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(Exhaustive) Symbolic Regression and model selection by minimum description length

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Pairing an exhaustive enumeration of functions with an information-theoretic score fixes symbolic regression and beats standard formulas on three astrophysics problems.

desk verdict A clear self-review of ESR+MDL with honest caveats, but the abstract's 'superior to literature standards' claims ride on a subjective complexity prior and an unproven dedup completeness. read the letter →

arxiv 2507.13033 v1 pith:NI7CSWWH submitted 2025-07-17 astro-ph.IM astro-ph.COastro-ph.GAcs.LG

Harry Desmond This is my paper
classification astro-ph.IMastro-ph.COastro-ph.GAcs.LG
keywords symbolicregressionminimumdescriptionlengthexhaustivesearchmodelselectionBayesianevidenceinformationtheorycosmologyMOND
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

Symbolic regression searches for equations with random genetic algorithms and then ranks candidates on an accuracy-versus-complexity Pareto front; this review argues that both steps are defective, because the search has an unknown chance of missing the best function and the final selection needs an arbitrary heuristic. The proposed fix is to enumerate every function built from a user-chosen operator basis up to a maximum tree complexity and score each one by minimum description length, the number of nats needed to transmit the function plus the residuals of the data. The paper reports that on three mature astrophysics problems the exhaustive search finds many functions with lower description length than the accepted literature formulas: the Friedmann equation, the MOND interpolating functions, and common inflaton potentials. The paper's central claim is that, within the user-defined operator budget, these standard formulas are not the best-supported simple descriptions of the data, and that an objective, reproducible ranking of every function is in reach.

What carries the argument

The load-bearing object is the exhaustive generator paired with the description-length score. ESR grows every possible tree template, labels nodes by arity, decorates the trees with every permutation of operators from a user basis set up to a maximum node count, then simplifies the list to unique representatives using comparison rules that include tree reordering, parameter permutations, simplifications, reparametrisation invariance, and parameter combinations. Each unique function is then scored by $$L(D)=-\log L(\hat{\$\theta$})+k\log n-\frac{p}{2}\log 3+\sum_j\log c_j+\sum_i\left[\frac{1}{2}\log I_{ii}+\log|\hat\theta_i|\right],$$ where $k\log n$ charges for the operators, the $c_j$ charge for natural-number constants from simplification, and the final sum codes the maximum-likelihood parameters at optimal precision using the observed Fisher information. The paper shows this equals the negative of the Bayesian evidence (up to a normalisation constant) provided the functional prior is $-\log P(f_i)=k\log n+\sum_\alpha\log c_\alpha$, making the ranking a specific, and replaceable, subjective choice.

What would settle it

Run ESR-MDL on a noiseless synthetic dataset generated by a known function of complexity at most 10 built from the test operator basis; if the true generator or a member of its behavioural equivalence class is not at the top of the description-length ranking, the exhaustive-and-correct claim fails. A cheaper check is to scan the enumerated 'unique' function set for two structurally different trees that are numerically identical over a dense grid, since any such pair shows the duplicate-removal rules are incomplete and subsequent description lengths are mis-assigned.

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Extended reading notes

Core claim

The paper claims that the two longstanding defects of symbolic regression are curable together. The first defect is that genetic-programming searches have an unknown probability of missing the best function; the second is that ranking by a Pareto front of likelihood versus a 'complexity' heuristic leaves the final choice arbitrary. The remedy is Exhaustive Symbolic Regression (ESR): generate every tree of operators from a user-specified basis up to a maximum node count, reduce the list to behaviourally unique functions with comparison rules, fit the free parameters, and rank every unique function by its description length, the number of nats needed to transmit the function and the residuals. On the benchmark dataset feynman_I_6_2a, only ESR finds the true generator, a standard normal, in its simplest form at complexity 7, while the comparison algorithms either miss it or find it overparametrised. Applied to cosmology, the MDL function for the expansion rate is $H^2(z)=\theta_0(1+z)^2$ for cosmic chronometers and $H^2(z)=\theta_0(1+z)^{1+z}$ for Pantheon+ supernovae, preferred over the simple Friedmann equation by 7.12 and 4.91 nats, with 38 and 36 competitor functions ranked above it. For the radial acceleration relation the best functions beat the classic MOND interpolating functions but usually lack a deep-MOND limit; for inflation, the $k\log n$ prior favours $\exp(-\exp(\exp(\exp(\phi))))$ at complexity 6, while a Katz prior trained on known inflationary potentials favours $\theta_0(\theta_1+\log(\phi)^2)$ or $\theta_0\phi^{\theta_1/\phi}$. The author's conclusion is that in all three applications many functions superior to the literature standards exist, and that the method supplies a principled, reproducible ranking in which no good function within the budget is missed.

