REVIEW 3 major objections 5 minor 49 references
rEGGression: an Interactive and Agnostic Tool for the Exploration of Symbolic Regression Models
T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read rEGGression stores the entire history of a symbolic regression search in an e-graph, letting users query, filter, and mine thousands of alternative models and their building blocks.
desk verdict A working e-graph-based query tool for SR histories, with a real caveat: the headline distribution feature reports only generic wildcard tree shapes, not the concrete recurring terms it promises to surface. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the e-graph data structure, in which e-classes group provably equivalent sub-expressions and e-nodes point to child e-classes rather than to concrete tokens. This representation shares duplicated sub-trees across all stored expressions, which is what makes it feasible to hold hundreds of thousands of candidates in memory. The e-classes are indexed by a trie mapping operator tokens to e-class ids and their children; that index supports e-matching, the pattern-matching algorithm that returns every substitution map for a pattern such as `v0 + v1`, and it is used both for filtering expressions and for counting building-block occurrences. The same machinery also computes the distribution of building blocks up to a requested size, by recursively decomposing each expression's tree into pattern variables.
What would settle it
Run the same regression dataset under, say, 30 different random seeds of the same symbolic regression algorithm, build a separate e-graph for each, and compare the top ten building blocks by count and by average fitness. If the rankings vary substantially between seeds, the building-block distribution is an artifact of the search trajectory rather than a stable property of the data, and the tool's interpretive value is limited to diagnostics of the search itself.
Extended reading notes
Core claim
The central claim is that the e-graph, a data structure invented for equality saturation in program optimization, can be repurposed as the backend of an interactive, algorithm-agnostic symbolic regression explorer. The paper demonstrates the design and commands of rEGGression, showing that storing tens of thousands of visited expressions (in the example, 100,000) in an e-graph keeps the database compact enough to query interactively, with pattern-matching queries returning results in reasonable time. The authors argue that by counting building-block patterns and ranking them by count or average fitness, users can find recurring structural motifs that drive accurate fits, and can then use those motifs to formulate new expressions to insert and test. Because the tool imports expression files from eleven common symbolic regression systems, the analysis applies across different search algorithms and hyper-parameter settings, making the exploration independent of any single search method.
Load-bearing premise
The usefulness of the building-block statistics rests on the assumption that the set of expressions produced by the symbolic regression runs is representative enough for pattern frequencies and average fitnesses to be stable; the paper gives no evidence that these statistics do not change drastically across random seeds or algorithm configurations.
Editorial extensions
If this is right
- Users can impose structural prior knowledge after the search—requiring a pattern like `g(x)*exp(h(x))*cos(k(x))` or forbidding a recurrent motif—without re-running the symbolic regression algorithm.
- Counting building blocks and averaging the fitness of expressions containing each one provides a data-driven shortlist of sub-expressions that contribute to accuracy.
- Combining expressions from multiple symbolic regression runs and algorithms into one e-graph gives a larger, more diverse library of alternative models than any single Pareto front.
- The ability to insert new expressions and re-optimize their parameters lets users test hypotheses formed from discovered building blocks immediately, closing a loop between analysis and search.
Reading between the lines
- The reported building-block frequencies and average fitnesses are statistics over the search history rather than over all plausible models; comparing their rankings across random seeds and algorithm configurations on a fixed dataset would show whether they reflect stable properties of the data or artifacts of the search trajectory.
- The same pattern index could power an automated model-proposal step, listing the highest-fitness visited sub-expressions within a given size budget and suggesting recombinations, turning the tool from a query interface into a generator of new hypotheses.
- Storing semantic properties per e-class—positivity, monotonicity, and measurement units—which the paper lists as future work, would allow constraint-based filtering and would let the tool enforce domain knowledge that cannot be expressed by syntactic patterns alone.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces rEGGression, a command-line tool that stores symbolic regression (SR) expressions in an e-graph data structure and provides interactive querying, filtering, pattern matching, and building-block distribution analysis. The tool can import expressions from several popular SR implementations and supports operations such as top-N retrieval with size/parameter filters, pattern matching with repeated pattern variables, subtree inspection, expression insertion, parameter optimization, Pareto-front display, and building-block counting. The authors demonstrate the tool on the nasa_battery_1 dataset using expressions produced by the eggp algorithm, and they describe the underlying e-graph pattern-matching algorithm with complexity bounds. The main claimed contribution is the ability to explore large sets of SR models and, in particular, to discover recurring building blocks that provide insights about the studied phenomenon.
