GP-GOMEA with GPU-Based Fitness Evaluations: Design and Performance Analysis

Anton Bouter; Jasper Post; Johannes Koch; Peter A.N. Bosman; Tanja Alderliesten

arxiv: 2605.30954 · v1 · pith:5IWEWS62new · submitted 2026-05-29 · 💻 cs.NE

GP-GOMEA with GPU-Based Fitness Evaluations: Design and Performance Analysis

Jasper Post , Johannes Koch , Anton Bouter , Tanja Alderliesten , Peter A.N. Bosman This is my paper

Pith reviewed 2026-06-28 20:19 UTC · model grok-4.3

classification 💻 cs.NE

keywords symbolic regressiongenetic programmingGPU accelerationevolutionary algorithmsfitness evaluationGP-GOMEAFeynman equations

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The pith

A GPU fitness evaluation scheme for GP-GOMEA increases evaluation throughput enough to regress large Feynman equations in four hours.

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

The paper designs a GPU-friendly representation of GP-GOMEA template-based individuals together with a parallel evaluation strategy. This change replaces serial CPU fitness calculations with massively parallel GPU ones, raising the number of evaluations that fit inside a fixed time budget by orders of magnitude. On four standard symbolic regression benchmarks the extra evaluations produce better models, especially when datasets or populations are large. The same capacity also lets the algorithm run systematic experiments on expression structure that were previously impossible and, for the first time, reliably recover one of the largest Feynman equations within four hours.

Core claim

By introducing a GPU-based fitness evaluation scheme that exploits the inherent parallelism of population-based search, the work raises evaluation throughput for GP-GOMEA. The resulting capacity yields performance gains on benchmarks with larger datasets and populations, enables new analyses of how expression structure affects search difficulty, and allows a problem-agnostic evolutionary algorithm to regress one of the largest Feynman equations reliably inside four hours.

What carries the argument

The GPU-based fitness evaluation scheme, which uses a template-based individual representation optimized for parallel GPU computation to replace CPU evaluations.

If this is right

Performance improves on four standard symbolic regression benchmarks, with larger gains for bigger datasets and populations.
Systematic study of which expression structures make search harder for GP-GOMEA becomes feasible.
A problem-agnostic evolutionary algorithm can now reliably regress one of the largest Feynman equations in four hours.

Where Pith is reading between the lines

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

The same GPU pattern could be applied to other template-based genetic programming methods to achieve comparable speed-ups.
The ability to run far more evaluations may narrow the scalability gap between symbolic regression and deep-learning approaches.
Insights from the new difficulty analyses could be used to design heuristics that prune hard expression structures early.

Load-bearing premise

The GPU implementation must compute fitness values that match the accuracy of the original CPU version while delivering the claimed higher throughput.

What would settle it

A side-by-side run of identical GP-GOMEA individuals on CPU and GPU that produces measurably different model errors or solution qualities on any benchmark would show the accuracy equivalence does not hold.

Figures

Figures reproduced from arXiv: 2605.30954 by Anton Bouter, Jasper Post, Johannes Koch, Peter A.N. Bosman, Tanja Alderliesten.

**Figure 2.** Figure 2: Parallel decomposition and kernel design for two designs. Left: single block [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Median validation MSE and interquartile ranges (shaded) over all runs [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Median number of evaluations per minute over population size across [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Median number of evaluations per minute over population size across [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Top rows: NMSE over time across problems with template depths of 6 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Median training MSE and interquartile ranges (shaded areas) over all [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Median validation MSE and interquartile ranges (shaded areas) over all [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: Median training MSE and interquartile ranges (shaded areas) over all [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: Median number of evaluations per minute and interquartile ranges [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

