The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands

TL;DR

This paper addresses the gap in theory for $L_p$-error rates of randomized quasi-Monte Carlo self-normalized importance sampling (RQMC-SNIS) when integrands are unbounded on unbounded domains. It first establishes $L_p$-rates for plain RQMC integration under a broad class of transport maps with quadratic-growth-type tail controls, using a radially defined projection to maintain dimension-independent bounds. Building on these results, it derives the $L_p$-error rate for RQMC-SNIS, showing an $O(N^{-eta+\epsilon})$ convergence with $\beta=1-p\max\{M_{\omega f,\tau},0\}/\alpha$, and demonstrating that for QMC-friendly $\omega f$ the rate can approach $O(N^{-1})$; this general framework covers shifted, scaled, and nonlinear transforms. Numerical experiments on a Bayesian inverse problem and Bayesian logistic regression corroborate the theory and highlight the practical advantages and limitations of different transport maps and proposals, particularly the robustness of linear $t$-proposals in higher dimensions.

Abstract

Self-normalized importance sampling (SNIS) is a fundamental tool in Bayesian inference when the posterior distribution involves an unknown normalizing constant. Although $L_1$-error (bias) and $L_2$-error (root mean square error) estimates of SNIS are well established for bounded integrands, results for unbounded integrands remain limited, especially under randomized quasi-Monte Carlo (RQMC) sampling. In this work, we derive $L_p$-error rate $(p\ge1)$ for RQMC-based SNIS (RQMC-SNIS) estimators with unbounded integrands on unbounded domains. A key step in our analysis is to first establish the $L_p$-error rate for plain RQMC integration. Our results allow for a broader class of transport maps used to generate samples from RQMC points. Under mild function boundary growth conditions, we further establish \(L_p\)-error rate of order \(\mathcal{O}(N^{-β+ ε})\) for RQMC-SNIS estimators, where $ε>0$ is arbitrarily small, $N$ is the sample size, and \(β\in (0,1]\) depends on the boundary growth rate of the resulting integrand. Numerical experiments validate the theoretical results.

Figures (4)

• Figure 1: A comparison on the first components of the two projection operators in $\mathbb{R}^2$ with $r=10$ for both operators and $\delta=0.1$ for our proposed operator.
• Figure 2: The $L_p$-error for different proposals with different $\kappa$ and $d = 5$.
• Figure 3: The $L_p$-error for different proposals with $\kappa = 1$ and $d = 30$.
• Figure 4: The $L_p$-error for different proposals with Pima dataset.

Theorems & Definitions (37)

• Remark 1
• Remark 2
• Lemma 3.1
• proof
• Remark 3
• Lemma 3.2
• proof
• Remark 4
• Lemma 3.3
• proof