A Better Multi-Objective GP-GOMEA -- But do we Need it?

Joe Harrison; Tanja Alderliesten. Peter A.N. Bosman

REVIEW 2 major objections 7 minor 27 references

A Better Multi-Objective GP-GOMEA -- But do we Need it?

T0 review · 2 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For accuracy versus expression size, single-objective GP-GOMEA with a logging archive beats its multi-objective counterpart.

desk verdict Solid empirical contribution that improves MO GP-GOMEA, but the headline SO-over-MO recommendation rests on an untested initialization assumption and the abstract overstates it. read the letter →

arxiv 2507.03777 v1 pith:EIVXLCHP submitted 2025-07-04 cs.NE

Joe Harrison , Tanja Alderliesten. Peter A.N. Bosman This is my paper · ORCID

classification cs.NE

keywords GP-GOMEAsymbolicregressionmulti-objectiveoptimizationhypervolumeexpressionsizebalancedclusteringsubexpressionreuse

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks when the multi-objective form of Modular GP-GOMEA is actually worth using in symbolic regression. It proposes two new clustering methods, BKRR and BKmRR, plus a duplicate-fitness mutation step, and shows that these improvements raise the multi-objective variant's average hypervolume, the standard measure of trade-off front quality. Even with all improvements, however, a single-objective GP-GOMEA that maximizes $R^2$ and uses a multi-objective elitist archive only to log non-dominated solutions still achieves a higher average hypervolume on the accuracy-versus-expression-size trade-off across five real-world datasets within a three-hour budget. The paper concludes that for most accuracy-plus-complexity objectives the single-objective setup should be the first choice, and that multi-objective search earns its cost mainly when the secondary objective is uncorrelated with accuracy, as with maximizing the number of cosine operators.

What carries the argument

The comparison rests on two working parts. First, the MO elitist archive: in both configurations a non-dominated archive is maintained, but in the single-objective configuration it has no influence on selection or variation, so it acts purely as a logger of the accuracy-size trade-off. Second, the proposed balanced clustering scheme BKmRR: extreme clusters are formed by taking the top $n/k$ individuals for each objective, then the remaining $k-2$ clusters are filled by round-robin assignment of the closest unassigned individual, so every cluster is equal in size and the best solutions per objective sit in dedicated single-objective clusters. The duplicate-fitness mutation step, which mutates individuals whose objective values duplicate until an active node changes, is what makes both configurations faster to explore.

What would settle it

Run the same five datasets with size-uniform initialization, so no individual at generation zero has expression size 1, and compare average hypervolume of single-objective-with-archive versus multi-objective with BKmRR; if MO then matches or exceeds SO, the SO-first conclusion is specific to this initialization and time budget.

Watch

Extended reading notes

Core claim

The central discovery is that the best-performing way to obtain a Pareto front of accuracy and expression size with Modular GP-GOMEA is to not run multi-objective optimization at all: a single-objective run that maximizes $R^2$ and merely records non-dominated solutions in an elitist archive consistently beats the genuinely multi-objective algorithm in average hypervolume on all five datasets. The reason is initialization-dependent: the half-and-half initializer already places minimal-size expressions on the front within the first generations, so the multi-objective variant's continued minimization of expression size consumes search effort without moving the front, while the single-objective run spends all its effort raising accuracy and incidentally discovers larger expressions along the way. The paper also shows that the new BKmRR clustering, which reserves equal-sized extreme clusters for each objective and fills the remaining clusters round-robin, improves multi-objective GP-GOMEA over the original clustering, and that a de-duplicated size objective can promote functional subexpression reuse on synthetic problems.

Load-bearing premise

The whole comparison depends on initialization: if the first generation already contains expressions of the smallest possible size, as happens with half-and-half initialization, then the multi-objective run wastes effort trying to shrink them further, and with a different initializer the single-objective advantage might vanish.

