Transport Quasi-Monte Carlo

TL;DR

This work tackles the challenge of applying quasi-Monte Carlo methods to general target distributions by introducing a transport-map framework that pushes the uniform reference measure to the target $p$. The transport map is built as a base transformation $G$ followed by multiple autoregressive layers, with a QMC-friendly parametrization that guarantees regularity and tractable Jacobians. The authors prove that, under mild boundary-growth conditions on the integrand, the scrambled net estimator achieves RMSE $O(n^{-1+oldsymbol{cvarepsilon}})$, and provide constructive sufficiency conditions for $G$ and the layer transforms. Practical recommendations include a sandwich-style elementwise transform based on Beta-mixture CDFs, dimension reduction via PCA on the relative score to identify low-dimensional structure, and importance sampling corrections when the pushforward is imperfect. Empirical results on posterior sampling tasks and high-dimensional Bayesian inference demonstrate improved accuracy and training efficiency for RQMC over standard MC and highlight the benefits of dimension reduction in complex models.

Abstract

Quasi-Monte Carlo (QMC) is a powerful method for evaluating high-dimensional integrals. However, its use is typically limited to distributions where direct sampling is straightforward, such as the uniform distribution on the unit hypercube or the Gaussian distribution. For general target distributions with potentially unnormalized densities, leveraging the low-discrepancy property of QMC to improve accuracy remains challenging. We propose training a transport map to push forward the uniform distribution on the unit hypercube to approximate the target distribution. Inspired by normalizing flows, the transport map is constructed as a composition of simple, invertible transformations. To ensure that RQMC achieves its superior error rate, the transport map must satisfy specific regularity conditions. We introduce a flexible parametrization for the transport map that not only meets these conditions but is also expressive enough to model complex distributions. Our theoretical analysis establishes that the proposed transport QMC estimator achieves faster convergence rates than standard Monte Carlo, under mild and easily verifiable growth conditions on the integrand. Numerical experiments confirm the theoretical results, demonstrating the effectiveness of the proposed method in Bayesian inference tasks.

Figures (7)

• Figure 1: A sequence of transformations maps an RQMC point set to the target distribution. Starting with RQMC samples in the unit cube, the base transform $G$ maps the samples to the entire space. Subsequently, the transformations $\tau^1,\ldots,\tau^K$ are applied sequentially to transform the samples to the target distribution. The log determinant of the Jacobian for the overall transformation is the sum of the log determinant of the Jacobian for each individual transformation.
• Figure 2: Illustration of the transformation ${\mathbf{z}}=\tau^k({\mathbf{x}})$ and its Jacobian. The input vector ${\mathbf{x}}\in\mathbb{R}^d$ is first transformed by a linear transformation ${\mathbf{y}}=L{\mathbf{x}}+b$. Next, each component $y_i$ of ${\mathbf{y}}$ is transformed independently by an elementwise transformation $T_i$. The log determinant of the Jacobian of $\tau^k$ is computed using the chain rule: the contribution from the linear transformation is $\sum_{j=1}^d \log L_{jj}$ because $L$ is a lower-triangular matrix, and the contribution from $T$ is $\sum_{j=1}^d\log \dot T_j(y_j)$, where $\dot T_j$ denotes the derivative of $T_j$.
• Figure 3: Outline of the proof. We first review the high-level condition \ref{['assump: growth']} for an integrand $h:[0,1]^d\to\mathbb{R}$ that ensures RQMC achieves an RMSE of $O(n^{-1+\varepsilon})$ for any $\varepsilon>0$. Next, we introduce two conditions \ref{['assump: f']} and \ref{['assump: tau']} that ensure $h=f\circ\tau$ satisfies \ref{['assump: growth']}. Finally, we provide sufficient conditions on $f$, the base transformation $G$, and the elementwise transformations $T^k_j$ ($1\leq k\leq K,1\leq j\leq d$) such that $f$ and the overall transport map $\tau=\tau^{K}\circ\cdots\circ\tau^1\circ G$ satisfy \ref{['assump: f']} and \ref{['assump: tau']}.
• Figure 4: MSE for estimating $\mathbb{E}_{{\mathbf{x}}\sim p}\left[x_j\right]$ versus sample size $n$ for various target distributions $p$ from posteriordb. Blue dots represent the MSE of plain MC, and orange stars represent the MSE of RQMC. Details of the experiments are provided in Section \ref{['sec: posteriordb']}.
• Figure 5: MSE for estimating $\mathbb{E}_{{\mathbf{x}}\sim p}\left[x_j^2\right]$ versus sample size, under the same settings as Figure \ref{['fig: posteriordb']}.

Theorems & Definitions (17)

• Theorem 4.1: Adapted from Theorem 5.7 of owen2006halton
• Theorem 4.2: Growth condition of $f\circ\tau$
• Lemma 4.3: Faa di Bruno formula
• proof : Proof of Theorem \ref{['thm: f tau']}
• Example 4.1: Inverse Gaussian CDF as the base transformation
• Example 4.2: Logit function as the base transformation
• Remark 4.1: Inequality \ref{['equ: assump T bound 1']}
• Remark 4.2: Inequality \ref{['equ: assump T bound']}
• Lemma 4.4
• Remark 4.3