Load-bearing premise

The system's guarantee depends on its simplification and duplicate-removal rules catching every structurally different function that behaves identically; if any equivalent pair slips through, the enumeration is not exhaustive and the description lengths, which depend on tree structure, are assigned to the wrong representatives.

Editorial extensions

If this is right

  • On feynman_I_6_2a, an algorithm whose Pareto front stalls at complexity 5 would miss the true data generator, whereas ESR finds it at complexity 7, so stochastic-search failures become measurable rather than unknown.
  • Because MDL puts accuracy and complexity in the same units, no second heuristic is needed to pick a function from a Pareto front, and the location of the minimum tells the user whether the complexity cap was high enough.
  • The Friedmann equation is not the MDL-preferred $H(z)$ function for either dataset: 38 functions beat it for cosmic chronometers and 36 for Pantheon+, but the winning functions share its low-redshift Taylor expansion, so better data are required to distinguish them.
  • For the radial acceleration relation, most MDL-best functions have a Newtonian limit but plateau at low $g_{\rm bar}$ instead of following the deep-MOND $x^{-1/2}$ limit, and the paper shows the same would occur on MOND-generated mock data, so the RAR does not currently single out MOND's functional form.
  • For inflation, the ranking is prior-sensitive: the $k\log n$ prior crowns $\exp(-\exp(\exp(\exp(\phi))))$, while a Katz prior trained on known inflationary potentials favours physically more plausible functions, and literature standards such as Starobinsky, quadratic, and quartic place far down the list.

Reading between the lines

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

  • The method turns 'which fitting function is best?' into a finite computation only up to the complexity cap, so the cap and operator basis should be reported alongside every ranking; the exponential scaling means the practical ceiling is near complexity 10, and claims about functions beyond it are out of scope.
  • Because the description-length score is a Bayesian evidence with an explicit functional prior, the headline preferences of 7.12 and 4.91 nats should be read as conditional on the $k\log n$ prior; readers who care about physical plausibility can and should re-run the ranking under the Katz prior, as the inflation section does.
  • The same exhaustive enumeration could serve as a recall benchmark for stochastic methods: given a dataset and operator basis, run a genetic algorithm many times and record what fraction of the top-MDL functions it finds, converting 'unknown probability of missing the best function' into a measured coverage statistic.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes Exhaustive Symbolic Regression (ESR), an algorithm that enumerates all functions built from a user-specified operator basis up to a maximum node-count complexity, together with model selection by the minimum description length (MDL) principle. The MDL score, Eq. 4.1, trades off accuracy against a complexity penalty that the paper shows is equivalent to a Bayesian evidence calculation with a particular functional prior. The paper benchmarks ESR on the feynman_I_6_2a dataset, where it recovers the standard-normal generator, and then applies the method to three astrophysics problems: the cosmic expansion history (Friedmann equation), the radial acceleration relation (MOND interpolating functions), and inflationary potentials. In each application the paper claims that ESR identifies many functions superior to the literature standards.

