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REVIEW 4 major objections 4 minor 1 cited by

Diffusion-Based Symbolic Regression

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read DDSR, a mask-based discrete diffusion model for symbolic regression, recovers known ground-truth equations at almost double the rate of the deep-learning baseline DSR (46.5% vs 24.8% at zero noise) while writing simpler expressions.

desk verdict Strong empirical results for a diffusion-based SR method, but the paper's own appendix contradicts its reverse-process description and no code is provided, so the central claim is not yet verifiable. read the letter →

arxiv 2505.24776 v1 pith:KZWTEUED submitted 2025-05-30 cs.LG

Zachary Bastiani , Robert M. Kirby , Jacob Hochhalter , Shandian Zhe This is my paper
classification cs.LG
keywords symbolicregressiondiscretediffusiontokenmaskingreinforcementlearningGRPOrisk-seekingpolicySRBenchexpressiongeneration
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 is the task of finding a compact mathematical expression that fits a table of measurements, and the paper aims to show that discrete diffusion is a better generator for it than the autoregressive neural network used by the leading deep-learning method, DSR. Its proposed system, DDSR, corrupts an expression by masking one token at a time and trains a Transformer to reconstruct masked tokens, with the whole loop driven by a reinforcement-learning reward for data fit. On SRBench's 133 problems with known ground-truth solutions, DDSR reports a symbolic solution rate of 46.5% at zero noise versus 24.8% for DSR, stays ahead at most noise levels, and produces expressions with average simplified complexity 17.7 — far shorter than TPSR's 61.4 or GP-GOMEA's 35.4. If the claims hold, scientists who want interpretable laws rather than black-box models get a way of searching equations that combines the accuracy of deep learning with the readability of short formulas.

What carries the argument

The load-bearing object is the random mask-based discrete diffusion process: a forward chain that masks exactly one token of the expression matrix $X_0$ per step, $X_t = Q_t \cdots Q_1 X_0$ with $Q_t = I - \mathrm{diag}(e_{q_t})$, and a reverse chain that reconstructs masked tokens from the Transformer's predicted distribution $q(X_0)$. Two further mechanisms carry the learning. The token-wise GRPO update assigns each token of a top-$\alpha$ expression the relative reward $A_i = R(\tau^{(i)}) - R_\alpha$, clips the likelihood ratio $h_{\theta k t}$ to $[1-\epsilon, 1+\epsilon]$, and adds a KL penalty to a reference model, so the objective in equation (6) — the reward-scaled log-likelihood of the true expression under the predicted distribution — is optimized inside a trust region. The long short-term risk-seeking policy maintains a candidate pool $S_\alpha \leftarrow S_\alpha \cup S^k_\alpha$ that keeps the top expressions from all epochs, which the paper credits with balancing short-term and long-term exploitation.

What would settle it

Run the generation loop in Algorithm 2 on a toy expression of length 5 and instrument how many tokens remain masked after each backward step: the claimed one-token-at-a-time reconstruction requires the mask count to fall by exactly one per iteration. If applying equation (5) after sampling the masked tokens produces a state with one fewer mask, the description holds; if the mask count drops by more than one per step, or the sampled tokens get overwritten, the reverse process differs from the paper's account and the reported results would depend on the actual implementation.

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Extended reading notes

Core claim

The central claim is that a discrete diffusion model built on random token masking can serve as an effective generator for symbolic regression when trained with reinforcement learning rather than likelihood maximization alone. DDSR's forward process hides exactly one token of the expression at each step, blurring structure gradually instead of perturbing every token at once as the standard D3PM discrete diffusion does; the reverse process starts from a fully masked sequence and reconstructs tokens step by step under a Transformer that predicts the complete token-distribution matrix from the partially masked input. Training uses a token-wise adaptation of GRPO: only the top $\alpha$% of sampled expressions by reward enter the update, each token's likelihood is weighted by the expression's relative reward, and updates are clipped to a trust region with a KL penalty toward a reference model. A long short-term risk-seeking policy widens the candidate set to include top expressions from every prior epoch, preventing hard-to-sample high-reward expressions from being forgotten. The paper reports that this combination roughly doubles DSR's symbolic solution rate at zero noise (46.5% vs 24.8%), beats most genetic-programming methods while writing far simpler expressions, and places DDSR on the Pareto frontier of the black-box problems, with ablations attributing separate gains to each component.

