[Paper Review] A Bootstrap Method for Error Estimation in Randomized Matrix Multiplication

Miles E. Lopes, Shusen Wang+1 more|arXiv (Cornell University)|Aug 6, 2017

Stochastic Gradient Optimization Techniques48 references3 citations

TL;DR

This paper proposes a bootstrap method to estimate the error in randomized matrix multiplication, using an extrapolation technique to efficiently estimate the accuracy of sketching methods as a function of reduced dimension. The method provides problem-specific error estimation without significant computational overhead, offering practical guidance on the cost-accuracy tradeoff in large-scale matrix computations.

ABSTRACT

In recent years, randomized methods for numerical linear algebra have received growing interest as a general approach to large-scale problems. Typically, the essential ingredient of these methods is some form of randomized dimension reduction, which accelerates computations, but also creates random approximation error. In this way, the dimension reduction step encodes a tradeoff between cost and accuracy. However, the exact numerical relationship between cost and accuracy is typically unknown, and consequently, it may be difficult for the user to precisely know (1) how accurate a given solution is, or (2) how much computation is needed to achieve a given level of accuracy. In the current paper, we study randomized matrix multiplication (sketching) as a prototype setting for addressing these general problems. As a solution, we develop a bootstrap method for \emph{directly estimating} the accuracy as a function of the reduced dimension (as opposed to deriving worst-case bounds on the accuracy in terms of the reduced dimension). From a computational standpoint, the proposed method does not substantially increase the cost of standard sketching methods, and this is made possible by an "extrapolation" technique. In addition, we provide both theoretical and empirical results to demonstrate the effectiveness of the proposed method.

Motivation & Objective

• To address the lack of practical methods for estimating the actual accuracy of randomized matrix multiplication in real-world applications.
• To provide a problem-specific error estimation technique that avoids overly pessimistic worst-case theoretical bounds.
• To develop a computationally efficient method that does not substantially increase the cost of standard sketching algorithms.
• To enable users to determine the appropriate sketch size for a desired accuracy level in large-scale matrix problems.
• To establish a framework for error estimation applicable beyond matrix multiplication, extending to other randomized numerical linear algebra methods.

Proposed method

• The method introduces a bootstrap-based approach to estimate the entry-wise error (ℓ∞-norm) of randomized matrix multiplication as a function of the sketch size t.
• It leverages an extrapolation technique to reduce computational cost, reusing existing sketching computations to estimate error across multiple sketch sizes.
• The approach models the error distribution using resampling from the original sketch, enabling estimation of quantiles such as the 95th percentile of error.
• Theoretical analysis shows that the bootstrap estimator is consistent under mild assumptions, with bounds derived using concentration inequalities and the Lévy-Prokhorov metric.
• The method relies on a key technical lemma relating the Lévy-Prokhorov distance to quantile differences, ensuring reliable error quantification.
• The algorithm is designed to be plug-and-play with existing sketching pipelines, requiring only minor modifications to standard sketching code.

Experimental results

Research questions

• RQ1 How can we estimate the actual error of a randomized matrix multiplication for a given input matrix pair, rather than relying on worst-case theoretical bounds?
• RQ2 What is the relationship between sketch size t and the resulting error in practice, and how can this be estimated efficiently?
• RQ3 Can a bootstrap method be designed to estimate error in randomized matrix multiplication without significantly increasing computational cost?
• RQ4 How accurate is the proposed bootstrap method in estimating error quantiles such as the 95th percentile across different matrix dimensions and sketch sizes?
• RQ5 To what extent does the method’s performance depend on the structure of the input matrices, such as condition number or coherence?

Key findings

• The proposed bootstrap method provides accurate estimation of the ℓ∞-norm error in randomized matrix multiplication across various sketch sizes, with empirical results showing strong agreement with true error.
• The method achieves this accuracy with minimal computational overhead, as the extrapolation technique reuses existing sketching computations to estimate error across multiple t values.
• Theoretical analysis confirms that the bootstrap estimator is consistent under standard assumptions, with error bounds that scale favorably with matrix dimensions and sketch size.
• The method outperforms traditional worst-case bounds, which are often overly conservative and fail to reflect problem-specific error behavior.
• Empirical validation on synthetic and real-world data shows that the bootstrap method reliably estimates error quantiles, enabling users to select optimal sketch sizes for desired accuracy levels.
• The approach is robust to variations in matrix condition number and input structure, demonstrating practical utility across diverse numerical linear algebra workloads.