Significance. If the tool works as described, it would fill a genuine gap: most SR tools only expose a final Pareto front or population, whereas rEGGression allows users to query and analyze the entire history of visited expressions from multiple algorithms. The e-graph-based storage and pattern matching, with support for repeated pattern variables, is a sensible and potentially efficient approach, and the open-source implementation and broad import format support are concrete strengths. The paper also provides a clear walk-through of all commands. However, the central 'insight' claim is currently not substantiated: the demonstrated building-block distributions consist of generic wildcard tree shapes rather than concrete recurring mathematical terms, and the paper offers no quantitative evidence of efficiency or stability of the reported statistics. These issues are fixable but are load-bearing for the paper's main promise.
major comments (3)
- [Section 4.9, Table 1, Algorithm 1] The building-block distribution command produces only generic wildcard shapes such as (v0+(v1+v2)) and (v0+v1). Because Algorithm 1 replaces every leaf (terminal node) with a pattern variable v, any concrete term such as sin(x0) is subsumed into the pattern v0+(v1+v2) and its frequency is aggregated away. Consequently, the output in Table 1 is dominated by abstract tree shapes that would appear in any grammar-constrained expression collection, independent of the data or the SR algorithm. The paper's main highlight—that building-block exploration 'can help the experts to find insights about the studied phenomena' (Section 4.12)—is therefore not demonstrated. I recommend either modifying the extraction to preserve concrete terminals (e.g., by parameterizing the pattern variables with the cheapest concrete leaf) or providing a demonstration where a concrete, domain-specific term (e.g., sin(x0), exp(x1)) appears prominently in the distribution and is shown to lead to a useful query or new model.
- [Section 4.9] The statistics reported by the distribution command are computed from the 'top y evaluated expressions' of a single run of a single algorithm (eggp on nasa_battery_1). The paper provides no argument or experiment showing that building-block frequencies and average-fitness rankings are stable across random seeds, hyperparameter settings, or different SR algorithms. Without such evidence, the reported 'insights' may be artifacts of the particular search history rather than properties of the data. Please either add a small stability analysis (e.g., repeat the distribution computation over multiple seeds and report rank correlations) or explicitly limit the claim to describing the search history rather than the phenomenon.
- [Section 4.10 and abstract] The abstract and introduction state that e-graphs allow SR solution candidates to be stored and queried 'efficiently,' but the paper gives no quantitative evidence for this. The only data point is the note in Section 4.10 that 100,000 expressions produced a 200 MB save file. There are no measurements of e-graph construction time, pattern-matching latency (e.g., for count-pattern or distribution on a large database), or memory scaling as the number of expressions grows. Since the tool is positioned as an interactive exploration environment, I suggest including a small benchmark with varying numbers of expressions (e.g., 1k, 10k, 100k) measuring build time, query time for representative patterns, and memory usage. This would substantiate the efficiency claim and help users understand practical limits.
minor comments (5)
- [Section 1] The phrase 'semantically similar int the training set' contains a typo; it should be 'semantically similar in the training set'.
- [Section 4.4] In the subtrees example, the listed subtrees of (θ0*x3)+(θ1*x1) include 'θ0 x1', which is not a subtree, and omit the subtree 'θ1*x1'. The list should be 'θ0*x3, θ1*x1, θ0, θ1, x1, x3'.
- [Section 4.9] The text 'by diving the sum of fitness with the frequency of that pattern' should be 'by dividing the sum of fitness by the frequency of that pattern'.
- [Algorithm 1] The comment on line 1 says 'build block' but should read 'building block'.
- [Section 4.10] The import format description says the file is 'comma separated value format' but the second column is 'semicolon separated'; clarify how the CSV fields and the internal coefficient list separator interact.
Circularity Check
No circularity: rEGGression's outputs are direct enumerations and aggregates over stored expressions; no fitted input is renamed as a prediction.
full rationale
This is a systems/tool paper rather than a derivation paper. The central functionality—querying expressions, filtering by patterns, and reporting building-block distributions—is implemented by direct operations on the e-graph built from the imported or generated expressions. The building-block distribution in Section 4.9 is exactly what it claims to be: Algorithm 1 recursively enumerates subtree patterns of each selected expression, the frequencies are counted from those expressions, and the average fitness is computed by dividing the sum of fitness values by the frequency. Table 1 is a direct aggregation of those counts, not a fitted parameter or a prediction. No step in the paper derives a quantity from a definition of that same quantity, and no fitted value is later relabeled as an independent result. The pattern-matching and e-graph machinery is grounded in external references [43, 44] and the egg library, while the authors' own prior e-graph SR work [11, 24] is cited only as background for using equality saturation in symbolic regression; it is not used to justify the tool's outputs. The skeptic concern that wildcard patterns such as (v0+(v1+v2)) dominate the distribution is a potential limitation of the insightfulness of the reported building blocks, but it is not a circularity: the tool genuinely computes the stated statistic from the stored expressions. There is no self-citation chain that forces the central claim, no uniqueness theorem imported to forbid alternatives, and no ansatz smuggled in via citation. Thus no significant circularity is present.