Load-bearing premise

The printed description of the backward denoising loop is internally inconsistent — it says the model samples all masked tokens at once, which conflicts with the claimed one-token-at-a-time reconstruction — and the whole method depends on that loop actually working as intended.

Editorial extensions

If this is right

  • Deep learning for symbolic regression does not need an RNN: a diffusion generator trained with reward-weighted denoising reports roughly double DSR's symbolic solution rate at zero noise, so the autoregressive front end is replaceable.
  • Accuracy and interpretability can come together: with average simplified complexity 17.7 versus 35.4 for GP-GOMEA and 61.4 for TPSR, DDSR sits on the Pareto frontier of R-squared versus model size on SRBench's black-box problems.
  • Token-wise trust-region updates speed up training: GRPO converges about 30 epochs earlier than the standard risk-seeking policy gradient on the reported learning curves, making reinforcement learning for expressions more sample-efficient.
  • Surviving expressions from past epochs are a reusable asset: the long short-term risk-seeking buffer can be added to other deep symbolic regression trainers, since it only changes which expressions enter the policy update.
  • Diffusion-based expression models are plug-in components for ensembles: the paper argues DDSR can replace or complement DSR inside the uDSR pipeline, combining its high solution rate with the ensemble's coverage.

Reading between the lines

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

  • The claimed 35.7-point accuracy gain over D3PM on the Strogatz set suggests the one-token-per-step mask is the structurally critical choice; a natural test is to run DDSR with a two-token or adaptive-rate mask on the same problems to see whether the gain comes from mask sparsity or from preserving tree locality.
  • The paper's conclusion concedes weaker robustness to high noise, and its Table 1 shows DSR without constant tokens overtaking DDSR at 10% noise (16.61% vs 10.69%); one could test whether a noise-scaled masking rate or reward shaping restores the advantage in that regime.
  • The LST buffer is effectively a replay set for exploitation; a variant that resamples the buffer with priority weighting rather than keeping only the global top $\alpha$% might train faster still, though that is not explored in the paper.
  • The runtime measurement (14,441 s vs 27,131 s for DSR with the same token space) indicates the mask-based schedule halves wall-clock cost; if that holds, masked-diffusion generators are a promising backbone for the supervised foundation-model regime the paper names as future work.
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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

4 major / 4 minor

Summary. The paper proposes DDSR, a random mask-based discrete diffusion model for symbolic regression, trained with token-wise Group Relative Policy Optimization (GRPO) and a long short-term risk-seeking policy. The method is evaluated on SRBench against eighteen baselines, reporting improved symbolic solution rates and accuracy relative to DSR, and simpler expressions than GP-based methods. The authors also provide ablations of the diffusion model, the GRPO component, and the long short-term policy.

Significance. The empirical evaluation is extensive and well-structured, with eight runs per problem, ablations of each component, and a broad baseline set. If the method works as described, the application of masked discrete diffusion to symbolic regression with RL training is a potentially useful contribution. However, the significance is substantially undermined by the unclear and internally inconsistent description of the generation process, which prevents a reader from verifying that the reported results are produced by the claimed diffusion-based method.

