Searching the Space of Feed-Forward Neural-Network Weight-Update Rules with Fixed Depth Symbolic Regression

Charles Brum; Edward Finkelstein

REVIEW 4 major objections 4 minor 14 references

A lightweight symbolic search can discover neural-network weight-update rules that beat hyperparameter-tuned standard optimizers on 25 of 30 small benchmark tasks.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 06:26 UTC pith:C24IZAOC

load-bearing objection Interesting idea, honest write-up, but the 25/30 claim isn't established because the search and the evaluation share the same single-seed MSE. the 4 major comments →

arxiv 2607.21855 v1 pith:C24IZAOC submitted 2026-07-23 cs.LG

Searching the Space of Feed-Forward Neural-Network Weight-Update Rules with Fixed Depth Symbolic Regression

Charles Brum , Edward Finkelstein This is my paper

classification cs.LG

keywords symbolic regressionweight-update rulesoptimizer discoverygenetic programmingneural networksmeta-learningadaptive optimizers

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that symbolic regression — searching a fixed-depth space of arithmetic expressions built from optimizer-style quantities — can discover compact, explicit weight-update rules for small feed-forward networks that outperform established optimizers whose hyperparameters have been tuned per task. Across 30 benchmark/network combinations, the search found a better rule in 25 cases, with an aggregate 44.47% reduction in mean squared error over the improved cases. The authors argue this makes symbolic regression a lightweight alternative for discovering optimizer variants, and that useful update dynamics often have low-complexity symbolic forms. The point matters because optimizer design is normally done by hand; if the search space is this rich, parts of that design work can be automated and the results stay interpretable.

Core claim

The central claim is that within a space of depth-5 symbolic expressions over operands derived from common optimizers — gradients, momentum velocities, running averages of squared gradients, and bias-corrected moment estimates — genetic-programming search finds update rules of the form w_{t+1} = f(operands) that beat the best hyperparameter-tuned baseline (SGD, heavy-ball, NAG, AdaGrad, RMSProp, AdaDelta, Adam, AdamW) in 25 of 30 benchmark/network combinations, with an aggregate MSE reduction of 44.47% over the improved cases. The discovered rules are compact but structurally diverse, often combining adaptive normalization, momentum-like terms, and nonlinear or rational expressions; four rec

What carries the argument

The engine is fixed-depth symbolic regression: a genetic programming search over expression trees of fixed depth 5, built from a fixed operator set (unary: -, ln, exp, cos, sin, sqrt, asin, acos, tanh; binary: +, -, *, /, ^) and a hand-chosen set of leaf operands that includes the current weight, gradient, momentum velocity, running and cumulative gradient statistics, and Adam-style bias-corrected moments. The operands are what carry prior optimizer knowledge into the search; the fixed depth and tight grammar keep candidate rules compact and interpretable, while seeding each new benchmark's population with the best rule from the previous benchmark lets the search transfer structure across ta

Load-bearing premise

The central claim rests on the assumption that measuring each candidate by its final MSE after exactly 10 epochs of training from a single random initialization, on the same benchmark data used to report results, is a faithful measure of optimizer quality — and that the Table 3 grid searches make the baselines genuinely 'best hyperparameter-tuned'.

What would settle it

Train the 25 discovered rules on the same benchmark/network pairs for 100 epochs, across 10 independent weight initializations, and compare the median final MSE against the same baselines re-tuned on validation data; if the discovered rules' advantage does not persist or the baselines catch up, the 25-of-30 result is an artifact of 10-epoch, single-seed evaluation rather than a property of the update rules.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

If the claim holds, symbolic regression is a viable, low-cost way to discover optimizer variants in small-network settings, without hand-designing update rules.
The existence of many algebraically distinct yet equally effective rules suggests large equivalence classes of update dynamics, which may explain why search succeeds despite structural diversity.
The recurring expression signatures involving accumulated gradient statistics indicate that useful optimizers can be built from running-moment quantities plus nonlinear transforms — motifs that are not explicitly enforced.
The concentration of failures in the largest architecture implies discovered rules' competitiveness degrades as model complexity grows, so scaling is the immediate open question.
Because rules are explicit and compact, they can be inspected, simplified, and potentially transferred to related tasks via seeding.