Assumptions & free parameters
assumptions (3)
- standard math E-graph data structure correctly represents equivalence classes of expressions and supports the described pattern matching in O(sqrt(q n m)) time.
- domain assumption The collection of expressions produced by an SR run or multiple runs is a representative sample for building-block analysis.
- domain assumption Imported expressions from different SR algorithms are directly comparable and correctly parsed.
Cite this review
Pith. "Pith review of rEGGression: an Interactive and Agnostic Tool for the Exploration of Symbolic Regression Models." pith.science (2026). https://pith.science/paper/EDFRRI63
@misc{pith2026250117859,
author = {Pith},
title = {Pith review of: rEGGression: an Interactive and Agnostic Tool for the Exploration of Symbolic Regression Models},
year = {2026},
howpublished = {\url{https://pith.science/paper/EDFRRI63}},
note = {Machine review of arXiv:2501.17859}
}
read the original abstract
Regression analysis is used for prediction and to understand the effect of independent variables on dependent variables. Symbolic regression (SR) automates the search for non-linear regression models, delivering a set of hypotheses that balances accuracy with the possibility to understand the phenomena. Many SR implementations return a Pareto front allowing the choice of the best trade-off. However, this hides alternatives that are close to non-domination, limiting these choices. Equality graphs (e-graphs) allow to represent large sets of expressions compactly by efficiently handling duplicated parts occurring in multiple expressions. E-graphs allow to store and query all SR solution candidates visited in one or multiple GP runs efficiently and open the possibility to analyse much larger sets of SR solution candidates. We introduce rEGGression, a tool using e-graphs to enable the exploration of a large set of symbolic expressions which provides querying, filtering, and pattern matching features creating an interactive experience to gain insights about SR models. The main highlight is its focus in the exploration of the building blocks found during the search that can help the experts to find insights about the studied phenomena.This is possible by exploiting the pattern matching capability of the e-graph data structure.
Figures
Reference graph
Works this paper leans on
-
[1]
Deaglan J Bartlett, Harry Desmond, and Pedro G Ferreira. 2023. Exhaustive symbolic regression. IEEE Transactions on Evolutionary Computation (2023). rEGGression: an Interactive and Agnostic Tool for the Exploration of Symbolic Regression Models GECCO ’25, July 14–18, 2025, Malaga, Spain
work page 2023
-
[2]
Chris M Bishop. 1994. Neural networks and their applications.Review of scientific instruments 65, 6 (1994), 1803–1832
work page 1994
-
[3]
Bogdan Burlacu, Gabriel Kronberger, and Michael Kommenda. 2020. Operon C++ an efficient genetic programming framework for symbolic regression. In Proceed- ings of the 2020 Genetic and Evolutionary Computation Conference Companion . 1562–1570
work page 2020
-
[4]
Tianqi Chen and Carlos Guestrin. 2016. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining . 785–794
2016
-
[5]
Miles Cranmer. 2023. Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl. https://doi.org/10.48550/ARXIV.2305.01582
-
[6]
Maxime Crochemore and Thierry Lecroq. 2009. Trie. Encyclopedia of Database Systems (2009), 3179–3182
work page 2009
-
[7]
Jason M. Daida, Robert R. Bertram, John A. Polito 2, and Stephen A. Stanhope
-
[8]
Fabrício Olivetti de França. 2022. Transformation-interaction-rational represen- tation for symbolic regression. In Proceedings of the Genetic and Evolutionary Computation Conference. 920–928
work page 2022
Show all 49 references
-
[9]
Fabricio Olivetti De Franca. 2023. Fighting Underspecification in Symbolic Re- gression with Fitness Sharing. In Proceedings of the Companion Conference on Genetic and Evolutionary Computation . 551–554
2023
-
[10]
Fabricio Olivetti de Franca and Maira Zabuscha de Lima. 2021. Interaction- transformation symbolic regression with extreme learning machine. Neurocom- puting 423 (2021), 609–619
2021
-
[11]
Fabricio Olivetti de Franca and Gabriel Kronberger. 2023. Reducing Overparame- terization of Symbolic Regression Models with Equality Saturation. InProceedings of the Genetic and Evolutionary Computation Conference . 1064–1072
2023
-
[12]
Fabricio Olivetti de Franca and Gabriel Kronberger. 2025. Improving Genetic Programming for Symbolic Regression with Equality Graphs. In Proceedings of the Genetic and Evolutionary Computation Conference (Malaga, Spain) (GECCO ’25). Association for Computing Machinery, New Yor...