Editorial extensions

If this is right

For the accuracy-versus-expression-size trade-off under a three-hour budget, a single-objective Modular GP-GOMEA with an MO logging archive is the better default, because it reaches a higher average hypervolume faster.
The proposed BKmRR clustering improves MO Modular GP-GOMEA over the original clustering: it statistically outperforms BKRR on all five datasets.
Mutating individuals that have duplicate per-objective fitness raises average hypervolume for both the single-objective and multi-objective variants.
Using a de-duplicated size objective during search increases the quality of the final non-dominated front on four of five synthetic datasets designed to contain subexpression reuse.
When the secondary objective is uncorrelated with accuracy and not already well optimized at initialization, such as the number of cosine operators, the multi-objective variant significantly beats the single-objective one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's own account implies that the SO-first recommendation could reverse under size-uniform initialization, where the initial population consists of full, larger trees; the authors flag this possibility but do not run the experiment.
A practical reading is that MO GP-GOMEA's value depends on whether half-and-half initialization already supplies near-optimal values of the secondary objective, so one can check that condition before choosing MO over SO.
The de-duplicated size objective likely generalizes to other complexity measures that count repeated structure, such as operator-cost sums or node-depth measures, since it targets the same repeated-subexpression waste.
The generation-versus-time comparison in the paper suggests that with longer budgets MO optimization may catch up; a direct extrapolation test would compare the two algorithms at, say, twenty-four hours rather than three.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper is worth your time for one question: when is the extra machinery of multi-objective GP-GOMEA actually buying you something? The authors propose two clustering improvements (BKRR and BKmRR), a duplicate-fitness mutation rule, and a de-duplicated size objective. The BKmRR clustering and the mutation give clear hypervolume gains over the previous MO version, and the de-duplicated size objective helps on 4/5 synthetic reuse benchmarks. That part is solid, incremental algorithmic work.

The more interesting result is comparative: with a half-and-half initializer, single-objective GP-GOMEA with an MO archive used only for logging beats the MO variant on accuracy-vs-size hypervolume, even after their improvements. The mechanism is believable: initialization already saturates the size objective at its minimum, so MO wastes effort minimizing size while SO spends all its time on accuracy. They then show the pattern flips when the secondary objective is uncorrelated with accuracy and far from the front at initialization (cosine count). That is a genuinely useful practitioner rule.

The soft spots are real but not fatal. No code or data is shipped, so the numbers are hard to check. More importantly, the headline claim in the abstract—'consistently finds a better average hypervolume'—is contingent on the half-and-half initializer. The authors acknowledge in the Discussion that a size-uniform initialization, as in [18] or [11], could reverse the ranking, and they do not run that experiment. Since their own mechanism says the SO advantage comes from size being saturated early, this is the load-bearing assumption, and it is untested. The 3-hour budget compounds the issue; they admit MO can catch up with longer budgets. The paper also omits NSGA-II and other standard MO baselines beyond the GOMEA family, so the scope is 'within GP-GOMEA,' which is fine but should be stated.

Overall: a fair, well-explained empirical study with a slightly overbroad abstract and one untested init assumption. I'd send it to review—the components and the SO-vs-MO analysis deserve referee time—and I'd ask the authors to either test size-uniform init or soften the claim.

Referee Report

2 major / 7 minor

Summary. Harrison et al. study whether a multi-objective (MO) version of Modular GP-GOMEA is preferable to a single-objective (SO) version with an MO archive for symbolic regression. They introduce two clustering methods (BKRR and BKmRR), a duplicate-fitness mutation scheme, and a de-duplicated size objective intended to promote subexpression reuse. Using five real-world and five synthetic datasets, they report that SO Modular GP-GOMEA with an MO archive achieves a higher average hypervolume than the MO variants when accuracy is traded against expression size within a three-hour budget, but that MO wins when the secondary objective is uncorrelated with accuracy, such as maximizing the number of cosine operators. The authors attribute the SO dominance to half-and-half initialization already producing minimal-size individuals, and they explicitly note that alternative initialization could change the outcome.

Significance. If the results hold, the paper provides useful practical guidance: for correlated secondary objectives, a single-objective GP-GOMEA with an MO archive can be a cheaper and often better choice, while MO search is valuable when the secondary objective is not already well covered by initialization. The empirical design is a strength: multiple datasets, ten repetitions, corrected Wilcoxon tests, and a deliberately chosen counterexample objective. The proposed BKmRR clustering and duplicate-fitness mutation improve MO GP-GOMEA and are worthy contributions in their own right. The main limitation is that the central SO-over-MO recommendation is tied to one initialization scheme and is not tested under the size-uniform initialization that the authors themselves identify as a potential confound. The absence of code or data also makes the quantitative claims hard to verify independently.

major comments (2)