Where Pith is reading between the lines

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

One extension the authors leave implicit is using the discovered rules as initializers for a second round of search on the same task, which could reveal whether the signatures in Table 7 are local optima or reflect a deeper structure.
Another extension: apply the same operand set to convolutional or recurrent architectures; the paper only tests fully-connected feed-forward nets, so whether these motifs transfer to other topologies is unknown.
The four recurring signatures could be turned into a prior for a Bayesian or constrained search, potentially shrinking the effective search space by orders of magnitude.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Interesting idea, honest write-up, but the 25/30 claim isn't established because the search and the evaluation share the same single-seed MSE.

read the letter

Two things to know about arXiv:2607.21855. First, the core idea is legitimate: use fixed-depth symbolic regression over a grammar of optimizer-derived operands (gradient, momentum, adaptive moments, etc.) to evolve explicit weight-update rules, and test them on small MLPs fitting symbolic-regression benchmarks. Second, the headline result — 25/30 wins and 44.47% aggregate MSE reduction — is probably inflated by the evaluation protocol, so treat it as a promising existence proof, not a measured claim.

What's actually new: the specific application of the authors' fixed-depth postfix grammar to optimizer discovery, with a custom operand set. That is a real extension of Bello et al. and Chen et al., and the paper is honest about the lineage. The method finds rules that are compact and structurally diverse; the observation that many combine momentum-like and normalization terms is plausible and could be useful for future search design. The paper also ships code and data links, and the limitations section is direct about the small scale.

The soft spots are in the comparison. The same 10-epoch final MSE used to select candidate rules is used to report the winner. With populations up to 100 and early stopping after 100 'better' expressions, the best-of-many will beat a single baseline on the same seed even if its expected performance is no better. No repeated seeds, confidence intervals, or held-out tasks are reported, so the 25/30 count likely includes selection noise. The baseline grid is also skewed: Adam and AdamW are capped at learning rate 1e-4, while plain GD gets up to 3e-3. On 10-epoch small problems, that makes the 'best hyperparameter-tuned' baselines weaker than a real tuning exercise would find.

None of this is fatal — the paper is explicitly preliminary, and the authors say further validation is needed. But the central numerical claim needs repeated-seed evaluation, a fairer baseline grid, and ideally a couple of held-out tasks before it can support the conclusion that symbolic regression reliably beats tuned optimizers.

Who's it for: someone working on meta-learning or optimizer search might want to read it as a data point, not as a benchmark result. It deserves a serious referee — the topic is relevant and the approach is concrete — but the referee should insist on the validation fix.

Recommendation: engage with it, but require revision before acceptance. I wouldn't cite the 25/30 result as evidence until the evaluation is cleaned up.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes using fixed-depth symbolic regression to discover explicit neural-network weight-update rules. Candidate rules are expressed as expression trees of depth 5 over operators and operands derived from standard optimizers (gradient, momentum, AdaGrad, RMSProp, AdaDelta, Adam). A genetic-programming search is run for each of 30 benchmark/network combinations, with fitness defined as the final MSE after 10 epochs of training. The paper reports that the discovered rule beats the best grid-searched baseline in 25 of 30 combinations, with an aggregate MSE reduction of 44.47% over the improved cases, and notes qualitative regularities in the discovered expressions. The authors frame the work as a preliminary demonstration that symbolic regression can find compact optimizer variants.

Significance. If the empirical claim were established, the paper would be a useful contribution to meta-learning and optimizer discovery, complementing prior work such as Bello et al. and Chen et al. by using a lightweight symbolic search rather than learned controllers. The manuscript is clearly written, provides code links, and defines the search space and evaluation protocol in detail. However, the headline quantitative claim is not supported by the current evaluation protocol: the discovered rules are selected and reported on the same single-seed 10-epoch MSE used as the search fitness, and the baseline grids are uneven, with Adam/AdamW learning rates capped at values that are likely too small for the task. The paper is honest in its limitations section, but the abstract and conclusion still assert the 25/30 result as a finding. The contribution is therefore a plausible but unverified existence proof.

major comments (4)