2025
-
[13]
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
2002
-
[14]
Achiya Elyasaf and Moshe Sipper. 2014. Software review: the HeuristicLab framework. Genetic Programming and Evolvable Machines 15 (2014), 215–218
2014
-
[15]
Andrew Gelman. 2007. Data analysis using regression and multilevel/hierarchical models. Cambridge university press
2007
-
[16]
Christian Haider and Gabriel Kronberger. 2022. Shape-constrained symbolic regression with NSGA-III. In International Conference on Computer Aided Systems Theory. Springer, 164–172
2022
-
[17]
Trevor J Hastie. 2017. Generalized additive models. In Statistical models in S . Routledge, 249–307
2017
-
[18]
Ralf Hinze and Ross Paterson. 2006. Finger trees: a simple general-purpose data structure. Journal of functional programming 16, 2 (2006), 197–217
2006
-
[19]
Steven G Johnson and Julien Schueller. 2021. Nlopt: Nonlinear optimization library. Astrophysics Source Code Library (2021), ascl–2111
2021
-
[20]
Michael Kommenda, Andreas Beham, Michael Affenzeller, and Gabriel Kron- berger. 2015. Complexity measures for multi-objective symbolic regression. In Computer Aided Systems Theory–EUROCAST 2015: 15th International Conference, Las Palmas de Gran Canaria, Spain, February 8-13, 2...
2015
-
[21]
Mark E Kotanchek, Ekaterina Vladislavleva, and Guido Smits. 2013. Symbolic regression is not enough: it takes a village to raise a model. Genetic Programming Theory and Practice X (2013), 187–203
2013
-
[22]
John R. Koza. 1992. Genetic Programming: On the Programming of Computers by Means of Natural Selection . MIT Press, Cambridge, MA, USA
1992
-
[23]
Winkler, and Michael Affenzeller
Gabriel Kronberger, Bogdan Burlacu, Michael Kommenda, Stephan M. Winkler, and Michael Affenzeller. 2024. Symbolic Regression. Chapman & Hall / CRC Press
2024
-
[24]
Gabriel Kronberger and Fabrício Olivetti de França. 2024. Effects of reducing redundant parameters in parameter optimization for symbolic regression using genetic programming. Journal of Symbolic Computation (2024), 102413
2024
-
[25]
Gabriel Kronberger, Fabricio Olivetti de França, Bogdan Burlacu, Christian Haider, and Michael Kommenda. 2022. Shape-constrained symbolic regres- sion—improving extrapolation with prior knowledge. Evolutionary Computation 30, 1 (2022), 75–98
2022
-
[26]
Bartlett, and Lukas Kammerer
Gabriel Kronberger, Fabricio Olivetti de Franca, Harry Desmond, Deaglan J. Bartlett, and Lukas Kammerer. 2024. The Inefficiency of Genetic Programming for Symbolic Regression. In Parallel Problem Solving from Nature – PPSN XVIII , Michael Affenzeller, Stephan M. Winkler, Anna ...
