# LLM-Guided Evolutionary Program Synthesis for Quasi-Monte Carlo Design

arXiv: https://arxiv.org/abs/2510.03650 · HTML: https://mlsanity.com/paper/2510.03650/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 19 references that are in the map.
- First posted on arXiv 4 Oct 2025.

## Builds on (most cited first, 4 of 4)

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Message-Passing Monte Carlo: Generating low-discrepancy point sets via Graph Neural Networks](https://mlsanity.com/paper/2405.15059/) (2024)
- [Computing Star Discrepancies with Numerical Black-Box Optimization Algorithms](https://mlsanity.com/paper/2306.16998/) (2023)
- [Constructing Optimal L∞ Star Discrepancy Sets](https://mlsanity.com/paper/2311.17463/) (2023)

## Abstract (arXiv)

Low-discrepancy point sets and digital sequences underpin quasi-Monte Carlo (QMC) methods for high-dimensional integration. We cast two long-standing QMC design problems as program synthesis and solve them with an LLM-guided evolutionary loop that mutates and selects code under task-specific fitness: (i) constructing finite 2D/3D point sets with low star discrepancy, and (ii) choosing Sobol' direction numbers that minimize randomized QMC error on downstream integrands. Our two-phase procedure combines constructive code proposals with iterative numerical refinement. On finite sets, we rediscover known optima in small 2D cases and set new best-known 2D benchmarks for N >= 40, while matching most known 3D optima up to the proven frontier (N <= 8) and reporting improved 3D benchmarks beyond. On digital sequences, evolving Sobol' parameters yields consistent reductions in randomized quasi-Monte Carlo (rQMC) mean-squared error for several 32-dimensional option-pricing tasks relative to widely used Joe--Kuo parameters, while preserving extensibility to any sample size and compatibility with standard randomizations. Taken together, the results demonstrate that LLM-driven evolutionary program synthesis can automate the discovery of high-quality QMC constructions, recovering classical designs where they are optimal and improving them where finite-N structure matters. Data and code are available at https://github.com/hockeyguy123/openevolve-star-discrepancy.git.