Significance. If the claims hold, ESR+MDL would be a valuable contribution to symbolic regression: it replaces stochastic search with exhaustive enumeration up to a complexity cap, and it replaces ad hoc Pareto-front selection with an information-theoretic scoring rule. The paper has genuine strengths: the underlying code and pre-computed function sets are publicly available; the feynman_I_6_2a benchmark is a credible positive control; and the mock-data experiment for the MOND radial acceleration relation is an honest out-of-sample check that tempers the astrophysical conclusions. The MDL framework is also usefully connected to Bayesian model selection through Eq. 4.3-4.4. However, the central guarantee of exhaustiveness depends on an unproven completeness property of the function-simplification rules, and the quantitative superiority claims are sensitive to the choice of functional prior, which the paper itself notes is subjective.

major comments (3)
  1. [Sec. 3(a)] The exhaustiveness guarantee is stated unconditionally: Sec. 3(a) says the simplification and comparison rules 'establish the unique functions', and Sec. 4 says 'we are guaranteed not to miss any function'. No completeness proof is given for the rule set (tree reordering, parameter permutations, simplifications, reparametrisation invariance, parameter combinations). For operator bases containing exp, log, sin, sqrt, and pow, behavioral identities are abundant, and any unrecognized equivalence would cause the same function to be scored under multiple tree representations with different node counts k in Eq. 4.1, making the description length and the ranking representation-dependent. This directly affects load-bearing claims such as the numbers of functions better than the Friedmann equation (38 for CCs and 36 for SNe, Sec. 5(a)). Please either supply a completeness argument for the equivalence rules, or state the guarantee as conditional on that rule set and empirically test the sensitivity to missed equivalences.
  2. [Sec. 4, Eq. 4.4] The MDL score is not prior-free. Eq. 4.4 makes explicit that the description length corresponds to the Bayesian evidence only for the specific functional prior -log P(f_i) = k log n + sum_j log c_j, and the text immediately notes that priors are subjective and proposes the Katz back-off prior as a replacement. The quantitative claims in Sec. 5(a) (preferences of 7.12 and 4.91 nats, and the counts 38 and 36 functions better than the Friedmann equation) are therefore statements about one choice of prior, as is the abstract's assertion that the algorithm identifies 'many functions superior to the literature standards'. The inflation application itself shows the sensitivity: the MDL function changes from exp(-exp(exp(exp(phi)))) under k log n to theta_0 (theta_1 + log(phi)^2) under the Katz prior (Sec. 5(c)). The paper should report how the rankings, preference strengths, and counts change under alternative priors, or at minimum qualify all such claims as prior-dependent.
  3. [Sec. 5(a)] The headline conclusion that ESR finds functions 'superior to the literature standards' is potentially misleading in the Friedmann case. The authors themselves state that the MDL functions in Eq. 5.2 have the same Taylor expansion as the Friedmann equation to second order and are preferred because they are simpler. Thus the reported preferences of 7.12 and 4.91 nats are largely a model-selection penalty on complexity rather than evidence that the Friedmann form is empirically inadequate. The abstract and conclusions should carry this qualification; without it, a reader may reasonably interpret 'superior' as meaning better-fitting in a likelihood sense.
minor comments (4)
  1. [Sec. 3(a)] The statement that 'the operator basis set and maximum complexity are the only things that must be specified by the user' is inconsistent with Sec. 4, where the choice of functional prior (k log n versus Katz) and its hyperparameters also affects the results. Please rephrase to say these are the only function-space choices, with model-selection choices discussed separately.
  2. [Sec. 3(b)] The conclusion that 'most SR algorithms fail even on very simple problems' is based on a single benchmark dataset (feynman_I_6_2a). Consider softening the generalization or citing a broader comparative study to support it.
  3. [References] Reference [3] should give the full author name (D. E. Goldberg), reference [4] contains a typo ('Wyley' for 'Wiley'), and reference [18] gives the edition year incorrectly ('Wiley 2nd edition' should likely be 2006, not 1991).
  4. [Sec. 5(b), Fig. 3] The figure caption refers to blue/red/cyan/magenta slopes; please ensure the printed version is readable for color-blind readers by adding line styles or labels to the data series.

Circularity Check

1 steps flagged · score 4.0 of 10

The exhaustive-search guarantee is imported from the authors' own prior work, but the applications are openly benchmarked and the MDL prior is acknowledged as subjective, so the circularity is partial rather than total.

  1. self citation load bearing [Sec. 3(a) (function simplification and uniqueness) and Sec. 4 (restatement of exhaustiveness)]
    "We then simplify the functions and remove duplicates using a set of function-comparison rules (tree reordering, parameter permutations, simplifications, reparametrisation invariance, parameter combinations). This establishes the unique functions... Full details may be found in [9]."

