Symbolic Quantile Regression for the Interpretable Prediction of Conditional Quantiles

Cas Oude Hoekstra; Floris den Hengst

arxiv: 2508.08080 · v3 · pith:NIREPFO2new · submitted 2025-08-11 · 💻 cs.LG · cs.NE· stat.AP

Symbolic Quantile Regression for the Interpretable Prediction of Conditional Quantiles

Cas Oude Hoekstra , Floris den Hengst This is my paper

Pith reviewed 2026-05-21 23:02 UTC · model grok-4.3

classification 💻 cs.LG cs.NEstat.AP

keywords symbolic regressionquantile regressioninterpretable machine learningconditional quantileswhite-box modelsregressionmachine learning

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The pith

Symbolic Quantile Regression extends symbolic regression to predict conditional quantiles at any point in the outcome distribution with transparent models.

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

The paper develops Symbolic Quantile Regression to allow symbolic regression techniques to target specific quantiles rather than just the mean of the target variable. This is achieved by replacing the standard error loss with a quantile loss during the model search process. A reader would care because many important predictions, such as risk assessments or median outcomes, require understanding the full distribution, and interpretable methods have been limited to averages until now. Experiments demonstrate that the resulting models outperform other transparent techniques and approach the accuracy of black-box models. The approach is further validated by using it to compare extreme and central predictions in an airline fuel consumption dataset.

Core claim

By adapting the loss function in symbolic regression to a quantile-based one, Symbolic Quantile Regression generates human-interpretable mathematical expressions that estimate conditional quantiles. Extensive testing shows these models exceed the performance of other interpretable methods and match strong black-box baselines while preserving transparency. This enables explanations of how predictors affect different parts of the target distribution.

What carries the argument

Symbolic Quantile Regression (SQR), the adaptation of symbolic regression's search process to minimize quantile loss instead of mean squared error, producing white-box expressions for any desired quantile level.

If this is right

Transparent models become available for predicting medians, upper or lower tails, and other quantiles in addition to averages.
Domain experts can directly read and compare expressions for different quantiles to understand shifting variable influences.
High-stakes applications such as safety or finance can use interpretable quantile predictions without relying on opaque systems.
The range of problems addressable by symbolic regression expands from mean estimation to full distributional analysis.

Where Pith is reading between the lines

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

Applying SQR to datasets with known physical constraints could test whether the discovered expressions respect those constraints at multiple quantiles.
Comparing SQR models across quantiles might surface subgroup differences or biases that average-based models obscure.
Integration with ensemble methods or post-processing could further improve accuracy while retaining interpretability.

Load-bearing premise

That replacing the loss function with a quantile variant in symbolic regression keeps both the accuracy and the interpretability of the generated models intact for various quantiles and data sets.

What would settle it

Running SQR on a held-out dataset where the quantile prediction errors are markedly higher than those of a black-box model, or where the expressions become too complex for domain experts to interpret easily.

read the original abstract

Symbolic Regression (SR) is a well-established framework for generating interpretable or white-box predictive models. Although SR has been successfully applied to create interpretable estimates of the average of the outcome, it is currently not well understood how it can be used to estimate the relationship between variables at other points in the distribution of the target variable. Such estimates of e.g. the median or an extreme value provide a fuller picture of how predictive variables affect the outcome and are necessary in high-stakes, safety-critical application domains. This study introduces Symbolic Quantile Regression (SQR), an approach to predict conditional quantiles with SR. In an extensive evaluation, we find that SQR outperforms transparent models and performs comparably to a strong black-box baseline without compromising transparency. We also show how SQR can be used to explain differences in the target distribution by comparing models that predict extreme and central outcomes in an airline fuel usage case study. We conclude that SQR is suitable for predicting conditional quantiles and understanding interesting feature influences at varying quantiles.

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.

Desk Editor's Note private letter to a colleague

Adapting symbolic regression to pinball loss gives interpretable quantile models that perform well but risk inconsistent ordering across quantiles.

read the letter

The main thing to know is that the authors have extended symbolic regression to conditional quantiles by plugging the pinball loss into the evolutionary fitness function. This produces separate interpretable expressions for each target quantile level. They show in experiments that these models beat other transparent approaches and match a strong black-box method on accuracy while remaining fully transparent. The case study on airline fuel usage demonstrates the value by showing how feature influences change between central and extreme quantiles, which is a concrete use case for understanding distribution shifts. The evaluation appears to use standard datasets and reports positive results, which supports the claim that the adaptation works without losing too much predictive power. The transparency is preserved because the output is still a symbolic formula rather than a complex model. A potential weakness is that training independent models for each quantile does not enforce the non-crossing property required for proper quantile functions. If the evolved expressions cross on the input space, the predictions for different tau values would not be consistent, weakening both the performance interpretation and the explanatory power. It would be good to see if they measured this or if it rarely occurs in practice. The work is aimed at applied researchers who want white-box models for quantile estimation in domains where interpretability matters, such as safety-critical systems. Readers familiar with symbolic regression will see a natural extension, and those using quantile methods get a new option to consider. It is worth sending for peer review because the core idea is sound and the empirical comparison provides a starting point for further validation.

