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arxiv: 2605.23272 · v1 · pith:A2DILOMEnew · submitted 2026-05-22 · 💻 cs.LG · cs.AI

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

Boxiao Wang , Kai Li , Zhiwei Chen , Yang Huang , Runxiang Wang , Ziwen Zhang , Yifan Zhang , Jian Cheng This is my paper

Pith reviewed 2026-05-25 04:43 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords symbolic regressionparameter optimizationbi-level optimizationnon-convex optimizationstructure-aware fittingsemantics-guided evaluationequation discovery
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The pith

Exploiting structural and semantic priors in symbolic expressions improves parameter optimization and resolves the good-structure-bad-score problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Symbolic regression searches for equations by trying different structures in an outer loop while fitting their parameters in an inner loop. Nonlinear operators make the inner optimization non-convex, so fast local solvers often produce poor fits and low scores even for correct structures, which misleads the search. The paper introduces SAGE-Fit to exploit the dual native priors of symbolic expressions through tailored modules that improve fitting quality. This plug-and-play component raises evaluation fidelity and boosts the performance of existing symbolic regression systems. A sympathetic reader would care because it targets a core bottleneck that wastes search effort on otherwise valid equations.

Core claim

The central claim is that the 'Good Structure, Bad Score' phenomenon arises from non-convex parameter optimization in the inner loop of bi-level SR frameworks, and that a Structure-Aware and Semantics-Guided Evaluator (SAGE-Fit) can mitigate it by designing tailored modules that capitalize on the dual native priors of symbolic expressions, thereby enhancing evaluation fidelity and improving various SR systems.

What carries the argument

SAGE-Fit, a fitting framework that exploits structural and semantic priors unique to symbolic expressions through tailored modules for each property.

If this is right

  • Significantly enhances evaluation fidelity of candidate structures.
  • Universally improves the performance of various symbolic regression systems as a plug-and-play module.
  • Reduces the impact of poor local minima in parameter fitting.
  • Mitigates misguidance of the outer-loop search away from true equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may enable more accurate scientific knowledge discovery by ensuring correct equations are not discarded due to fitting issues.
  • Future work could explore integrating these priors directly into the search process rather than just evaluation.
  • Testing on problems with known ground truth would show whether score improvements correlate with better equation recovery rates.

Load-bearing premise

The dual native priors of symbolic expressions can be capitalized on via tailored modules to reliably improve fitting quality without introducing new optimization issues or biases.

What would settle it

A controlled experiment on synthetic data with known correct structures where standard BFGS yields poor scores but SAGE-Fit yields high scores and leads to correct selection in the outer loop.

Figures

Figures reproduced from arXiv: 2605.23272 by Boxiao Wang, Jian Cheng, Kai Li, Runxiang Wang, Yang Huang, Yifan Zhang, Zhiwei Chen, Ziwen Zhang.

Figure 1
Figure 1. Figure 1: “Needle-in-a-basin” phenomenon in SR. We select three representative cases from llmsrbench. For each task, we [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Non-convexity and multimodality in nonlinear SR: disconnected basins and local-optimum traps. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper identifies the 'Good Structure, Bad Score' issue in symbolic regression (SR) arising from bi-level optimization: the outer loop searches discrete equation structures while the inner loop fits continuous parameters, but non-convexity and reliance on fast local solvers (e.g., BFGS) frequently yield poor local minima and underestimated scores for correct structures. It proposes SAGE-Fit, an SR-native plug-and-play framework that exploits structural and semantic priors via tailored modules to improve inner-loop fitting quality and thereby guide the outer search more effectively. Extensive experiments are claimed to show universal performance gains across SR systems.

Significance. If the modules reliably improve fitting fidelity without new biases or optimization pathologies, the work would address a recognized bottleneck in SR pipelines and could be adopted broadly as a modular enhancement. The emphasis on SR-native priors (rather than generic optimizers) is a targeted contribution; reproducible code or parameter-free derivations would strengthen this, but none are noted in the provided abstract.

major comments (2)
  1. [Abstract] The central claim that tailored modules 'effectively mitigating this optimization bottleneck' rests on the assumption that structural and semantic priors can be capitalized without introducing new issues; however, the abstract supplies no derivation details, pseudocode, or verification that the modules achieve the claimed mitigation (e.g., no mention of how structure-aware components interact with BFGS or handle nonlinear operators).
  2. [Abstract] Experimental evidence is asserted ('extensive experiments demonstrate... universally improves') but no quantitative results, baselines, or ablation on the modules are visible in the abstract; this leaves the load-bearing claim that evaluation fidelity is enhanced without supporting numbers or controls.
minor comments (1)
  1. [Abstract] Clarify the precise definition of 'dual native priors' and how each module maps to structural vs. semantic properties.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below, focusing on the abstract's level of detail while noting that the full paper provides the requested technical content.

read point-by-point responses

  1. Referee: [Abstract] The central claim that tailored modules 'effectively mitigating this optimization bottleneck' rests on the assumption that structural and semantic priors can be capitalized without introducing new issues; however, the abstract supplies no derivation details, pseudocode, or verification that the modules achieve the claimed mitigation (e.g., no mention of how structure-aware components interact with BFGS or handle nonlinear operators).

    Authors: The abstract serves as a high-level summary and is constrained by length. Full derivation details, pseudocode for the structure-aware and semantics-guided modules, and their specific interactions with local solvers such as BFGS (including handling of nonlinear operators) are provided in Section 3 of the manuscript. We can revise the abstract to include a one-sentence high-level description of the modules if the editor requests. revision: partial

  2. Referee: [Abstract] Experimental evidence is asserted ('extensive experiments demonstrate... universally improves') but no quantitative results, baselines, or ablation on the modules are visible in the abstract; this leaves the load-bearing claim that evaluation fidelity is enhanced without supporting numbers or controls.

    Authors: The abstract summarizes the outcome of the experiments at a high level, as is conventional. Quantitative results, baseline comparisons, and module ablations are reported in detail in Sections 4 and 5, supported by tables and figures. We are willing to incorporate one or two key quantitative highlights into the abstract during revision to strengthen the summary. revision: partial

Circularity Check

0 steps flagged

No significant circularity; proposal is a design and empirical claim

full rationale

The paper describes a bi-level optimization setup in SR and proposes SAGE-Fit as a plug-and-play module that exploits structural and semantic priors via tailored components. No derivation chain, equations, or fitted parameters are shown that reduce by construction to the inputs; the central claim is an engineering improvement evaluated experimentally rather than a tautological prediction or self-referential definition. No self-citations or uniqueness theorems are invoked in the provided text to bear load on the result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5763 in / 773 out tokens · 19269 ms · 2026-05-25T04:43:10.583526+00:00 · methodology

discussion (0)

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Reference graph

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