[Abstract; Section 3.7; Section 4] The central claim that SO GP-GOMEA with an MO archive 'consistently' outperforms MO GP-GOMEA for the size/accuracy trade-off is load-bearing on an untested initialization premise. The paper's own Section 4 states: 'Initialising the individuals differently, e.g., uniformly in terms of size as done in [18] or [11], may potentially positively influence the performance of MO GP-GOMEA, but this requires further research.' Because half-and-half initialization already places individuals at the minimum expression size in the first generations, the comparison does not exercise MO's ability to discover the size/accuracy trade-off from a non-saturated starting distribution. The abstract and the conclusion ('SO optimization ... as a first step for most scenarios') should be qualified to the tested initialization and time budget, or an experiment with size-uniform initialization should be added.
[Section 2.4; Section 3.1] The SO variant is described in Section 2.4 as using the MO elitist archive only for logging, with the archive not influencing the evolutionary process 'in any way.' However, Section 3.1 states that runs terminate when the MO elitist archive does not change for 100 consecutive generations. The archive therefore controls the stopping criterion and, consequently, the number of generations and the final reported hypervolume of the SO runs. This apparent contradiction should be resolved, either by clarifying that the archive affects only termination or by adjusting the description in Section 2.4.

minor comments (7)

[Section 3.4, Table 5] The sentence 'MO GP-GOMEA is outperformed by SO GP-GOMEA in all datasets' is not supported by the numbers as presented: Table 5 reports BKmRR hypervolume 0.900 ± 0.005 on Tower, while Table 4 reports SO hypervolume 0.899 ± 0.004. Please verify the comparison and rephrase.
[Section 3.4, Table 5 caption] The caption should state explicitly that the MO experiments in this table use a population size five times larger than the SO experiments, and how the SO population size was chosen for this comparison.
[Algorithm 2] Lines 21 and 22 of Algorithm 2 are identical ('Identify clusters as extreme or middle clusters'); one of the two lines should be removed.
[Section 3.5] The sentence 'Symbolic expressions in GP are often modelled as binary-unary trees' appears twice in this section; it is a duplicate from Section 2.1 and should be deleted.
[Section 3.7] There is a typo: 'intitializes' should be 'initializes'.
[General] No code or data repository is provided. For a claims-heavy empirical paper, making the implementation and datasets available would substantially improve reproducibility.
[Table 12] The SO standard deviations for the cosine objective are very large (e.g., Air 0.446 ± 0.159, Concrete 0.468 ± 0.201); reporting confidence intervals or effect sizes would help readers assess the stability of the MO advantage.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the central SO-versus-MO comparison is empirical and self-contained.

full rationale

The paper's central claim is an empirical comparison of average hypervolume between SO Modular GP-GOMEA with a logging-only MO archive and MO Modular GP-GOMEA with proposed clustering improvements. The results are measured on five real-world datasets and five synthetic datasets with statistical testing; no prediction is derived from a fitted parameter or from the definition of the objectives. The proposed BKRR and BKmRR clustering methods are algorithmic proposals tested by experiment, not quantities inferred from the outcome they are used to explain. The Discussion's acknowledgment that different initialization (e.g., size-uniform) could change the ranking is an honest limitation of generality, not a circular step, because the paper does not define SO superiority into the setup. Self-citations to the authors' prior Modular GP-GOMEA [6] and MO GP-GOMEA [20] supply baseline algorithms and synthetic benchmark generators, but the present paper re-executes these algorithms and compares against independent datasets, so the citations are not load-bearing substitutes for evidence. No uniqueness theorem is imported, and no known result is renamed. The only mild concern is that the recommendation is contingent on half-and-half initialization, which the paper itself flags; this is a correctness or robustness risk, not circularity.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central comparison is empirical, so the main free parameters are experimental settings (population, clusters, time budget) and ad hoc objective definitions (de-duplicated size, complexity scores). The paper does not introduce new physical or mathematical entities, and its claims are not derivations requiring axioms beyond standard GP-GOMEA background.

free parameters (5)

Population size = 4096
Fixed by hand for all experiments; affects the comparison and the 5x population robustness check.
Number of clusters k = 5
Chosen for the MO variant; the paper notes cluster division determines extreme vs middle clusters.
Time budget = 3 hours
Stopping criterion that limits all runs; the SO-vs-MO ranking is time-budget dependent, acknowledged by the authors.
De-duplicated size discount = subtract all duplicate non-leaf nodes
Ad hoc objective modification in Section 3.6 to encourage reuse; no principled derivation is given.
Operator complexity scores = Table 13 values
Hand-assigned scores in Appendix A used only for the complexity-measure experiment in Section 3.7.2.

assumptions (3)

domain assumption Standard assumptions of GP-GOMEA as described in prior work [22]
The paper builds on the fixed-template, FOS, and optimal mixing machinery without re-deriving it.
domain assumption Datasets are representative of symbolic regression tasks
Five real-world and five synthetic datasets are used; the synthetic ones are assumed to have known ground truth from [6].
standard math Hypervolume with normalized objectives is a valid comparison metric
The paper uses average hypervolume as primary metric, with normalization and reference point choices described in Section 3.1.