Referee Report

2 major / 2 minor

Summary. The paper introduces Symbolic Quantile Regression (SQR), which adapts symbolic regression by substituting the pinball loss into the fitness function to generate interpretable symbolic expressions for conditional quantiles at chosen levels τ. It reports that SQR outperforms other transparent models and performs comparably to a strong black-box baseline across evaluations while preserving transparency, and demonstrates explanatory use by comparing extreme versus central quantile models in an airline fuel-usage case study.

Significance. If the performance claims are robust and the estimated quantile functions satisfy the required monotonicity property, SQR would provide a meaningful advance by extending symbolic regression to quantile estimation in a transparent manner, enabling fuller distributional insights in high-stakes domains without sacrificing interpretability.

major comments (2)

[Method / Experiments] The method trains a separate symbolic regression model for each target quantile τ by direct substitution of the pinball loss. Because the models are evolved independently, nothing in the genetic operators, selection, or simplification enforces Q_τ1(x) ≤ Q_τ2(x) for τ1 < τ2 on the data support. The evaluation sections report no crossing rates, rearrangement post-processing, or joint multi-τ objective. This directly undermines the claim that the family of functions correctly represents conditional quantiles and weakens both the performance and interpretability assertions.
[Experiments] The central performance claim (outperformance of transparent models and comparability to black-box baselines) rests on evaluation results whose details—exact data splits, baseline implementations, hyper-parameter search protocols, and statistical tests—are not fully specified. Without these, it is impossible to rule out that post-hoc choices inflate the reported advantages.

minor comments (2)

[Abstract / Introduction] The abstract and introduction could more explicitly define the pinball loss and the precise fitness function used inside the SR loop.
[Figures / Tables] Figure captions and table headers should state the exact quantile levels τ examined and the number of independent runs performed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and describe the changes we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Method / Experiments] The method trains a separate symbolic regression model for each target quantile τ by direct substitution of the pinball loss. Because the models are evolved independently, nothing in the genetic operators, selection, or simplification enforces Q_τ1(x) ≤ Q_τ2(x) for τ1 < τ2 on the data support. The evaluation sections report no crossing rates, rearrangement post-processing, or joint multi-τ objective. This directly undermines the claim that the family of functions correctly represents conditional quantiles and weakens both the performance and interpretability assertions.

Authors: We agree that independent evolution of models for each τ does not automatically enforce monotonicity across quantiles, which is a recognized limitation in many quantile regression approaches. While our empirical results showed limited crossings on the evaluated datasets, we did not report crossing rates or apply rearrangement. In the revision we will add an analysis of observed crossing rates, discuss the monotonicity issue explicitly, and outline a simple rearrangement post-processing option that can be applied when strict ordering is required for a given application. revision: partial
Referee: [Experiments] The central performance claim (outperformance of transparent models and comparability to black-box baselines) rests on evaluation results whose details—exact data splits, baseline implementations, hyper-parameter search protocols, and statistical tests—are not fully specified. Without these, it is impossible to rule out that post-hoc choices inflate the reported advantages.

Authors: We accept that the current manuscript lacks sufficient detail for full reproducibility. The revised version will include an expanded experimental section (or appendix) that specifies the exact train/validation/test splits for each dataset, the precise implementations and hyper-parameter grids used for all baselines, the search protocol applied to SQR, and the results of appropriate statistical tests (including p-values or confidence intervals) for the performance comparisons. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method is a direct adaptation evaluated externally

full rationale

The paper defines SQR by substituting the standard pinball loss into the symbolic regression fitness function to target conditional quantiles at each τ independently. This is a straightforward methodological extension rather than a self-referential definition or fitted parameter renamed as a prediction. Performance and interpretability claims rest on empirical evaluation against external datasets and baselines, with no equations or self-citations that reduce the central results to the method's own inputs by construction. The derivation chain is self-contained and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard assumptions of symbolic regression plus the untested premise that quantile loss can be substituted without harming interpretability or search efficiency.

free parameters (1)

quantile level
User-specified value (e.g., 0.5 or 0.9) that defines which point in the conditional distribution is targeted.

axioms (1)

domain assumption Symbolic regression search can be driven by a quantile loss instead of squared error while retaining white-box properties.
Invoked when the method is defined in the abstract as an extension of SR to quantiles.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

SQR estimates the conditional quantiles ... by minimizing an established QR loss known as the pinball loss together with a loss for the interpretability of the expression.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We compare SQR with ... on 122 regression data sets ... without compromising transparency.

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