Hutch++: Optimal Stochastic Trace Estimation

TL;DR

Hutch++ is a new randomized algorithm for estimating the trace of a matrix using fewer matrix-vector multiplications than previous methods, achieving near-optimal efficiency and improved accuracy, especially for positive semidefinite matrices.

Contribution

Introduction of Hutch++, a variance-reducing, low-rank approximation-based trace estimator that improves efficiency over Hutchinson's method and is nearly optimal among all matrix-vector query algorithms.

Findings

Hutch++ requires O(1/ε) matrix-vector products for (1±ε) accuracy.

Hutch++ outperforms Hutchinson's estimator in experiments.

Theoretical analysis shows near-optimal complexity for matrix trace estimation.

Abstract

We study the problem of estimating the trace of a matrix $A$ that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a $(1 \pm \epsilon)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$ using just $O(1/\epsilon)$ matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires $O(1/\epsilon^2)$ matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory mainly requires $A$ to be positive semidefinite, we provide generalized guarantees for general square matrices, and show empirical gains in such applications.