LLM-Guided Evolutionary Program Synthesis for Quasi-Monte Carlo Design
This work addresses the problem of automating the discovery of high-quality QMC constructions for researchers and practitioners in numerical integration, though it appears incremental as it builds on existing evolutionary and program synthesis methods.
The authors tackled the problem of designing quasi-Monte Carlo (QMC) point sets and sequences by framing it as program synthesis and using an LLM-guided evolutionary algorithm. They achieved new best-known 2D benchmarks for N ≥ 40, matched most known 3D optima up to N ≤ 8, and reduced randomized QMC error by evolving Sobol' parameters for 32-dimensional option-pricing tasks compared to standard parameters.
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.