[§4.3 and Table 8] The fitness used to guide evolution is the 10-epoch final MSE on the same benchmark and random initialization that is later reported as 'Best Found Rule MSE' in Table 8. Evolution runs until 100 expressions outperform the baseline or improvement stagnates, so the reported value is the best of many single-seed evaluations. On a fixed seed, the best of many candidates will tend to beat the baseline even if the expected performance is no better. To support the 25/30 claim, the authors must evaluate the discovered rules on multiple independent initializations (ideally held-out data) and report means/confidence intervals, comparing against baselines selected under the same protocol.
[Table 3] The baseline grid-search is lopsided. Adam and AdamW are restricted to η ≤ 1e-4, while GD reaches 3e-3, heavy-ball/NAG reach 1e-3, and AdaGrad reaches 3e-2. For 10-epoch training on small regression tasks, η = 1e-4 is typically far below useful Adam settings, so the 'best hyperparameter-tuned established optimizer' is artificially weak. Please expand the Adam/AdamW grids (e.g., include 1e-4, 3e-4, 1e-3, 3e-3) or allocate equal tuning budgets across optimizers and report the per-optimizer best.
[§4.2 and Table 4] Both the baseline results in Table 4 and the discovered-rule results in Table 8 appear to be single runs with one random initialization. No repeated seeds, variance estimates, or significance tests are reported. Under this protocol, the observed margins (e.g., 3.51 vs. 2.46 in the first row) cannot be distinguished from initialization noise. The authors should run both baselines and discovered rules over several seeds and report medians or means with error bars; this is necessary before claiming systematic superiority.
[§6, Abstract, and Conclusion] Section 6 explicitly states that the study is preliminary and lists missing generalization tests (larger architectures, longer training, classification, standard benchmarks). The abstract and conclusion nevertheless state unconditionally that symbolic regression 'found an update rule outperforming' baselines in 25/30 cases. This wording presents selected single-seed results as established fact. The claims should either be qualified to the specific evaluation protocol or, preferably, backed by the multi-seed validation recommended above.

minor comments (4)

[Table 1] The AdaDelta row reports tree depth 1 with 2 inputs, which seems inconsistent with the expression given in Table 2; please verify the tree-depth counting and input list. Also standardize the spelling of AdaDelta (the text uses 'Adadelta' in places).
[Table 4] The 'Best Rule' column uses names like 'basic' without definition in the caption. For AdaDelta and AdaGrad rows, no η is listed; clarify in the caption or table whether those hyperparameters are not applicable or were searched but not selected.
[§4.3] The procedure for seeding populations with rules found in previous experiments is not fully specified. Which previous benchmark is used, and does this create dependence across the 30 runs? Also define what 'improvements in the best fitness became negligible' means quantitatively (e.g., tolerance and patience).
[References] Several references lack complete publication details (e.g., [8] and [11] have no venue or DOI). The arXiv identifiers in [1], [2], [4], [10] would be more useful if expanded with year and version. Please harmonize the reference style.

Circularity Check

0 steps flagged

No equation-level circularity; the headline numbers are selected search outcomes rather than out-of-sample predictions.

full rationale

The paper makes no analytic derivation whose output is equivalent to its input by construction. The symbolic search evaluates candidate update rules with exactly the metric that is later reported: §2.3 states that each candidate is scored by 'updating the weights of a randomly initialized feed-forward neural network for an integer number of epochs using the generated weight-update rule, and computing its loss thereafter,' and §4.3 uses the same final MSE after 10 epochs for selection and for the reported 'Best Found Rule MSE.' This creates selection bias, not circularity: the search is selecting for low MSE and then reporting that low MSE, but it is not fitting a parameter and then predicting a closely related quantity. The paper explicitly frames the results as an empirical search rather than a validated prediction, and the Limitations section acknowledges that 'the discovered expressions are selected based on empirical performance rather than theoretical convergence guarantees' and that generalization to larger architectures and longer training horizons remains untested. The self-citations [6,7] supply the genetic-programming grammar and method, but the benchmark results are newly generated in this paper; no load-bearing conclusion rests on an unverified uniqueness theorem or on a prior result that already contains the present claim. The baselines are tuned by grid search, but this is a comparison protocol rather than a fitted input renamed as a prediction. The 44.47% aggregate reduction is computed only over the cases that improved, which is a reporting choice that inflates the headline but is not definitionally equivalent to the search objective. Overall, the central claim is an empirical finding with recognized selection and generalization limitations, and I find no specific circular step requiring a score above 2.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

No new physical entities are postulated. The load-bearing choices are procedural: fixed depth 5, 10-epoch fitness, early stopping, baseline grid ranges, and constant values inside evolved rules are not independently derived.

