Neural Low-Discrepancy Sequences

2025-10-04

AI Summary

Existing message-passing Monte Carlo methods generate low-discrepancy point sets but cannot guarantee that *every* prefix forms a low-discrepancy sequence (LDS), limiting their applicability in online/incremental sampling scenarios—such as numerical integration, robot motion planning, and scientific machine learning. To address this, we propose NeuroLDS, the first neural-network-based framework for LDS generation that directly maps integer indices to spatial points and jointly optimizes discrepancy across *all* prefixes. Our approach employs a two-stage training strategy: first, supervised learning to approximate classical constructions (e.g., Sobol’ sequences), followed by unsupervised fine-tuning to minimize prefix-wise star discrepancy. Experiments demonstrate that NeuroLDS significantly outperforms conventional methods under multiple discrepancy metrics and achieves superior generalization and practical utility in numerical integration accuracy, robot motion planning coverage, and physics-informed modeling tasks.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Neuro-Symbolic Learning
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Low-discrepancy points are designed to efficiently fill the space in a uniform manner. This uniformity is highly advantageous in many problems in science and engineering, including in numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. Whereas most previous low-discrepancy constructions rely on abstract algebra and number theory, Message-Passing Monte Carlo (MPMC) was recently introduced to exploit machine learning methods for generating point sets with lower discrepancy than previously possible. However, MPMC is limited to generating point sets and cannot be extended to low-discrepancy sequences (LDS), i.e., sequences of points in which every prefix has low discrepancy, a property essential for many applications. To address this limitation, we introduce Neural Low-Discrepancy Sequences ($NeuroLDS$), the first machine learning-based framework for generating LDS. Drawing inspiration from classical LDS, we train a neural network to map indices to points such that the resulting sequences exhibit minimal discrepancy across all prefixes. To this end, we deploy a two-stage learning process: supervised approximation of classical constructions followed by unsupervised fine-tuning to minimize prefix discrepancies. We demonstrate that $NeuroLDS$ outperforms all previous LDS constructions by a significant margin with respect to discrepancy measures. Moreover, we demonstrate the effectiveness of $NeuroLDS$ across diverse applications, including numerical integration, robot motion planning, and scientific machine learning. These results highlight the promise and broad significance of Neural Low-Discrepancy Sequences. Our code can be found at https://github.com/camail-official/neuro-lds.

Problem

• Generating low-discrepancy sequences using neural networks
• Overcoming MPMC's limitation to static point sets
• Minimizing prefix discrepancies for sequential applications

Innovation

• Neural network maps indices to low-discrepancy points
• Two-stage learning combines supervised and unsupervised training
• Framework generates sequences where every prefix minimizes discrepancy