Message-Passing Monte Carlo: Generating low-discrepancy point sets via Graph Neural Networks

TL;DR

This paper tackles the problem of generating low-discrepancy point sets for quasi-Monte Carlo methods by introducing Message-Passing Monte Carlo (MPMC), the first machine learning approach that uses Graph Neural Networks to transform input points into highly uniform configurations. The method encodes a point set as a graph, applies deep message-passing, and decodes outputs clipped to $[0,1]^d$, with training driven by the differentiable $L_2$-discrepancy via Warnock’s formula. It extends to high dimensions using random projections of Hickernell’s discrepancy and supports problem-dependent tailoring of point sets through projection focus. Empirically, MPMC achieves state-of-the-art or near-optimal discrepancies in low dimensions for moderate $N$, and demonstrates significant gains in high-dimensional settings (e.g., 32D Asian option pricing) compared to standard QMC constructions. The results suggest a promising new paradigm for generating tailor-made, high-quality sampling sets with wide-ranging impact on numerical integration, finance, and computer graphics.

Abstract

Discrepancy is a well-known measure for the irregularity of the distribution of a point set. Point sets with small discrepancy are called low-discrepancy and are known to efficiently fill the space in a uniform manner. Low-discrepancy points play a central role in many problems in science and engineering, including numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. In this work, we present the first machine learning approach to generate a new class of low-discrepancy point sets named Message-Passing Monte Carlo (MPMC) points. Motivated by the geometric nature of generating low-discrepancy point sets, we leverage tools from Geometric Deep Learning and base our model on Graph Neural Networks. We further provide an extension of our framework to higher dimensions, which flexibly allows the generation of custom-made points that emphasize the uniformity in specific dimensions that are primarily important for the particular problem at hand. Finally, we demonstrate that our proposed model achieves state-of-the-art performance superior to previous methods by a significant margin. In fact, MPMC points are empirically shown to be either optimal or near-optimal with respect to the discrepancy for low dimension and small number of points, i.e., for which the optimal discrepancy can be determined. Code for generating MPMC points can be found at https://github.com/tk-rusch/MPMC.

Figures (11)

• Figure 1: Two different low-discrepancy point sets with $N=64$: Korobov lattice (left), and Sobol' (right).
• Figure 2: Schematic drawing of our proposed approach to transform (random) input points $\{{\bf X}_i\}_{i=1}^N$ into low-discrepancy points $\{\hat{{\bf X}}_i\}_{i=1}^N$. Both the input and output point sets are actual instances of our proposed model, with $N=64$ and $d=2$ in this example.
• Figure 3: Schematic of the proposed model to learn low-discrepancy points. First, (random) input points $\{{\bf X}_i\}_{i=1}^N$ are encoded to a high dimensional representation. Second, the encoded representations are passed through a deep GNN \ref{['eq:MPMC_GNN']}, where the underlying computational graph is constructed based on nearest neighbors using the positions of the initial input points. Finally, the node-wise output representations of the final GNN layer are decoded and clamped yielding new $d$-dimensional points $\{\hat{{\bf X}}_i\}_{i=1}^N$ in $[0,1]^d$.
• Figure 4: Star-discrepancy $D^*$ of Halton, Sobol', lifted Sobol', Subset Selection, Hammersley, Fibonacci, and MPMC for increasing number of points $N=20,\dots,\textcolor{black}{1020}$ in $d=2$.
• Figure 5: $\mathcal{L}_2$-discrepancy of Halton, Sobol', Subset Selection, Hammersley, Fibonacci, and MPMC for increasing number of points $N=20,\dots,1020$ in $d=2$.