Technical Summary: Neural Low-Discrepancy Sequences (NEUROLDS)

1. Problem Statement

Low-discrepancy (LD) points are essential in scientific computation for efficiently filling space in a uniform manner, benefiting applications ranging from numerical integration and uncertainty quantification to computer vision and machine learning. While Quasi-Monte Carlo (QMC) methods utilizing LD points often achieve superior convergence rates ($O(N^{-1})$) compared to standard Monte Carlo ($O(N^{-1/2})$), a critical distinction exists between LD sets and LD sequences.

• LD Sets: Finite collections of points optimized for a specific, pre-specified number of samples $N$. Recent methods like Message-Passing Monte Carlo (MPMC) excel here, achieving record-low discrepancy for fixed $N$. However, they lack extensibility; adding points to an optimized set typically disrupts the uniformity, requiring re-optimization for every new $N$.
• LD Sequences: Infinite constructions where every prefix of length $N$ maintains low discrepancy. Classical examples include Sobol' and Halton sequences. While extensible, these classical constructions often exhibit performance fluctuations, particularly at non-power-of-two sample sizes, and may suffer from structural correlations in higher dimensions.

The central challenge addressed by this work is the lack of a machine learning-based framework capable of generating extensible LD sequences that maintain minimal discrepancy across all prefixes, overcoming the fixed-$N$ limitation of MPMC while outperforming classical number-theoretic sequences.

2. Methodology: NEUROLDS

The authors propose Neural Low-Discrepancy Sequences (NEUROLDS), a deterministic framework that maps indices to points in the unit hypercube $[0, 1]^d$ using a neural network. The approach is inspired by classical constructions but replaces fixed digit expansions with learnable functions.

Architecture

The generator $f_\theta: \{1, \dots, N\} \to [0, 1]^d$ operates as follows:

1. Index Encoding: An index $i$ is mapped to a sinusoidal positional encoding vector $\psi_i \in \mathbb{R}^{1+2K}$, analogous to Fourier features. This exposes multiple frequency scales of the index to the network, mimicking the role of base-$b$ digit expansions in Halton or Gray-coded direction numbers in Sobol' sequences.
2. Neural Mapping: The encoded features are passed through an $L$-layer Multi-Layer Perceptron (MLP) with ReLU activations and a final sigmoid activation to output the point $X_i \in [0, 1]^d$.

Two-Stage Optimization

To ensure stability and performance, NEUROLDS employs a two-stage training process:

1. Pre-training (Supervised Approximation): The network is initialized by regressing onto a classical reference sequence (e.g., Sobol' or Halton) using Mean Squared Error (MSE). This step provides an essential inductive bias, stabilizing the learning process and preventing the network from collapsing into degenerate solutions (e.g., points clustering in a corner).
2. Fine-tuning (Unsupervised Discrepancy Minimization): The network is further refined by minimizing differentiable $L_2$-based discrepancy losses over all sequence prefixes. The loss function is defined as:
   $$\mathcal{L}_{disc}(\theta) = \sum_{P=2}^{N} w_P \cdot D^2_\bullet(\{X_i\}_{i=1}^P)$$
   where $D^2_\bullet$ represents various kernel discrepancy measures (e.g., star, symmetric, centered) and $w_P$ are weights (uniform in the primary experiments). This ensures that the sequence maintains low discrepancy not just for the final set of $N$ points, but for every intermediate prefix.

3. Key Contributions

• First ML-based Finite LDS Generator: NEUROLDS is the first framework to utilize machine learning to generate finite low-discrepancy sequences rather than just sets, addressing the extensibility gap in previous ML-driven QMC methods.
• Superior Discrepancy Performance: The framework demonstrates that neural networks can learn to generate sequences with significantly lower discrepancy than classical constructions (Sobol', Halton) and randomized variants across various dimensions and discrepancy metrics.
• Extensibility: Unlike MPMC, which must be retrained for every new sample size, NEUROLDS generates points sequentially. The model learns a mapping from indices to points, allowing for adaptive sample size increases without restarting the construction.
• Weighted Integration: The method supports weighted discrepancy losses, allowing it to adapt to anisotropic integrands where certain dimensions are more influential than others (e.g., in the Borehole function case study).

4. Experimental Results

The authors evaluate NEUROLDS across three distinct application domains:

A. Discrepancy Minimization

In dimensions $d=1$ to $4$, NEUROLDS consistently achieves the lowest discrepancy values across all sequence lengths for symmetric ($D^2_{sym}$), star ($D^2_{star}$), and centered ($D^2_{ctr}$) metrics.

• Comparison: It outperforms Sobol', Halton, and Scrambled Sobol' (averaged over 32 runs).
• Visuals: Unlike Sobol' sequences which can exhibit structured alignments or clustering artifacts, NEUROLDS points appear more irregular while maintaining even coverage.

B. Quasi-Monte Carlo Integration

Using the 8-dimensional Borehole function (a benchmark for uncertainty quantification), NEUROLDS was tested for numerical integration.

• Setup: The model was trained with coordinate-weighted discrepancy losses to emphasize influential variables identified by sensitivity analysis.
• Result: NEUROLDS consistently delivered superior accuracy (lower absolute error) compared to Sobol', Halton, and a greedy Nelder-Mead baseline across nearly all tested sample sizes $N$.

C. Robot Motion Planning

The framework was applied to Probabilistic Roadmaps (PRM) and Rapidly-exploring Random Trees (RRT) for a kinematic chain navigating a semi-circular tunnel.

• Context: Sequential sampling is critical here; early bias can starve the planner of exploring narrow passages.
• Result: NEUROLDS achieved the highest success rates across all passage widths. To match the average performance of NEUROLDS, Sobol' required 2.50$\times$ more samples, Halton 1.55$\times$, and uniform sampling 2.27$\times$.

D. Scientific Machine Learning

NEUROLDS was used to train neural networks approximating the solution to a 2D Black-Scholes PDE (multi-asset basket option pricing).

• Result: Networks trained on NEUROLDS samples achieved lower mean-squared error (MSE) in predicting fair prices compared to those trained on uniform random points, Sobol' sequences, and greedy baselines, regardless of the specific $L_2$-discrepancy metric used for training.

Ablation Studies

• Pre-training Necessity: Direct optimization without pre-training on Sobol' led to degenerate solutions (points collapsing to a corner). Pre-training is essential for stability.
• Architecture: While an autoregressive GNN and LSTM were tested, the index-conditioned MLP proved more robust and computationally efficient. LSTMs offered marginal discrepancy improvements but at prohibitive computational costs.
• Frequency Parameter ($K$): Increasing the sinusoidal frequency parameter $K$ improved the stability of the discrepancy profile, reducing fluctuations.

5. Significance and Claims

The paper claims that NEUROLDS represents a significant advancement in the field of Quasi-Monte Carlo methods by successfully bridging the gap between the flexibility of machine learning and the rigorous requirements of low-discrepancy sequences.

• Overcoming Limitations: It solves the "fixed-$N$" limitation of MPMC, enabling the generation of extensible sequences that maintain high uniformity as sample sizes grow.
• Broad Applicability: The results highlight the promise of neural architectures in diverse fields, including numerical integration, robotics, and scientific machine learning.
• Hybrid Approach: The authors modestly note that their reliance on classical Sobol' or Halton sequences for pre-alignment underscores the continued importance of classical number-theoretic methods. These classical constructions provide the necessary stability and inductive bias for the neural network to learn effective discrepancy-minimizing rules.

The work suggests that while classical constructions remain foundational, neural networks can learn to surpass them in terms of discrepancy minimization and adaptability, offering a new paradigm for generating sampling points in computational science.