Simultaneous Model-Based Evolution of Constants and Expression Structure in GP-GOMEA for Symbolic Regression

Johannes Koch; Peter A.N. Bosman; Tanja Alderliesten

arxiv: 2606.02236 · v1 · pith:UMCMHWZNnew · submitted 2026-06-01 · 💻 cs.NE

Simultaneous Model-Based Evolution of Constants and Expression Structure in GP-GOMEA for Symbolic Regression

Johannes Koch , Tanja Alderliesten , Peter A.N. Bosman This is my paper

Pith reviewed 2026-06-28 11:59 UTC · model grok-4.3

classification 💻 cs.NE

keywords symbolic regressiongenetic programmingmodel-based evolutionary algorithmsconstant optimizationmixed discrete-continuous optimizationGP-GOMEA

0 comments

The pith

Merging real-valued GOMEA with GP-GOMEA optimizes constants and structure simultaneously for better symbolic regression.

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

This paper proposes integrating constant optimization directly into the evolution process of GP-GOMEA by merging it with the real-valued version of GOMEA. The method evolves both the discrete structure of symbolic expressions and their real-valued constants at the same time rather than treating constants separately. Comparisons against other constant-handling techniques, including ephemeral random constants and post-evolution tuning, show the integrated approach generally yields higher accuracy. A reader would care because symbolic regression aims to produce compact, interpretable models from data, and better joint optimization can improve fit without added complexity.

Core claim

The paper claims that merging the real-valued variant of GOMEA with GP-GOMEA enables simultaneous optimization of constants and expression structure, and that this integrated method generally performs best compared to other forms of handling constants such as linear scaling, restarts, and constant tuning after GP optimization.

What carries the argument

The merged GP-GOMEA that performs simultaneous model-based evolution of constants and expression structure.

If this is right

The simultaneous method outperforms variants that use ephemeral random constants or tune constants only after evolution.
Joint optimization of structure and constants produces more accurate expressions while keeping them compact.
The approach works well when combined with techniques such as linear scaling and restarts.
Results confirm that well-integrated handling of mixed discrete-continuous variables improves outcomes on symbolic regression.

Where Pith is reading between the lines

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

The same simultaneous-optimization principle could be tested in other evolutionary algorithms that mix discrete structures with continuous parameters.
Real-world applications in scientific modeling might benefit from fewer separate tuning phases if the joint method scales to noisy or high-dimensional data.
Other genetic programming systems that currently fix constants early could be re-examined to see whether co-evolution reduces the need for post-processing.

Load-bearing premise

That interactions between expression structure and constants are strong enough that separate optimization steps will miss better solutions.

What would settle it

Applying the merged algorithm and the strongest non-simultaneous constant-handling variant to the same benchmark problems and finding no accuracy advantage for the merged version on most cases.

Figures

Figures reproduced from arXiv: 2606.02236 by Johannes Koch, Peter A.N. Bosman, Tanja Alderliesten.

**Figure 1.** Figure 1: The genotype (left) of a single individual in GP-RV-GOMEA and how it relates to the encoded expression (right). Shaded values are introns that do not affect the semantic meaning of the expression. Interleaving Scheme Using this mixed representation, optimization then follows the approach described in [25] with additional modifications specific to GP. The approach is outlined in Algorithm 1. After initiali… view at source ↗

**Figure 2.** Figure 2: The constant optimization landscape can be multi-modal, both with and without linear scaling (LS). The error for every constant combination was computed using (the same) 500 instances sampled from U(−10, 10) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: The training and testing MSE scores for the synthetic problems on a logarithmic scale. The bar corresponds to the median MSE before tuning, while the circle and horizontal line correspond to the median and interquartile range (IQR) after postprocessing. Note that the MSE was capped at the target value of 1e − 8, which is highlighted with a vertical line. The results are shown in [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 4.** Figure 4: The number of constants used and expression sizes for the synthetic problems. The bar corresponds to the median before post-processing, while the circle and horizontal line correspond to the median and IQR after post-processing [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: The SHAP[18] values show how different aspects influence expression size for the synthetic problems. The different methods are highlighted on the color map, for the other binary aspects the color indicates whether it was enabled or not. 4.3 Real-world Problems: GP-RV-GOMEA vs ERCs and Coefficient Mutation In this experiment, we consider the problems listed in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: The training and testing R 2 scores for the real-world problems. The bar corresponds to the median R 2 before tuning, while the circle and horizontal line correspond to the median and IQR after post-processing [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: The median and IQR for the test R 2 scores over evaluations spent with linear scaling and restarts enabled. The results for the R2 scores displayed in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: The number of constants used and expression sizes for the real-world problems. The bar corresponds to the median before post-processing, while the circle and horizontal line correspond to the median and IQR after post-processing [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: The SHAP[18] values show how different aspects influence expression size for the real-world problems. The different methods are highlighted on the color map, for the other binary aspects the color indicates whether it was enabled or not. previous experiment, the effect of tuning constants after GP is most noticeable for ERCs and in the absence of LS and restarts, but small compared to the effect of constan… view at source ↗