    The central methodological claim, 'with ESR we are guaranteed not to miss any function', depends on the completeness of the deduplication and simplification rules: only if every behavioural equivalence is recognized can the enumeration be exhaustive and the function counts in the applications be meaningful. The paper offers no completeness proof; it asserts that the rules 'establish the unique functions' and refers to [9], a paper whose authors include the present author. The external feynman_I_6_2a benchmark is genuine out-of-sample evidence for one successful recovery, but it does not verify exhaustiveness or the completeness of the equivalence rules. Thus the load-bearing guarantee is effectively inherited from a self-citation rather than derived or independently checked.

full rationale

The paper is not fundamentally circular: the MDL formula of Eq. 4.1 is derived from an explicit coding argument plus a stated functional prior, and the equivalence with Bayesian evidence (Eqs. 4.2-4.4) is shown rather than assumed. The astronomical applications compare external datasets (cosmic chronometers, Pantheon+ supernovae, Planck constraints) and the authors transparently note that the MDL preference for simpler functions is a feature of the chosen prior, which they describe as subjective. The mock-data test in Sec. 5(b) is an honest control: it shows that even MOND-generated data would not exhibit the deep-MOND limit among the best ESR functions, which directly addresses the obvious objection that the method is rigged against MOND. The main circularity concern is narrower: the exhaustive-search guarantee rests on the completeness of the function-comparison rules, and that completeness is supported only by a reference to the authors' own prior work [9]. Since the feynman_I_6_2a benchmark demonstrates a success but not exhaustiveness, and no machine-checked proof of the simplification rules is provided, the core 'guaranteed not to miss any function' premise is partially self-citational. This warrants a score of 4: some self-citation is load-bearing, but the central applications still contain substantial independent, externally falsifiable content.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central claims rest on a handful of user-chosen inputs and unproved background assumptions. The user inputs are the complexity cap (typically 10), the per-application operator basis sets, the functional prior (k log n versus Katz), and the unreported Katz hyperparameters. The background assumptions are standard for the field: a Gaussian Laplace approximation, Shannon-Fano coding, correctly specified astrophysical likelihoods, slow-roll inflation kinematics, and MOND's asymptotic limits. The most fragile item is the completeness of the simplification rules in Sec. 3(a), because the entire exhaustiveness guarantee depends on it. No invented physical entities are required.

free parameters (4)
  • maximum complexity cap = 10 (typical); 13 planned for ESR 2.0
    Chosen by computational budget. The exhaustiveness guarantee and all headline functions are confined to functions at or below this cap; exponential scaling prevents pushing much beyond 10 (Sec. 3(a)).
  • operator basis set per application = e.g. {x=1+z, theta, inv, +, -, x, /, pow} for H(z); Sets A and B for inflation
    Chosen by hand; results are basis-dependent. For inflation the MDL function changes with the basis set (theta0(theta1 + log(phi)^2) for Set A versus theta0 phi^(theta1/phi) for Set B under the Katz prior).
  • Katz back-off model hyperparameters = not reported
    Two hyperparameters (combination length and minimum occurrence count for direct probability estimation) control the learned prior; no values are given, so the Katz-based inflation results cannot be reproduced from the text alone.
  • functional prior choice (k log n versus Katz) = k log n for H(z) and RAR; both variants for inflation
    The rankings, the nat preferences, and the counts of 'functions better than the literature standards' are all evaluated under a chosen prior; the paper notes priors are subjective (Sec. 4).
assumptions (7)
  • domain assumption Completeness of the function-equivalence rules used to deduplicate the search
    Sec. 3(a): the exhaustiveness guarantee holds only if tree reordering, parameter permutations, simplifications, reparametrisation invariance and parameter combinations identify all behaviorally identical functions; no proof of completeness is given.
  • standard math Laplace approximation to the posterior in Eq. 4.3
    Sec. 4: the claimed equivalence between the description length and the Bayesian evidence assumes the posterior is approximately Gaussian around the maximum-posterior point.
  • standard math Residual description cost equals negative log-likelihood (Shannon-Fano coding)
    Sec. 4: L(D|H) = -log Lhat is taken as the optimal coding cost for residuals; this presumes a correctly specified likelihood for the data.
  • domain assumption Single-field slow-roll inflation with V(phi) constrained only by (A_s, n_s, r)
    Sec. 5(c): the inflaton-potential application reduces all data to three CMB numbers and assumes the slow-roll dynamics of Eq. 5.4.
  • domain assumption MOND's Newtonian and deep-MOND limits define the admissible interpolating-function class
    Sec. 5(b): functions are judged MOND-compatible by the limits nu -> 1 and nu -> x^{-1/2}; the finding that few best functions have the deep-MOND limit is interpreted against this yardstick.
  • domain assumption The likelihoods of the CC, Pantheon+ and Planck datasets are correctly specified
    Secs. 5(a) and 5(c): the nat differences (7.12 and 4.91) and the recovered constraints depend on the Gaussian likelihood and the covariance matrices being right.
  • ad hoc to paper The functional prior -log P(f_i) = k log n + sum_j log c_j (Eq. 4.4)
    Sec. 4: the MDL-Bayesian equivalence fixes this prior, and all H(z) and RAR rankings use it. The paper concedes priors are subjective and provides the Katz alternative only for the inflation application.