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Cite this review

Pith. "Pith review of A Better Multi-Objective GP-GOMEA -- But do we Need it?." pith.science (2026). https://pith.science/paper/EIVXLCHP

@misc{pith2026250703777,
  author       = {Pith},
  title        = {Pith review of: A Better Multi-Objective GP-GOMEA -- But do we Need it?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EIVXLCHP}},
  note         = {Machine review of arXiv:2507.03777}
}

read the original abstract

In Symbolic Regression (SR), achieving a proper balance between accuracy and interpretability remains a key challenge. The Genetic Programming variant of the Gene-pool Optimal Mixing Evolutionary Algorithm (GP-GOMEA) is of particular interest as it achieves state-of-the-art performance using a template that limits the size of expressions. A recently introduced expansion, modular GP-GOMEA, is capable of decomposing expressions using multiple subexpressions, further increasing chances of interpretability. However, modular GP-GOMEA may create larger expressions, increasing the need to balance size and accuracy. A multi-objective variant of GP-GOMEA exists, which can be used, for instance, to optimize for size and accuracy simultaneously, discovering their trade-off. However, even with enhancements that we propose in this paper to improve the performance of multi-objective modular GP-GOMEA, when optimizing for size and accuracy, the single-objective version in which a multi-objective archive is used only for logging, still consistently finds a better average hypervolume. We consequently analyze when a single-objective approach should be preferred. Additionally, we explore an objective that stimulates re-use in multi-objective modular GP-GOMEA.

Figures

Figures reproduced from arXiv: 2507.03777 by the authors.

**Figure 1.** Example of the GOM procedure. Orange subscript [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 3.** Example of clusters (similar symbols) and corre [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Individuals in the MO elitist archives in all 10 ex [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Two issues in MO GP-GOMEA with the original [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: Example of a balanced clustering from our proposed [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: Comparison of average hypervolume versus gener [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: Comparison of average hypervolume versus time [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 9.** Figure 9: Scatter plot of objective values at the start (left) and [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: Scatter plot of objective values at the start (left) [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: Illustration of (functional) subexpression (re-)use. [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [18]

Pablo Ramos Criado, D Barrios Rolanía, Daniel Manrique, and Emilio Serrano

work page

[2] [11]

Guillaume Lample and François Charton. 2019. Deep learning for symbolic mathematics. arXiv preprint arXiv:1912.01412 (2019)

arXiv 2019

[3] [1]

Peter AN Bosman. 2010. The anticipated mean shift and cluster registration in mixture-based EDAs for multi-objective optimization. In Proceedings of the 12th annual conference on Genetic and evolutionary computation . 351–358

work page 2010

[4] [2]

Anton Bouter, Ngoc Hoang Luong, Cees Witteveen, Tanja Alderliesten, and Peter A. N. Bosman. [n. d.]. The multi-objective real-valued gene-pool optimal mixing evolutionary algorithm. In Proceedings of the Genetic and Evolutionary Computation Conference (New York, NY, USA, 2017-07). ACM, 537–544. https: //doi.org/10.1145/3071178.3071274

arXiv 2017

[5] [3]

Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and TAMT Meyarivan. 2002. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation 6, 2 (2002), 182–197. A Better Multi-Objective GP-GOMEA - But do we Need it? GECCO ’25 Companion, July 14–18, 2025, Malaga, Spain

work page 2002

[6] [4]

Finale Doshi-Velez and Been Kim. 2017. Towards a rigorous science of inter- pretable machine learning. arXiv preprint arXiv:1702.08608 (2017)

arXiv 2017

[7] [5]

Ilan Gronau and Shlomo Moran. 2007. Optimal implementations of UPGMA and other common clustering algorithms. Information processing letters 104, 6 (2007), 205–210

work page 2007

[8] [6]

Bosman, and Tanja Alderliesten

Joe Harrison, Peter A.N. Bosman, and Tanja Alderliesten. 2025. Thinking Outside the Template with Modular GP-GOMEA. arXiv preprint arXiv2505.01262 (2025)

arXiv 2025