read the original abstract

Genetic programming (GP) approaches are among the state-of-the-art for symbolic regression, the task of constructing symbolic expressions that fit well with data. To find highly accurate symbolic expressions, both the expression structure and any contained real-valued constants, are important. GP-GOMEA, a modern model-based evolutionary algorithm, is one of the leading algorithms for finding accurate, yet compact expressions. Yet, GP-GOMEA does not perform dedicated constant optimization, but rather uses ephemeral random constants. Hence, the accuracy of GP-GOMEA may well still be improved upon by the incorporation of a constant optimization mechanism. Existing research into mixed discrete-continuous optimization with EAs has shown that a simultaneous and well-integrated approach to optimizing both discrete and continuous parts, leads to the best results on a variety of problems, especially when there are interactions between these parts. In this paper, we therefore propose a novel approach where constants in expressions are optimized at the same time as the expression structure by merging the real-valued variant of GOMEA with GP-GOMEA. The proposed approach is compared to other forms of handling constants in GP-GOMEA, and in the context of other commonly used techniques such as linear scaling, restarts, and constant tuning after GP optimization. Our results indicate that our novel approach generally performs best and confirms the importance of simultaneous constant optimization during evolution.

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

Merging real-valued GOMEA into GP-GOMEA lets the algorithm optimize expression structure and constants together, and the experiments show this beats the usual constant-handling tricks on the benchmarks they ran.

read the letter

The core advance here is taking the model-based mixing operator from real-valued GOMEA and folding it into GP-GOMEA so that constants get tuned during the same generations that change the tree structure. That is a clean, direct extension of prior GOMEA work rather than a wholly new idea, but applying it to this specific GP setting is new.

The paper does the obvious right things: it compares the joint approach against ephemeral random constants, post-hoc tuning, linear scaling, and restarts, all inside the same GP-GOMEA framework. The results line up with the background claim that simultaneous handling helps when structure and values interact. The experimental design tests exactly the distinction the authors care about.

The soft spots are the usual ones for this kind of work. The benchmarks are standard symbolic regression suites, so we still need to see how it behaves on noisier or higher-dimensional data. The abstract claims the new method “generally performs best,” but without the full tables it is hard to judge effect sizes or whether a few hard instances drive the average. Statistical testing and run counts are not visible from what we have.

This is a solid incremental paper for the evolutionary computation crowd that already uses GOMEA or cares about constant handling in GP. It is not going to change how most people do symbolic regression tomorrow, but the comparison is fair and the method is reproducible from the description.

I would send it to peer review. The central experiment is well-posed and the implementation rests on existing, verified GOMEA machinery, so referees can focus on whether the gains hold up under closer scrutiny rather than on whether the idea makes sense.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes extending GP-GOMEA by merging it with the real-valued GOMEA variant, enabling simultaneous model-based evolution of both expression structure (discrete) and constants (continuous). It empirically compares this integrated approach against ephemeral random constants, post-evolution constant tuning, linear scaling, and restarts on symbolic regression tasks, concluding that the novel simultaneous method generally performs best and underscoring the value of joint optimization when interactions exist between structure and constants.

Significance. If the empirical results hold under scrutiny, the work provides concrete evidence supporting integrated discrete-continuous optimization in model-based evolutionary algorithms for symbolic regression, a domain where constant accuracy directly affects model quality. It builds directly on established GOMEA linkage-learning machinery without introducing new free parameters or self-referential definitions, and the experimental design explicitly contrasts simultaneous versus sequential constant handling.

minor comments (3)

Abstract and §4 (results): the statement that the novel approach 'generally performs best' should be accompanied by explicit reporting of the number of benchmarks, the statistical tests applied (e.g., Wilcoxon or Friedman with post-hoc correction), and effect-size measures; without these, the strength of the central empirical claim is difficult to assess from the summary tables alone.
§3.2 (method): the description of how the real-valued linkage model is merged with the GP-GOMEA dependency model would benefit from a short pseudocode fragment or diagram illustrating the joint sampling step, to make the integration reproducible.
Table captions and §4: ensure every table reports the exact number of independent runs and any parameter settings that differ from the cited GP-GOMEA baseline, to avoid ambiguity when readers attempt replication.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The feedback affirms the contribution of simultaneous discrete-continuous optimization via the merged GOMEA variants. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Minor self-citation on GOMEA background; empirical claim independent

full rationale

The paper's core contribution is an empirical comparison of constant-handling variants (ephemeral random constants, post-tuning, linear scaling, restarts, and the proposed simultaneous model-based optimization) on symbolic regression benchmarks. The performance claim is grounded in experimental results rather than any derivation that reduces to fitted inputs or self-referential definitions. The background statement on mixed discrete-continuous EAs is presented as established prior work and does not serve as a load-bearing uniqueness theorem or ansatz for the reported outcomes. No equations or algorithmic steps in the provided text exhibit self-definitional, fitted-prediction, or renaming circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption drawn from prior mixed discrete-continuous EA research that simultaneous optimization is superior when interactions exist; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Existing research into mixed discrete-continuous optimization with EAs has shown that a simultaneous and well-integrated approach to optimizing both discrete and continuous parts leads to the best results on a variety of problems, especially when there are interactions between these parts.
Invoked in the abstract to justify proposing the merged algorithm.

pith-pipeline@v0.9.1-grok · 5781 in / 1343 out tokens · 48650 ms · 2026-06-28T11:59:34.518988+00:00 · methodology

discussion (0)

Reference graph

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