针对小而宽数据集，进化出简洁可解释的回归公式。
Evolutional Math: Cross-Validated Island-Model Genetic Programming for Interpretable Symbolic Regression on Small, Wide Datasets
- 用交叉验证的R²替代相关性评估，避免过拟合陷阱。
- 多岛架构加迁移机制，保持搜索多样性，防止早熟收敛。
- 结构去重+参数精调，提升公式的简洁性与预测力。
基于遗传编程的符号回归在小而宽的数据集（如临床试验监测、生物统计学和工程试点研究中常见）上常失败，因会收敛到臃肿且过拟合的表达式，仅利用相关性而非真实预测能力。本文提出 Evolutional Math，一个开源遗传编程系统，通过四项设计实现该场景下的紧凑、可解释公式：首先，使用留出交叉验证折的R²作为适应度，而非训练集上的皮尔逊相关系数，消除单变量捷径；其次，采用多岛架构，各岛以不同算子子集（代数、对数、三角函数、全集）初始化，每M代进行环形拓扑迁移，防止搜索陷入单一公式空间区域；第三，结构去重机制将仅差常数的公式视为等价，使精英档案保留结构各异的候选者；第四，每次迁移后，前k名个体通过scipy L-BFGS-B进行数值常数精调，解耦结构搜索与参数优化。在合成基准（形式为log(x_i)*x_j/(x_k*c)、三角混合）及匿名临床站点监控数据集（24行，约290个数值特征）上测试，系统在数万次唯一公式评估内稳定恢复真实结构，R²达或超过0.99。参考实现已发布，采用非商业源代码可用许可。
原文摘要 · Abstract (English)
Symbolic regression via genetic programming routinely fails on small, wide datasets - a regime common in clinical-trial monitoring, biostatistics, and engineering pilot studies - by converging on bloated, overfit expressions that exploit correlation rather than prediction. We present Evolutional Math, an open-source genetic programming system that combines four design choices to yield compact, interpretable formulas in this regime. First, fitness is measured by R-squared on held-out cross-validation folds rather than Pearson correlation on the training set, eliminating single-variable shortcuts that correlate but mis-scale. Second, a multi-island architecture runs independent populations seeded with distinct operator subsets (algebraic, logarithmic, trigonometric, and full) with ring-topology migration every M generations, preventing the search from collapsing into one region of formula space. Third, a structural deduplication scheme treats formulas differing only in constants as equivalent, so the elite archive contains structurally distinct candidates rather than near-duplicate variants. Fourth, top-k individuals undergo numerical constant refinement via scipy L-BFGS-B after each migration phase, decoupling structure search from parameter fitting. We evaluate the system on synthetic benchmarks of the form log(x_i) * x_j / (x_k * c), trigonometric mixtures, and an anonymized clinical site-monitoring dataset with 24 rows and approximately 290 candidate numeric features. The system consistently recovers compact ground-truth structures with R-squared at or above 0.99 within tens of thousands of unique formula evaluations. A reference implementation is released under a noncommercial source-available license.
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