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

2026-04-21

AI Summary

This work addresses the slow $O(1/N)$ variance convergence of traditional Monte Carlo integration, which stems from reliance on independent sampling and hinders efficient integration of smooth functions. To overcome this limitation, the authors leverage determinantal point processes (DPPs) to generate repulsive, non-independent samples that significantly accelerate convergence. They unify and extend two classical DPP-based integration schemes to continuous domains and introduce corresponding sampling algorithms that achieve a superlinear variance decay rate of $O(N^{-(1+1/d)})$. Furthermore, they design integrand-adapted DPPs that preserve unbiasedness while enhancing performance. Both theoretical analysis and empirical results demonstrate that the choice of DPP adaptation critically determines convergence behavior, highlighting the approach’s theoretical novelty and practical utility.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data integration and cleaning
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ 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 $ω$. 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.

Problem

• Monte Carlo integration
• determinantal point processes
• variance reduction
• numerical integration
• repulsive point processes

Innovation

• Determinantal Point Processes
• Monte Carlo Integration
• Variance Reduction
• Repulsive Sampling
• Adaptive DPP