提出新准则EIC,让符号回归找到更稳定、更合理的公式。
Beyond Accuracy and Complexity: The Effective Information Criterion for Structurally Stable Symbolic Regression
- 用信息通道模型衡量公式计算中的噪声放大,判断结构合理性
- 引入EIC后,生成模型训练效率提升2-4倍,泛化性能提高22.4%
- 70%人类专家偏好与EIC结果一致,验证其可解释性价值
符号回归(SR)传统上在准确率与复杂度间权衡,隐含假设简单公式更合理。我们指出该假设不足:现有算法常找到看似简洁但结构不合理、数值不稳定的公式,缺乏物理意义。受真实物理定律结构稳定性启发,提出有效信息准则(EIC),将公式视为信息通道,量化递归计算中固有舍入噪声的放大程度,无需真值即可区分物理合理与病态结构。分析显示人类公式与SR结果存在显著结构稳定性差距。将EIC融入SR流程:启发式搜索中引导算法走向稳定区域,获得更优帕累托前沿;生成模型中,基于EIC过滤使预训练样本效率提升2-4倍,泛化决定系数R²提升22.4%。108位专家评估表明,EIC与人类偏好匹配率达70%,证实结构稳定性是人眼可解释性的关键前提。
原文摘要 · Abstract (English)
Symbolic regression (SR) traditionally balances accuracy and complexity, implicitly assuming that simpler formulas are structurally more rational. We argue that this assumption is insufficient: existing algorithms often exploit this metric to discover accurate and compact but structurally irrational formulas that are numerically ill-conditioned and physically inexplicable. Inspired by the structural stability of real physical laws, we propose the Effective Information Criterion (EIC) to quantify formula rationality. EIC models formulas as information channels and measures the amplification of inherent rounding noise during recursive calculation, effectively distinguishing physically plausible structures from pathological ones without relying on ground truth. Our analysis reveals a stark structural stability gap between human-derived equations and SR-discovered results. By integrating EIC into SR workflows, we provide explicit structural guidance: for heuristic search, EIC steers algorithms toward stable regions to yield superior Pareto frontiers; for generative models, EIC-based filtering improves pre-training sample efficiency by 2-4 times and boosts generalization R2 by 22.4%. Finally, an extensive study with 108 human experts shows that EIC aligns with human preferences in 70% of cases, validating structural stability as a critical prerequisite for human-perceived interpretability.
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