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Cite this review

Pith. "Pith review of (Exhaustive) Symbolic Regression and model selection by minimum description length." pith.science (2026). https://pith.science/paper/NI7CSWWH

@misc{pith2026250713033,
  author       = {Pith},
  title        = {Pith review of: (Exhaustive) Symbolic Regression and model selection by minimum description length},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NI7CSWWH}},
  note         = {Machine review of arXiv:2507.13033}
}
read the original abstract

Symbolic regression is the machine learning method for learning functions from data. After a brief overview of the symbolic regression landscape, I will describe the two main challenges that traditional algorithms face: they have an unknown (and likely significant) probability of failing to find any given good function, and they suffer from ambiguity and poorly-justified assumptions in their function-selection procedure. To address these I propose an exhaustive search and model selection by the minimum description length principle, which allows accuracy and complexity to be directly traded off by measuring each in units of information. I showcase the resulting publicly available Exhaustive Symbolic Regression algorithm on three open problems in astrophysics: the expansion history of the universe, the effective behaviour of gravity in galaxies and the potential of the inflaton field. In each case the algorithm identifies many functions superior to the literature standards. This general purpose methodology should find widespread utility in science and beyond.

Figures

Figures reproduced from arXiv: 2507.13033 by the authors.

Figure 1
Figure 1. Pareto front of mean square error against function complexity produced by six SR algorithms on the feynman_I_6_2a benchmark dataset. Only ESR and Operon identify that the data is drawn from a standard normal, and only ESR finds this function in its simplest form. as the PySR score for combining accuracy and complexity [15] is akin to making up a function for trading off Lˆ and p in ignorance of the evidence. How can… view at source ↗
Figure 2
Figure 2. Left: Top 150 ESR H(z) functions overplotted on the CC data (upper panel), and the residuals from Eq. 5.1 (lower panel). Right: Pareto fronts for both datasets with the literature standards (Friedmann equations) shown as separate symbols. Reproduced from [9]. required to get flat rotation curves. Classic IF choices are the “Simple” (ν(x) = 1/2 + (1/4 + 1/x) 1/2 ) and “RAR” (ν(x) = 1/(1 − exp(− √ x))) functions. This… view at source ↗
Figure 3
Figure 3. Left: Pareto front identified by ESR for the RAR compared to the three most common literature fits. Right: Logarithmic slopes of the 10 best ESR functions in the limit gbar → 0 (blue) and gbar → inf (red), and at the lower (cyan) and upper (magenta) edges of the data. Where these depend on free parameters of the function they are shown as bands indicating the 95% C.L. The MOND lower and upper slopes are shown by ver… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Left: Pareto front for inflaton potentials using operator basis set B, using fiducial MDL (solid) or Katz back-off model (dashed) function priors. Right: The potential favoured by the k log(n) prior, exp(− exp(exp(exp(ϕ)))), with the slow-roll region over which inflati…

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Cited by 1 Pith paper

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