major comments (4)
  1. [Section 3.1 and Algorithm 2] The reverse diffusion process is not executable as written. The text states that at each backward step t the model inputs X_t, predicts q(X_0), samples all masked tokens in X_t to form an intermediate X_0, and then applies Eq. (5) to obtain X_{t-1}; this would either reconstruct all masked tokens at once (contradicting the claim that tokens are reconstructed one at a time) or, if Eq. (5) is applied as a forward-masking operator, would re-mask tokens and increase the number of masked tokens rather than decrease it. Moreover, the only sampling procedure given in Appendix A.1 (Algorithm 2) is an autoregressive token-by-token sampler with valid-token constraints that does not use X_t, the predicted q(X_0), or Eq. (5) at all. Because no code is released, the reader cannot determine which procedure was actually implemented; this ambiguity is load-bearing because the diffusion-based generation is the paper's core contribution.
  2. [Eq. (5) in Section 3.1] The definition of the forward process is notationally inconsistent. In the same equation, Q_t is used both for the single-step mask operator I − diag(e_q_t) and for the cumulative product Q_t Q_{t−1} ... Q_1; the latter definition is self-referential. This makes it impossible to reproduce the forward process without additional interpretation.
  3. [Section 3.2 and Algorithm 1] The training objective applies the reward-weighted likelihood to a randomly selected diffusion step t, but the generation procedure in Algorithm 2 is independent of the diffusion process. If generation is actually performed by Algorithm 2, then the method is a transformer-based autoregressive sampler trained with an additional denoising loss, and the claim that DDSR is a diffusion-based generator is not validated by the reported experiments. The paper needs to clarify how the denoising objective in Eq. (6) relates to the sampling procedure used at test time.
  4. [Section 5.2 and Figure 4] The ablation results do not consistently support the claimed benefit of GRPO. On the Feynman dataset, DDSR-RSPG (55.6%) slightly outperforms DDSR-GRPO (55.2%), while on Strogatz DDSR-GRPO is better (96.4% vs 92.9%). The statement that 'GRPO accelerates training' by 30 epochs is not a substitute for a statistically supported accuracy comparison; the paper should either report significance tests or temper the claim.
minor comments (4)
  1. [Section 1] In the second contribution bullet, 'GPRO' should be spelled 'GRPO'.
  2. [Section 3.2] The entropy-gradient term is mentioned but its exact form is not specified; please provide the formula.
  3. [Section 3.3] The description of the candidate pool as containing 'all model versions seen so far' is inconsistent with Algorithm 1, which removes the bottom α% of expressions every epoch; please clarify whether the pool is a fixed-size replay buffer or a true history.
  4. [Table 1] Table 1 reports no confidence intervals; consider adding them to support the claim of statistical significance.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: rewards are external-data-driven NRMSE, and the claimed predictions do not reduce by construction to fitted inputs or a self-citation chain.

full rationale

The derivation chain is self-contained in the relevant sense. The reward in Eq. (3) is R(τ)=1/(1+NRMSE(τ,x,y)), computed from the measurement dataset D, not from the diffusion model's own outputs; the training objective in Eq. (6) weights the model's likelihood of proposed expressions by that external reward. The 'predictions' (generated expressions) are therefore evaluated against ground-truth data rather than being forced by construction. The only author-overlap citation is Bingo (Randall et al., 2022, co-authored by Hochhalter), and it is used as a baseline for comparison, not as load-bearing support for the method or as a uniqueness argument. The long short-term replay pool in Eq. (8) reuses the model's own past samples, but the rewards attached to those samples remain fixed external NRMSE values; this is standard self-training/on-policy RL, not the pattern of renaming a fitted parameter as a prediction. The reverse-process text in Section 3.1 has an internal consistency issue (sampling all masked tokens at once versus one-at-a-time reconstruction, and applying Eq. (5) to an intermediate X0 would produce Xt rather than Xt−1), but that is an implementation/correctness concern, not circularity. Absence of released code affects reproducibility but does not by itself establish circularity. Since no central claim reduces to its inputs by definition or by self-citation, the circularity score is 0.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The central claim rests primarily on domain assumptions about expression representation, reward design, and the validity of the masked diffusion and GRPO training loop. The most fragile assumption is the coherence of the reverse diffusion process, which is described ambiguously in Section 3.1. The method introduces no new physical or conceptual entities beyond the standard masked token representation for expressions.

