Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature

TL;DR

This paper investigates positive-weight kernel quadrature, demonstrating geometric bounds and algorithms that outperform Monte Carlo in spectral regimes with stable, positive weights.

Contribution

It introduces a geometric analysis of positive weights in kernel quadrature, deriving error bounds and a Frank--Wolfe algorithm for stable, spectral-accelerated integration.

Findings

Convex hull approximation yields sharper error bounds than equal-weight averaging.

Positive weights can achieve near-O(1/N) rates under exponential spectral decay.

A Frank--Wolfe algorithm maintains simplex weights with explicit error guarantees.

Abstract

Kernel quadrature can exploit RKHS spectral structure and outperform Monte Carlo on smooth integrands, but optimized quadrature weights are generally signed and may be numerically unstable. We study whether spectral acceleration remains possible when the weights are constrained to be positive, i.e., simplex weights. In the exact-target fixed-pool setting, an evaluated i.i.d. candidate pool of size $N$ is already available and the task is to reweight it so as to approximate the kernel mean embedding. We show that this positive reweighting problem is governed not by the equal-weight empirical average, but by the random convex hull generated by the pool. Our main geometric result shows that the mean of a bounded $d$-dimensional random vector can be approximated by a convex combination of $N$ i.i.d. samples at accuracy $O(d/N)$ with high probability, sharper than equal-weight averaging in the fixed-dimensional regime. We transfer this $d$-dimensional convex-hull approximation to full RKHS worst-case error through an augmented Mercer-truncation argument. The resulting positive-weight KQ bounds consist of a spectral tail term and a finite-sample convex-hull term, yielding Monte-Carlo-beating rates in favorable spectral regimes, including near-$O(1/N)$ rates up to logarithmic factors under exponential spectral decay. We also provide a constructive Frank--Wolfe algorithm that operates directly on the pool atoms, maintains simplex weights, and admits an explicit optimization-error bound.