arXiv:2605.23272cs.LGcs.AI2026-05

改进符号回归的参数优化,让正确方程获得真实分数。

When Good Equations Get Bad Scores: Improving Symbolic Regression Through Better Parameter Optimization

论文配图:When Good Equations Get Bad Scores: Improving Symbolic Regression Through Better Parameter Optimization
图 1 · 摘自论文原文
  • 利用符号表达式的结构与语义先验设计专用优化模块
  • 在多个数据集上显著提升方程评估准确性,避免误判正确结构
  • 可无缝集成到现有符号回归系统,适合追求高精度建模的研究者

符号回归(SR)通过从观测数据中提炼数学方程,在科学知识发现中发挥核心作用。现有方法普遍采用双层优化框架:外层搜索离散方程结构,内层优化连续参数。参数拟合质量直接影响结构评分,进而决定外层搜索方向。然而,非线性算子使内层优化高度非凸,依赖快速局部求解器(如BFGS)常陷入次优解,导致正确结构得分偏低,形成‘好结构、坏分数’现象,严重制约效率并误导搜索。为此,我们提出SAGE-Fit(结构感知与语义引导的符号回归评估框架),充分利用符号表达式的双重原生先验。通过针对结构与语义特性设计专用模块,有效缓解该优化瓶颈。大量实验表明,本方法作为即插即用模块,显著提升评估保真度,并普遍增强多种SR系统的性能。

原文摘要 · Abstract (English)

Symbolic Regression (SR) plays a central role in scientific knowledge discovery by distilling mathematical equations from observational data. Most existing SR methods function within a bi-level optimization framework: an outer loop that searches for the discrete equation structure, and an inner loop that optimizes the continuous parameters of that structure. Crucially, parameter-fitting quality directly determines a structure's score and thus the outer-loop search. However, nonlinear operators make the inner loop highly non-convex, and budget-driven reliance on fast local solvers (e.g., BFGS) often yields poor local minima and underestimated scores for correct structures. This ``Good Structure, Bad Score'' phenomenon becomes a key bottleneck, degrading efficiency and misguiding the search away from the true equation. To resolve this, we propose SAGE-Fit (Structure-Aware and Semantics-Guided Evaluator for Symbolic Regression), an SR-native fitting framework that exploits the dual native priors of symbolic expressions. By capitalizing on the structural and semantic priors unique to SR, we design tailored modules for each property, thereby effectively mitigating this optimization bottleneck. Extensive experiments demonstrate that our approach, as a plug-and-play module, significantly enhances evaluation fidelity and universally improves the performance of various SR systems.

符号回归优化算法科学发现

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