a claim that structured embeddings (aka "fast Johnson-Lindenstrauss transforms") based on FFTs provide a geometry preserving subspace embedding if O(dlogd) samples are used for a d-dimensional subspace. However, this is usually a pessimistic estimate, and O(d), say 2d samples, is known to suffice in most problems that arise in practice. A similar story holds for sparse sketches, for which beautiful theory is available, but in practice they often perform much better for a particular instance. The same is true even of deterministic algorithms, such as the column-pivoted QR factorization, specifically whether it gives a rank-revealing QR factorization. While the worst-case analysis suggests the algorithm can be exponentially unstable, decades of practical computation has shown that such examples are vanishingly rare. Clearly, in any of these problems, it would be highly useful to have a methodology to assess the correctness of a solution computed for the particular problem at hand.The objective of this proposal is to develop techniques for computing upper bounds on the error incurred by a particular instantiation of a randomized algorithm for solving a linear algebraic problem. Such bounds would allow users to combine the computational speed of randomized algorithms with the reliability of classical deterministic methods. It would also help users in safely balancing the needs of speed and precision. The approach we propose to follow here is close in spirit to the notion of "responsibly reckless" algorithms, recently coined by Jack Dongarra, the latest Turing Award laureate. The idea is to try a fast, but potentially unstable, algorithm, to obtain a potential solution. Then we assess the quality of the solution in a reliable fashion. Specifically, the purpose of this proposal is to develop fast and robust algorithms for this assessment, thereby certifying the correctness of the solution, or otherwise output a warning that the solution may be inaccurate. We will design algorithms, develop theory, and implement them in open-source software optimised for modern computing platforms.The historical development of finite element methods would help put our project in context. A priori analysis came first, and error bounds that were invaluable in guiding the design of finite element spaces were proven. However, these bounds were too pessimistic to usefully assess the error incurred in any particular instantiation, since the bound had to cover the most adversarial input. A posteriori analysis then emerged in response, as it applies specifically to a problem at hand and can, by drawing on quantities available post-computation, provide accurate estimates or bounds on the error. In this project we aim to achieve the same in the RNLA context.Specific directions we will pursue include: Subspace embedding, low-rank approximation, column-pivoted QR factorization, linear systems, eigenvalue problems and SVD, least-squares problems, and building Krylov subspaces efficiently.

过去二十年最令人兴奋的发展之一是随机数值线性代数（RNLA）的快速发展。这引起了计算科学的范式转变，并使矩阵计算达到了前所未有的规模。除了速度之外，RNLA 还经常提供具有卓越准确性和稳健性的解决方案。 RNLA 的成功案例包括低秩近似、最小二乘问题和迹估计。此外，该领域在线性系统、特征值问题和张量逼近方面也取得了最新进展。尽管有这些成功案例，但理论与实践之间往往存在很大差距。为了说明这一点，请考虑“子空间草图”任务，这是 RNLA 的一个关键思想。有理论支持这样一种说法：如果 O(dlogd) 个样本用于 d 维子空间，则基于 FFT 的结构化嵌入（又名“快速 Johnson-Lindenstrauss 变换”）可提供几何保留子空间嵌入。然而，这通常是一个悲观的估计，并且 O(d)（例如 2d 样本）已知足以解决实践中出现的大多数问题。类似的情况也适用于稀疏草图，对于这些草图，可以使用美丽的理论，但在实践中，它们通常在特定情况下表现得更好。即使对于确定性算法（例如列旋转 QR 分解）也是如此，特别是它是否提供显示排名的 QR 分解。虽然最坏情况的分析表明该算法可能呈指数级不稳定，但数十年的实际计算表明此类示例极其罕见。显然，在任何这些问题中，拥有一种方法来评估针对当前特定问题计算的解决方案的正确性将非常有用。该提案的目标是开发计算以下错误所产生的误差上限的技术用于解决线性代数问题的随机算法的特定实例。这样的界限将允许用户将随机算法的计算速度与经典确定性方法的可靠性结合起来。它还将帮助用户安全地平衡速度和精度的需求。我们建议在此遵循的方法在精神上与“负责任的鲁莽”算法的概念非常接近，该算法是由最新的图灵奖获得者 Jack Dongarra 最近提出的。这个想法是尝试一种快速但可能不稳定的算法来获得潜在的解决方案。然后我们以可靠的方式评估解决方案的质量。具体来说，该提案的目的是为此评估开发快速且稳健的算法，从而证明解决方案的正确性，或者以其他方式输出解决方案可能不准确的警告。我们将设计算法，发展理论，并在针对现代计算平台优化的开源软件中实现它们。有限元方法的历史发展将有助于我们的项目的背景。首先进行先验分析，并证明了对于指导有限元空间设计非常有价值的误差范围。然而，这些界限过于悲观，无法有效评估任何特定实例化中产生的错误，因为界限必须覆盖最具对抗性的输入。随后出现了后验分析作为回应，因为它专门适用于手头的问题，并且可以通过利用计算后可用的数量，提供准确的估计或误差范围。在这个项目中，我们的目标是在 RNLA 背景下实现相同的目标。我们将追求的具体方向包括：子空间嵌入、低秩近似、列枢轴 QR 分解、线性系统、特征值问题和 SVD、最小二乘问题以及构建克雷洛夫子空间有效。