arxiv: 2604.23256 · v1 · submitted 2026-04-25 · 💻 cs.NE · cs.AI· cs.LG· cs.SC

Recognition: unknown

Why Architecture Choice Matters in Symbolic Regression

Chakshu Gupta

Pith reviewed 2026-05-08 06:44 UTC · model grok-4.3

classification 💻 cs.NE cs.AIcs.LGcs.SC

keywords symbolic regressiontree architecturegradient descentexpressivenessoptimization landscaperecovery ratesoperator trees

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The pith

In symbolic regression the tree architecture determines which targets gradient descent recovers, not the structure's expressiveness.

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

The paper tests three fixed tree structures for symbolic regression, all using the same operators and targets but arranged differently for how variables enter the expression. Across more than 12,700 runs, recovery rates swing from 100 percent to 0 percent depending on the chosen tree, with the ranking reversing on different target functions. The most expressive structure fails on targets that a restricted alternative solves reliably, showing that the optimization landscape shaped by the architecture controls success. Balanced non-chain trees are never recovered at all, and altering an operator or its gradient profile can eliminate recovery entirely. These results matter because they show that simply increasing expressiveness does not make gradient-based discovery more reliable.

Core claim

Expressiveness guarantees that a solution exists in the search space but does not guarantee that gradient descent finds it. The most expressive tree fails on targets that restricted alternatives recover reliably, and the ranking of structures reverses across targets. Balanced tree shapes are never recovered, switching the operator changes which targets succeed, and reversing an operator's gradient profile collapses recovery entirely.

What carries the argument

The fixed tree architecture that places operators and variables at specific positions, which in turn shapes the loss landscape that gradient descent navigates when optimizing the weights.

If this is right

On some targets one structure recovers the formula at 100 percent while another scores 0 percent.
The ordering of which structure performs best reverses when the target function changes.
Changing which operators are available alters the set of targets that are successfully recovered.
Reversing the gradient profile of an operator eliminates recovery for targets that previously succeeded.
Balanced non-chain tree shapes are never recovered regardless of the target.

Where Pith is reading between the lines

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

Practitioners using gradient-based symbolic regression may need to test multiple fixed architectures rather than assuming a single choice will work across targets.
The results suggest that future methods could benefit from mechanisms that adapt the tree shape during search instead of committing to one structure upfront.
Similar architecture sensitivity may appear in other gradient-trained symbolic or neuro-symbolic models where expressiveness is traded against trainability.

Load-bearing premise

The differences in recovery rates across the three tree structures are caused by the architecture itself rather than by unstated choices of initialization, hyperparameters, or the particular set of target functions and operators.

What would settle it

Re-running the full set of experiments with different random initializations or altered hyperparameter schedules and checking whether the 0 percent recovery cases remain at 0 percent or begin to succeed.

Figures

Figures reproduced from arXiv: 2604.23256 by Chakshu Gupta.

**Figure 2.** Figure 2: Gradient ratio ∥∇𝑥 ∥/∥∇𝑦 ∥ during training (Eq. 6, 10- seed mean ± s.e.). The gradient trajectory during training ( view at source ↗

read the original abstract

Symbolic regression discovers mathematical formulas from data. Some methods fix a tree of operators, assign learnable weights, and train by gradient descent. The tree's structure, which determines what operators and variables appear at each position, is chosen once and applied to every target. This paper tests whether that choice affects which targets are actually recovered. Three structures are compared, all sharing the same operator and target language but differing in how variables enter the tree; one is strictly more expressive. Across over 12,700 training runs, one structure recovers a target at 100% while another scores 0%, and the ranking reverses on a different target. Expressiveness guarantees that a solution exists in the search space, but not that gradient descent finds it: the most expressive structure fails on targets that a restricted alternative solves reliably. Switching the operator changes which targets succeed; reversing its gradient profile collapses recovery entirely. Balanced (non-chain) tree shapes are never recovered. These findings show that the optimization landscape, not expressiveness alone, determines what gradient-based symbolic regression recovers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Architecture choice in gradient-based symbolic regression can cause total recovery failures or reversals even when a more expressive structure is available, based on 12k+ runs.

read the letter

The main point is that in these gradient-based symbolic regression setups, picking the tree structure isn't just about having enough expressiveness. One structure can recover a target 100% of the time while another gets zero, and the ranking flips on different targets. The most expressive option fails on things that restricted alternatives solve reliably, balanced trees never recover, and changing the operator or its gradient profile can kill performance entirely. The paper backs this with over 12,700 runs across three structures that share the same operators but differ in variable placement.

Referee Report

3 major / 2 minor

Summary. The manuscript presents an empirical study on gradient-based symbolic regression comparing three fixed tree architectures that share identical operator sets and target languages but differ in variable-entry topology (one strictly more expressive than the others). Across more than 12,700 training runs, it reports large differences in recovery rates, including complete reversals (100% vs. 0% on specific targets) and the total failure of balanced (non-chain) trees. The central claim is that the optimization landscape induced by architecture—not expressiveness alone—determines which targets gradient descent can recover, with additional observations that operator choice and gradient profile also affect outcomes.

