On the Limits of Interpretable Machine Learning in Quintic Root Classification

Rohan Thomas; Majid Bani-Yaghoub

math.NAFeb 26, 2026

On the Limits of Interpretable Machine Learning in Quintic Root Classification

Authors: Rohan Thomas, Majid Bani-Yaghoub

Organizations: Division of Computing, Analytics & Mathematics, School of Science and Engineering, University of Missouri-Kansas City, 5100 Rockhill Rd., Kansas City, Missouri 64110, USA

Abstract

Can Machine Learning (ML) autonomously recover interpretable mathematical structure from raw numerical data? We aim to answer this question using the classification of real-root configurations of polynomials up to degree five as a structured benchmark. We tested an extensive set of ML models, including decision trees, logistic regression, support vector machines, random forest, gradient boosting, XGBoost, symbolic regression, and neural networks. Neural networks achieved strong in-distribution performance on quintic classification using raw coefficients alone (84.3% + or - 0.9% balanced accuracy), whereas decision trees perform substantially worse (59.9% + or - 0.9%). However, when provided with an explicit feature capturing sign changes at critical points, decision trees match neural performance (84.2% + or - 1.2%) and yield explicit classification rules. Knowledge distillation reveals that this single invariant accounts for 97.5% of the extracted decision structure. Out-of-distribution, data-efficiency, and noise robustness analyses indicate that neural networks learn continuous, data-dependent geometric approximations of the decision boundary rather than recovering scale-invariant symbolic rules. This distinction between geometric approximation and symbolic invariance explains the gap between predictive performance and interpretability observed across models. Although high predictive accuracy is attainable, we find no evidence that the evaluated ML models autonomously recover discrete, human-interpretable mathematical rules from raw coefficients. These results suggest that, in structured mathematical domains, interpretability may require explicit structural inductive bias rather than purely data-driven approximation.

Figures & tables

Model	Description
Decision Trees (CART)	Interpretable models that produce explicit if-then rules, allowing direct examination of learned structure.
Logistic Regression	Linear classifier to test whether linear decision boundaries suffice for this problem.
Support Vector Machines (SVM)	RBF kernel to explore whether kernel methods could find separating boundaries.
Random Forest	Ensemble of decision trees to assess whether aggregation improves performance.
Gradient Boosting	Sequential ensemble method for comparison.
XGBoost	Optimized gradient boosting implementation.

Table 1: Overview of Machine Learning models used in this study to evaluate their interpretability

Polynomial	Accuracy (%)	Observations
Quadratic	$97.3\pm 4.2$	Discovered discriminant rule (100% in 2/5 trials)
Cubic	$94.4\pm 1.6$	Discovered highly accurate but complex heuristic
Quartic	$47.3\pm 5.8$	Consistently failed; relied on simple linear heuristic
Quintic	$53.5\pm 2.9$	Failed to find interpretable rules; relied on complex heuristics

Table 2: Summary of Symbolic Regression Performance by Polynomial Degree

Model	Balanced Accuracy (Raw Coefficients)
Neural Networks	83.8% $\pm$ 3.9%
Gradient Boosting	63.3% $\pm$ 3.8%
SVM (RBF)	62.9% $\pm$ 0.4%
Random Forest	62.5% $\pm$ 1.5%
Decision Trees (CART)	53.1% $\pm$ 3.1%
Logistic Regression	41.6% $\pm$ 0.9%

Table 3: Degree 5 Initial Screening Performance (Mean

\pm

95% CI)

Model	Raw Coefficients	With Crit8 1
Neural Networks	84.3% $\pm$ 0.9%	89.9% $\pm$ 1.4%
Decision Trees	59.9% $\pm$ 0.9%	84.2% $\pm$ 1.2%
Performance Gap	24.4 points	5.7 points

Table 4: Performance comparison of neural networks and decision trees when additional feature “crit8” added.

Metric	Value	Interpretation
NN Test Balanced Accuracy	87.0% $\pm$ 1.2%	Robust feature learning
Tree Test Fidelity	98.9% $\pm$ 0.1%	Successful logic transfer
Tree Standalone Accuracy	84.6% $\pm$ 1.2%	Generalizable rule extraction

Table 5: Quantitative results for the Knowledge Distillation process from Neural Networks (NN) to Decision Trees. The high fidelity indicates that the interpretable tree successfully approximates the neural model’s decision logic.

Feature Set	Neural Networks	Decision Trees
Raw coefficients	84.3% $\pm$ 0.9%	59.9% $\pm$ 0.9%
+ Sturm sequences	85.2% $\pm$ 1.1%	62.1% $\pm$ 4.1%
+ Newton sums	83.0% $\pm$ 1.3%	63.0% $\pm$ 4.2%
+ Critical points	89.3% $\pm$ 0.7%	84.2% $\pm$ 1.2%
+ Hybrid symbolic	80.5% $\pm$ 1.4%	61.4% $\pm$ 4.0%
All 63 combined	87.0% $\pm$ 1.2%	68.2% $\pm$ 4.5%

Table 6: Comparative performance across different mathematical feature sets for quintic polynomials. Bold values indicate the optimal feature configuration, where Critical points provide the most significant gain in interpretability and accuracy.

Panel of Figure 1 — Figure 1: Behavioral divergence between neural approximation (Top Row) and discrete mathematical execution (Bottom Row). Error bars indicate 95% Confidence Intervals. Left: Invariant decision trees maintain perfectly stable, scale-invariant bounds out-of-distribution, whereas neural approximations decay. Middle: Trees lock in boundaries with mere dozens of samples, while networks require thousands. Right: Both models degrade identically under noise, reflecting the fundamental alteration of the true polynomial roots.

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