Message-Passing Monte Carlo: Generating low-discrepancy point sets via Graph Neural Networks (2405.15059v2)

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.

• The paper introduces a novel GNN-based framework that generates low-discrepancy point sets optimized via message-passing techniques.
• The method outperforms traditional sequences like Sobol' and Halton in star-discrepancy, ensuring near-optimal uniformity.
• Extensive evaluations demonstrate superior performance in both low- and high-dimensional applications, notably in computational finance.

Overview of Message-Passing Monte Carlo: Generating Low-Discrepancy Point Sets via Graph Neural Networks

The paper "Message-Passing Monte Carlo: Generating Low-Discrepancy Point Sets via Graph Neural Networks" introduces a novel approach for generating low-discrepancy point sets using machine learning techniques, specifically Graph Neural Networks (GNNs). The primary objective of the research is to leverage the geometric nature of point distributions to achieve configurations with minimal discrepancy, a measure crucial for numerous applications in science and engineering.

Introduction and Motivation

Monte Carlo (MC) methods are well-established for approximating and simulating complex real-world systems. They rely on repeated random sampling, which, while effective, exhibits a convergence rate of $\mathcal{O}(N^{-1/2})$ in the number of samples $N$. This inefficiency necessitates a large number of samples to achieve high precision, rendering MC methods impractical for some complex problems. Quasi-Monte Carlo (QMC) methods address this by employing deterministic point sets that are uniformly distributed, thus achieving better convergence rates through low-discrepancy point sets. Despite advancements in QMC methods, constructing such point sets for specific dimensionalities and sample sizes remains challenging.

Methodology

The proposed method leverages GNNs within the framework of Geometric Deep Learning to address these challenges. The architecture constructs a computational graph based on nearest neighbors of the input points and processes these points through message-passing neural networks. The model is trained to minimize a closed-form solution of a specific discrepancy measure for its outputs.

Several key aspects of the methodology include:

• Training Set Construction: The input points for the model can be uniform random samples, points from existing low-discrepancy sequences (e.g., Sobol', Halton), or randomly perturbed low-discrepancy points.
• GNN Architecture: The GNN processes the input points using layers that iteratively update node features based on their neighbors. The graph structure is typically defined based on a radius from each point's nearest neighbors, ensuring a local connectivity structure that is advantageous for reducing discrepancy.
• Training Objective: The training objective leverages the $_2$-discrepancy, specifically Warnock’s formula, due to its computational feasibility and differentiability. For higher dimensions, modifications based on Hickernell’s $_p$-discrepancy using random projections are employed.

Empirical Evaluation

The efficacy of the proposed method is demonstrated through extensive empirical evaluations. Key findings include:

• Low-Dimensional Performance: MPMC points outperformed existing low-discrepancy point sets such as Sobol', Halton, subset selection, Hammersley, and Fibonacci in terms of star-discrepancy. The discrepancy values for MPMC points were significantly lower on average.
• Near-Optimal Performance: When compared to known optimal star-discrepancy sets for specific $N$ and $d$, MPMC points exhibited near-optimal performance, closely matching or achieving similar discrepancy values.
• High-Dimensional Efficacy: In a high-dimensional context, particularly a computational finance problem involving Asian call option pricing, MPMC points demonstrated significantly lower approximation errors than Sobol' and Halton sequences. This highlights the superior uniformity of MPMC points even in higher dimensions.

Ablation Studies

Several ablation studies were conducted to validate various components of the model:

• Graph Structure: The choice of radius for nearest-neighbor graphs had a notable impact on performance, with non-zero radius values significantly enhancing the model's ability to reduce discrepancy.
• GNN Architecture: Among different GNN architectures tested (GCN, GAT, MPNN), MPNNs yielded the lowest discrepancy values consistently.
• Input Point Sets: Performance was slightly better when using Sobol' or randomized Sobol' points as inputs compared to purely random points, though the best-performing models for each input type yielded similar results.

Implications and Future Work

The implications of this research are manifold:

• QMC Methods: MPMC points can enhance the effectiveness of QMC methods in various applications by providing superior low-discrepancy point sets.
• Computational Efficiency: The ability to generate near-optimal point sets in near-real time is especially beneficial for computationally intensive tasks.
• Adaptability: The model's framework allows for generating custom-made point sets tailored to specific dimensions important for particular problems.

Future research directions include extending MPMC to generate low-discrepancy sequences rather than fixed sets and applying the method to various practical applications in science and engineering to further validate its utility and performance.

Conclusion

The paper presents a pioneering machine learning approach for generating low-discrepancy point sets, demonstrating significant advancements over existing methods. By grounding the methodology in GNNs and optimizing for discrepancy measures computationally feasible for gradient-based learning, MPMC represents an innovative step forward in QMC point generation. The promising empirical results underscore the potential of this method to impact a wide range of scientific and engineering applications.