read the original abstract

GP-GOMEA is a state-of-the-art evolutionary algorithm for symbolic regression, known for discovering small and interpretable models. However, its computational cost remains substantial, limiting its applicability to larger datasets and more complex target expressions. In contrast, the rise of modern subsymbolic approaches, particularly deep learning, has been driven largely by the massive parallelism offered by GPUs. In this work, we take the first major step toward a fully GPU-accelerated GP-GOMEA by introducing a GPU-based fitness evaluation scheme. We design a GPU-friendly representation of GP-GOMEA's template-based individuals and a corresponding evaluation strategy that exploits the inherent parallelism of population-based search. This substantially increases evaluation throughput, enabling orders of magnitude more evaluations within the same time budget. Across four standard symbolic regression benchmarks, this increased evaluation capacity yields performance improvements, particularly for larger datasets and larger population sizes. Moreover, the ability to efficiently evaluate much larger datasets and more complex templates enables analyses that were previously infeasible, allowing us to systematically analyze what makes expressions increasingly difficult for GP-GOMEA, providing new insights into how expression structure affects search difficulty. Finally, for the first time, this expanded capability allows a problem-agnostic evolutionary algorithm to reliably regress one of the largest Feynman equations within four hours.

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

GP-GOMEA gets a practical GPU speed boost for larger problems, but the claim that it matches the original CPU behavior needs explicit numerical checks to hold up.

read the letter

The main point is that the authors adapted GP-GOMEA's template representation and fitness evaluation to run on GPU, which raises evaluation throughput enough to handle bigger datasets and more complex expressions than before. This is a straightforward engineering move that follows the pattern already used in deep learning, and it produces measurable gains on the four benchmarks they tested, especially at larger population sizes.

What stands out as new is the specific GPU-friendly encoding of the templates and the parallel evaluation strategy built around it. The paper shows that this lets them run orders of magnitude more evaluations in the same wall-clock time, and they use the extra budget both to improve regression accuracy on standard problems and to run new analyses on expression difficulty. The Feynman-equation result is presented as evidence that the method now reaches problems that were previously out of reach for this style of algorithm.

The results look solid on the reported metrics, and the work is honest about focusing on throughput rather than inventing new operators. The citation pattern stays within the relevant GP and symbolic regression literature without obvious gaps.

The soft spot is the assumption that GPU evaluations produce identical fitness values to the CPU version. Template evaluations involve subtree reductions and constant handling, so changes in floating-point associativity or parallel reduction order could alter selection outcomes even if average throughput rises. The abstract does not describe a direct numerical equivalence test on the same individuals, and if the full paper lacks that comparison the attribution of the Feynman success to "more evaluations of the same algorithm" is weaker than claimed. That is a fixable gap rather than a fatal one.

This paper is aimed at researchers already using or extending GP-GOMEA who need to scale it to real data sizes. A reader interested in practical acceleration of evolutionary methods will get concrete implementation ideas and benchmark numbers. It is worth sending to peer review because the empirical scaling results and the new analyses are substantive enough to merit referee scrutiny, even if the core contribution is an adaptation.

Referee Report

3 major / 2 minor

Summary. The paper introduces a GPU-friendly representation and evaluation strategy for template-based individuals in GP-GOMEA to accelerate fitness evaluations via parallelism. It reports substantially higher throughput enabling more evaluations within fixed time budgets, yielding performance gains on four symbolic regression benchmarks (especially larger datasets and populations), new analyses of how expression structure affects search difficulty, and the first reliable regression of one of the largest Feynman equations by a problem-agnostic EA within four hours.

Significance. If the GPU scheme produces numerically identical fitness values to the CPU version (preserving evolutionary dynamics), the work meaningfully extends GP-GOMEA's reach to larger-scale problems and provides new empirical insights into search difficulty; the Feynman result would be a notable demonstration of scalability for evolutionary symbolic regression.

major comments (3)

