Low-Discrepancy Set Post-Processing via Gradient Descent

TL;DR

The paper addresses the high computational cost of constructing low-discrepancy sets by introducing a projected gradient-descent post-processing method for $L_2$ discrepancies. Using an ADAM-based optimizer on a smoothed Warnock-like loss $d_2^*(P, au)$ and enforcing $[0,1]^d$-box constraints, the approach efficiently refines starting point sets (e.g., Fibonacci lattices, Sobol', SubsetL2) to achieve competitive or superior $L_2$ and $L_{ fty}$ discrepancies, with runtimes of seconds to minutes on a laptop. The authors demonstrate strong improvements in dimension 2 and competitive results in dimensions 3–5, including periodic and extreme discrepancies, while highlighting the importance of initialization and stability at larger $n$. The method offers a simple, accessible refinement tool that can complement existing constructions for numerical integration and uniform sampling, and the code is publicly available.

Abstract

The construction of low-discrepancy sets, used for uniform sampling and numerical integration, has recently seen great improvements based on optimization and machine learning techniques. However, these methods are computationally expensive, often requiring days of computation or access to GPU clusters. We show that simple gradient descent-based techniques allow for comparable results when starting with a reasonably uniform point set. Not only is this method much more efficient and accessible, but it can be applied as post-processing to any low-discrepancy set generation method for a variety of standard discrepancy measures.

Figures (7)

• Figure 1: The $L_\infty$ values for $n=20$ up to $n=1020$ for well-known low discrepancy sets. Projected Gradient Descent is in gray.
• Figure 2: The $L_2^*$ value immediately before restart for the last 100 restarts of a 400 restart test initialized with a uniformly chosen random set of $n=260$. The test concluded with a best $L_2^*$ value of 0.003018 and an $L_\infty^*$ value of 0.01344 for the set corresponding to this value.
• Figure 3: $L_2$ star discrepancy evolution for the Fibonacci set for $n=60,~240,~1020$ (left to right) when using the projected gradient descent algorithm. The learning rate is set to $0.0005$ for $n=60$ and $0.0001$ for the two others.
• Figure 4: Gradient descent on a set of 1750 points. The fluctuations are too great to allow convergence.
• Figure 5: Periodic $L_2$ discrepancy values during the gradient descent for $n=64,~128$ points with Sobol' initialization.