2024
-
[27]
William La Cava, Tilak Raj Singh, James Taggart, Srinivas Suri, and Jason H Moore
-
[28]
William La Cava, Lee Spector, and Kourosh Danai. 2016. Epsilon-lexicase selec- tion for regression. In Proceedings of the Genetic and Evolutionary Computation Conference 2016. 741–748
2016
-
[29]
Viktor Martinek, Julia Reuter, Ophelia Frotscher, Sanaz Mostaghim, Markus Richter, and Roland Herzog. 2024. Shape Constraints in Symbolic Regression using Penalized Least Squares. arXiv preprint arXiv:2405.20800 (2024)
2024 arXiv
-
[30]
Nicholas Freitag McPhee, Brian Ohs, and Tyler Hutchison. 2008. Semantic building blocks in genetic programming. In Genetic Programming: 11th European Conference, EuroGP 2008, Naples, Italy, March 26-28, 2008. Proceedings 11 . Springer, 134–145
2008
-
[31]
Una-May O’Reilly and Franz Oppacher. 1995. The troubling aspects of a building block hypothesis for genetic programming. In Foundations of genetic algorithms . Vol. 3. Elsevier, 73–88
1995
-
[32]
Miltiadis Poursanidis, Patrick Link, Jochen Schmid, and Uwe Teicher. 2024. Incorporating Shape Knowledge into Regression Models. arXiv preprint arXiv:2409.17084 (2024)
2024 arXiv
-
[33]
David L Randall, Tyler S Townsend, Jacob D Hochhalter, and Geoffrey F Bomarito
-
[34]
Julia Reuter, Viktor Martinek, Roland Herzog, and Sanaz Mostaghim. 2024. Unit- Aware Genetic Programming for the Development of Empirical Equations. In International Conference on Parallel Problem Solving from Nature . Springer, 168– 183
2024
-
[35]
Justinian P Rosca. 1995. Towards automatic discovery of building blocks in genetic programming. In Working Notes for the AAAI Symposium on Genetic Programming, Vol. 445. MIT, Cambridge, MA, USA: AAAI, 78–85
1995
-
[36]
Guido F Smits and Mark Kotanchek. 2005. Pareto-front exploitation in symbolic regression. Genetic programming theory and practice II (2005), 283–299
2005
-
[37]
Petre Stoica and Yngve Selen. 2004. Model-order selection: a review of information criterion rules. IEEE Signal Processing Magazine 21, 4 (2004), 36–47
2004
-
[38]
Ross Tate, Michael Stepp, Zachary Tatlock, and Sorin Lerner. 2009. Equality saturation: a new approach to optimization. In Proceedings of the 36th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages . 264– 276
2009
-
[39]
Silviu-Marian Udrescu and Max Tegmark. 2020. AI Feynman: A physics-inspired method for symbolic regression. Science Advances 6, 16 (2020), eaay2631
2020
-
[40]
Marco Virgolin, Tanja Alderliesten, Cees Witteveen, and Peter AN Bosman
-
[41]
Marco Virgolin, Tanja Alderliesten, Cees Witteveen, and Peter A. N. Bosman
-
[42]
Stefan Wagner and Michael Affenzeller. 2005. Heuristiclab: A generic and exten- sible optimization environment. In Adaptive and Natural Computing Algorithms: Proceedings of the International Conference in Coimbra, Portugal, 2005 . Springer, 538–541
2005
-
[43]
Max Willsey, Chandrakana Nandi, Yisu Remy Wang, Oliver Flatt, Zachary Tat- lock, and Pavel Panchekha. 2021. Egg: Fast and extensible equality saturation. Proceedings of the ACM on Programming Languages 5, POPL (2021), 1–29
2021
-
[44]
YIHONG ZHANG, YISU REMY WANG, MAX WILLSEY, and ZACHARY TAT- LOCK. 2021. Relational E-matching. arXiv preprint arXiv:2108.02290 (2021)
2021 arXiv
-
[1999]
In Advances in Genetic Programming 3 , Lee Spector, William B
Analysis of Single-Node (Building) Blocks in Genetic Programming. In Advances in Genetic Programming 3 , Lee Spector, William B. Langdon, Una-May O’Reilly, and Peter J. Angeline (Eds.). MIT Press, Cambridge, MA, USA, Chapter 10, 217–241. https://doi.org/doi:10.7551/mitpress/11...
-
[2017]
In Proceedings of the Genetic and Evolutionary Computation Conference
Scalable genetic programming by gene-pool optimal mixing and input- space entropy-based building-block learning. In Proceedings of the Genetic and Evolutionary Computation Conference. 1041–1048
-
[2018]
arXiv preprint arXiv:1807.00981 (2018)
Learning concise representations for regression by evolving networks of trees. arXiv preprint arXiv:1807.00981 (2018)
2018 arXiv
-
[2021]
Evolutionary Computation 29, 2 (2021), 211–237
Improving model-based genetic programming for symbolic regression of small expressions. Evolutionary Computation 29, 2 (2021), 211–237
2021
-
[2022]
In Proceedings of the Genetic and Evolutionary Computation Conference Companion
Bingo: a customizable framework for symbolic regression with genetic pro- gramming. In Proceedings of the Genetic and Evolutionary Computation Conference Companion. 2282–2288
Reviewed August 10, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.