Significance. If the experimental controls isolate architecture as the sole variable, the result is significant for gradient-based symbolic regression methods: it demonstrates that more expressive structures can systematically underperform on targets solved reliably by restricted alternatives, and that balanced trees are unrecoverable. The scale of the experiment (12,700 runs) provides empirical weight to the reversals and the landscape-dependence claim. This could guide architecture selection in practice and motivate further analysis of why certain topologies create harder optimization problems.

major comments (3)

[Experimental Setup / Methods] Experimental Setup / Methods: The manuscript does not explicitly state that a single shared hyperparameter vector (learning-rate schedule, initialization distribution, batch size, and gradient implementation) was used across all three architectures. Since the central claim attributes recovery-rate gaps (including 100% vs. 0% reversals) solely to tree topology, confirmation that no per-architecture tuning or differing effective step-sizes occurred is required to rule out confounds from initialization or optimization details.
[Results] Results section (recovery-rate tables/figures): While the abstract and results report dramatic differences and reversals across targets, the text does not indicate whether variance was measured across independent random seeds for each architecture-target pair or whether statistical tests were applied to the 100%/0% claims. Without this, the reliability of the ranking reversals cannot be fully assessed.
[Discussion / Operator Gradients] Discussion of operator gradients: The observation that reversing an operator's gradient profile collapses recovery is load-bearing for the landscape claim, yet the manuscript provides no explicit equations or pseudocode for how the gradient is computed through the tree for each architecture. This detail is necessary to verify that the gradient implementation itself does not differ across structures.

minor comments (2)

[Abstract] The abstract would be clearer if it named the exact number of targets, operators, and the precise definitions of the three tree structures (e.g., via a small diagram or equations) rather than describing them only qualitatively.
[Introduction / Methods] Notation for the three architectures is introduced late; a dedicated figure or table early in the paper comparing their variable-entry topologies would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. We address each major comment point by point below. Revisions have been made to the manuscript to provide the requested clarifications and details.

read point-by-point responses

Referee: [Experimental Setup / Methods] Experimental Setup / Methods: The manuscript does not explicitly state that a single shared hyperparameter vector (learning-rate schedule, initialization distribution, batch size, and gradient implementation) was used across all three architectures. Since the central claim attributes recovery-rate gaps (including 100% vs. 0% reversals) solely to tree topology, confirmation that no per-architecture tuning or differing effective step-sizes occurred is required to rule out confounds from initialization or optimization details.

Authors: We confirm that a single shared hyperparameter vector was used across all three architectures, with identical learning-rate schedules, initialization distributions, batch sizes, and gradient implementations. No per-architecture tuning or adjustments to effective step sizes were performed. We have added an explicit statement in the Methods section to document this shared configuration. revision: yes
Referee: [Results] Results section (recovery-rate tables/figures): While the abstract and results report dramatic differences and reversals across targets, the text does not indicate whether variance was measured across independent random seeds for each architecture-target pair or whether statistical tests were applied to the 100%/0% claims. Without this, the reliability of the ranking reversals cannot be fully assessed.

Authors: The total of over 12,700 training runs incorporates multiple independent random seeds for each architecture-target pair. We have revised the Results section to explicitly note that variance was measured across these repeated seeds and that the reported recovery rates reflect success fractions over the repetitions. Formal statistical tests were not applied, but the absolute reversals (100% vs. 0%) are consistent outcomes that do not require such tests to establish the ranking differences. revision: yes
Referee: [Discussion / Operator Gradients] Discussion of operator gradients: The observation that reversing an operator's gradient profile collapses recovery is load-bearing for the landscape claim, yet the manuscript provides no explicit equations or pseudocode for how the gradient is computed through the tree for each architecture. This detail is necessary to verify that the gradient implementation itself does not differ across structures.

Authors: We agree that explicit details on gradient computation are needed to support the claims. We have added equations and pseudocode in the Methods section describing the gradient computation through the tree for each architecture. These additions confirm that the implementation is consistent across structures, with observed differences attributable to topology. revision: yes

Circularity Check

0 steps flagged

Purely empirical comparison with no derivations or self-referential predictions

full rationale

The paper conducts an experimental study comparing three tree architectures for gradient-based symbolic regression across 12,700+ runs. It reports observed recovery rates (e.g., 100% vs 0% on specific targets) and notes that expressiveness does not guarantee optimization success. No equations, derivations, fitted parameters renamed as predictions, or self-citation chains are present in the abstract or described methodology. All claims rest on direct experimental outcomes rather than any reduction to inputs by construction. This is the standard case of a non-circular empirical paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

This is an empirical study; the central claim rests on experimental comparisons rather than new theoretical axioms or parameters. The three tree structures are the primary variables under test.

axioms (2)

domain assumption Gradient descent optimization behaves consistently across the compared tree structures when the same operators and targets are used.
The paper assumes the optimization procedure is held constant so that differences can be attributed to architecture.
domain assumption The selected targets and operators are representative of typical symbolic regression problems.
Generalization of the findings depends on this representativeness.

pith-pipeline@v0.9.0 · 5474 in / 1160 out tokens · 38830 ms · 2026-05-08T06:44:22.196733+00:00 · methodology

discussion (0)

Reference graph

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