On two ways to use determinantal point processes for Monte Carlo integration

Abstract

The standard Monte Carlo estimator ${\hat{I}}_N^{\mathrm{MC}}$ of $\int fd\ddot{I}‰$ relies on independent samples from $\ddot{I}‰$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $\ddot{I}‰$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.

NeurIPS 2019