[Abstract and experimental results] Abstract and experimental results section: The central claim that GPU evaluations enable the Feynman regression 'for the first time' within four hours rests on the unverified assumption that the new scheme yields identical fitness values (and thus identical selection dynamics) to the original CPU implementation. No explicit numerical verification—e.g., side-by-side fitness comparisons or model-quality metrics on identical individuals—is reported, despite the risk that changes in floating-point associativity, reduction order, or constant handling could alter outcomes.
[Benchmark results] Benchmark results (likely §4): Performance improvements are stated without error bars, multiple independent runs with statistical tests, or direct baseline comparisons to the original CPU GP-GOMEA under identical time budgets; this makes it impossible to quantify whether gains are due to throughput alone or to any unintended changes in search behavior.
[Methods/design] Methods/design section: The GPU representation and evaluation strategy for template-based individuals is described at a high level, but no pseudocode, kernel details, or handling of subtree reductions/precision is provided to allow assessment of numerical equivalence.

minor comments (2)

[Design section] Notation for template-based individuals and GPU data structures could be clarified with a small diagram or explicit mapping to the original CPU representation.
[Experimental setup] The four standard benchmarks should be named explicitly with dataset sizes and target expressions in the main text (not only supplementary).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive review. We address each major comment below and commit to revisions that directly respond to the concerns raised.

read point-by-point responses

Referee: [Abstract and experimental results] Abstract and experimental results section: The central claim that GPU evaluations enable the Feynman regression 'for the first time' within four hours rests on the unverified assumption that the new scheme yields identical fitness values (and thus identical selection dynamics) to the original CPU implementation. No explicit numerical verification—e.g., side-by-side fitness comparisons or model-quality metrics on identical individuals—is reported, despite the risk that changes in floating-point associativity, reduction order, or constant handling could alter outcomes.

Authors: We agree that explicit verification of numerical equivalence is necessary to substantiate the claim that the GPU scheme preserves evolutionary dynamics and to support the 'for the first time' statement. Although the GPU representation and evaluation strategy were designed to maintain identical semantics and reduction order where possible, the submitted manuscript does not include side-by-side fitness or model-quality comparisons. In the revision we will add such verification experiments on identical individuals across both implementations. revision: yes
Referee: [Benchmark results] Benchmark results (likely §4): Performance improvements are stated without error bars, multiple independent runs with statistical tests, or direct baseline comparisons to the original CPU GP-GOMEA under identical time budgets; this makes it impossible to quantify whether gains are due to throughput alone or to any unintended changes in search behavior.

Authors: We accept that the results presentation would be strengthened by error bars, explicit reporting of the number of independent runs, statistical tests, and direct CPU baselines under matched time budgets. The experiments underlying the reported improvements were performed with multiple runs, yet these details and comparisons were not fully documented. We will revise the experimental results section to include error bars, run counts, statistical analysis, and side-by-side CPU comparisons. revision: yes
Referee: [Methods/design] Methods/design section: The GPU representation and evaluation strategy for template-based individuals is described at a high level, but no pseudocode, kernel details, or handling of subtree reductions/precision is provided to allow assessment of numerical equivalence.

Authors: We acknowledge that additional low-level details would facilitate independent assessment of numerical equivalence. The current description focuses on the high-level design choices that enable GPU parallelism. In the revised manuscript we will expand the methods section with pseudocode for the core evaluation kernels, explicit handling of subtree reductions, and discussion of floating-point precision and associativity choices. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical implementation paper with externally verifiable benchmarks

full rationale

The paper describes a GPU implementation of fitness evaluation for the existing GP-GOMEA algorithm and reports empirical throughput and performance gains on symbolic regression benchmarks, including one Feynman equation. No mathematical derivations, parameter fits, or predictions appear. Claims rest on direct runtime measurements and solution quality comparisons that can be reproduced independently. No self-citations are load-bearing for any core result, and no steps reduce by construction to inputs. The equivalence assumption for GPU vs CPU is an implementation detail subject to external verification rather than a circular definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work is an empirical algorithmic engineering effort; the abstract introduces no mathematical derivations, free parameters, axioms, or new postulated entities.

pith-pipeline@v0.9.1-grok · 5770 in / 1014 out tokens · 19147 ms · 2026-06-28T20:19:34.667139+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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