Title: LLM-Guided Evolutionary Program Synthesis for Quasi-Monte Carlo Design
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Evolutionary Computation, Large Language Models (LLMs), Quasi-Monte Carlo (QMC), Star Discrepancy, Sobol' Sequences, Algorithmic Discovery, Quantitative Finance

Abstract: 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 known 3D optima up to the proven frontier (N ≤ 8) and reporting
improved 3D benchmarks beyond. On digital sequences, evolving Sobol’ parame-
ters 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 demon-
strate that LLM-driven evolutionary program synthesis can automate the discov-
ery 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 anonymous.

TL;DR: LLM-guided evolutionary program synthesis discovers new best-known 2D finite low-discrepancy sets (N≥40), improves 3D benchmarks beyond proven optimality, and evolves Sobol’ parameters that reduce rQMC error in 32-D option pricing.