free parameters (7)

fixed expression-tree depth = 5
Chosen by the authors to bound search; not derived from data; affects expressiveness and comparability.
max population size = 100
Set after preliminary experiments indicated larger populations gave similar solutions at greater cost (§4.3).
evolution stopping thresholds = unspecified
'Improvements became negligible' / early stop after 100 better expressions is a hand-set selection rule.
optimizer hyperparameter initial values = η=0.5, θ=0.01, γ=0.9, ϵ=1e-8, β1=0.9, β2=0.999
Initialized arbitrarily (Table 2); evolution can form constants but search starts from these values.
fitness epoch budget = 10
All rules restricted to exactly 10 epochs for computational feasibility (§4.1).
numeric constants in evolved expressions = not enumerated in text
The reported best rules are fitted to each benchmark; constants are products of selection, not externally validated.
baseline grid ranges = see Table 3
Choice of search ranges for each hand-designed optimizer determines what 'best hyperparameter-tuned' means; coarse grids can understate baselines.

axioms (5)

domain assumption The genetic-programming/fixed-depth grammar implementation from [6,7] is correct and behaves as claimed.
The search method is imported from the authors' prior papers; no independent verification or machine-checked proof is given here.
domain assumption Symbolic-regression fitness (MSE after 10 epochs on the benchmark data) is a valid enough measure of optimizer quality to select rules.
Used to score every candidate; if short-horizon MSE on one random init is noisy, selection is unreliable.
domain assumption Operands from standard optimizers are sufficient building blocks for competitive update rules.
The search space excludes other possible operands; conclusions are limited to this basis.
domain assumption The benchmark functions and network architectures are representative enough to support the stated conclusion.
Ten small symbolic-regression targets and three tiny MLPs; authors themselves note larger-scale validation is needed.
standard math Standard optimizer formulas in §1 are correctly implemented.
All baseline comparisons presuppose correct implementations of GD, heavy-ball, NAG, AdaGrad, RMSProp, AdaDelta, Adam, and AdamW.

pith-pipeline@v1.3.0-alltime-deepseek · 10953 in / 13426 out tokens · 134271 ms · 2026-08-01T06:26:12.173130+00:00 · methodology

0 comments

read the original abstract

We investigate whether symbolic regression can discover explicit neural network weight-update rules that outperform standard hand-designed optimizers on small symbolic regression benchmarks. Candidate update rules are represented as fixed-depth symbolic expressions over operands derived from common optimizers, including gradient, momentum, adaptive-gradient, and moment-estimate quantities. Across 30 benchmark/neural network combinations, the symbolic regression procedure found an update rule outperforming the best hyperparameter-tuned established optimizer in 25 cases, with an aggregate MSE reduction of 44.47\% over the improved cases. The discovered rules do not all share a single common symbolic form, but many combine adaptive normalization, momentum-like quantities, nonlinear transformations, and rational expressions. These results suggest that symbolic regression can serve as a lightweight mechanism for discovering compact optimizer variants, while also highlighting the need for larger-scale validation.

Figures

Figures reproduced from arXiv: 2607.21855 by Charles Brum, Edward Finkelstein.

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages

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Pith/arXiv arXiv 2024

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Gregor, K.: Finding online neural update rules by learning to remember (2020)

2020

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In: IEEE Workshop and Summer School on Evolutionary Computing

Hemberg, E., McPhee, N., O’Neill, M., Brabazon, A.: Pre-, in-and postfix gram- mars for symbolic regression in grammatical evolution. In: IEEE Workshop and Summer School on Evolutionary Computing. vol. 2008, pp. 18–22 (2008)

2008

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Real, E., Liang, C., So, D.R., Le, Q.V.: Automl-zero: Evolving machine learning algorithms from scratch (2020), https://arxiv.org/abs/2003.03384

Pith/arXiv arXiv 2020

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In: 2000 IEEE Symposium on Combinations of Evolutionary Computation and Neural Networks

Runarsson, T., Jonsson, M.: Evolution and design of distributed learning rules. In: 2000 IEEE Symposium on Combinations of Evolutionary Computation and Neural Networks. Proceedings of the First IEEE Symposium on Combinations of Evolutionary Computation and Neural Networks (Cat. No.00. pp. 59–63 (2000). https://doi.org/10.1109/ECNN.2000.886220

arXiv 2000

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arXiv 2001