free parameters (9)
  • risk_percent_alpha = 5
    Top 5% of sampled expressions are used for policy updates; this controls the exploration-exploitation trade-off and is a hand-chosen hyperparameter.
  • KL_coefficient_beta = 0.01
    Weight of the KL divergence regularization term in equation (7) that keeps the current model close to a reference model.
  • clip_epsilon = 0.2
    Trust region bound for per-token likelihood ratios in GRPO, from equation (7).
  • entropy_coefficient_lambda = 0.0005
    Scales the entropy gradient added to encourage exploration and prevent model collapse.
  • max_expression_depth = 32
    Maximum number of tokens and depth of the expression tree; limits the search space and affects runtime.
  • batch_size_B = 1000
    Number of expressions sampled in each epoch; affects gradient estimates and diversity.
  • reference_model_update_interval_G = 5
    Number of epochs between updates of the reference model θ_ref used in the KL regularization.
  • gradient_steps_per_epoch_C = 5
    Number of gradient steps taken per epoch in Algorithm 1.
  • oversampling = 3
    Listed in Table 2; likely the number of samples drawn per position during generation, though not defined in the text.
assumptions (5)
  • domain assumption The forward masking process in equation (5) defines a valid generative model that can be trained by maximizing the reconstruction likelihood p(X0 | φθ(Xt)).
    The method assumes that learning to fill in masked tokens under random masking is a sufficient training signal for generating high-reward expressions; this is the foundational assumption of the diffusion framework.
  • domain assumption Expressions can be represented as fixed-length BFS-ordered token sequences with a predefined token library and validity constraints.
    Section 3.1 and Appendix A assume this representation, including zero-padding and the validity rules in Algorithm 2.
  • domain assumption The reward R(τ) = 1/(1+NRMSE) is a suitable scalar objective for guiding symbolic regression.
    This reward is adopted from DSR without further justification; it blends accuracy and, weakly, expression quality.
  • ad hoc to paper The reverse generation process described in Section 3.1, in which masked tokens are sampled and then equation (5) is applied to obtain the next state, is implementable and produces valid expressions.
    This is the weakest assumption; the text is internally inconsistent about whether one token or all masked tokens are sampled at each backward step.
  • domain assumption GRPO's trust region and KL regularization guarantee stable RL training in this setting.
    The paper relies on the GRPO framework from DeepSeekMath literature without proving convergence properties for this application.

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Cite this review

Pith. "Pith review of Diffusion-Based Symbolic Regression." pith.science (2026). https://pith.science/paper/KZWTEUED

@misc{pith2026250524776,
  author       = {Pith},
  title        = {Pith review of: Diffusion-Based Symbolic Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZWTEUED}},
  note         = {Machine review of arXiv:2505.24776}
}
read the original abstract

Diffusion has emerged as a powerful framework for generative modeling, achieving remarkable success in applications such as image and audio synthesis. Enlightened by this progress, we propose a novel diffusion-based approach for symbolic regression. We construct a random mask-based diffusion and denoising process to generate diverse and high-quality equations. We integrate this generative processes with a token-wise Group Relative Policy Optimization (GRPO) method to conduct efficient reinforcement learning on the given measurement dataset. In addition, we introduce a long short-term risk-seeking policy to expand the pool of top-performing candidates, further enhancing performance. Extensive experiments and ablation studies have demonstrated the effectiveness of our approach.

Figures

Figures reproduced from arXiv: 2505.24776 by the authors.

Figure 1
Figure 1. Illustration of the forward and backward process of the random masked-based diffusion. White entries represent the masked tokens. Green entries represent tokens in the original expression (left), and the generated tokens (right). Specifically, let qt denote the token index to be masked at time step t, and let qt = {q1, . . . , qt} represent the set of all masked indices up to step t. Given X0, we sample qt and Xt as… view at source ↗
Figure 2
Figure 2. Performance on SRBench problems with ground-truth solutions. Error bars denote a 95% confidence interval. uDSR’s high performance is attributed to it being an ensemble method that combines several SR methodologies. 0%, 0.1%, 1%, and 10%. All experiments were conducted on A40s from the NCSA Delta cluster1 . Each A40 ran 8 trials in parallel. For each problem at each noise level, we ran DDSR eight times, with each run… view at source ↗
Figure 3
Figure 3. Mean R 2 score vs. model size on black￾box problems. The numerical values are reported in Appendix [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: DDSR ablations accuracy rate (%) for the Feynman and Strogatz datasets. 0 100 200 300 400 500 600 Epoch 0.6 0.7 0.8 0.9 1.0 Reward DDSR-GRPO DDSR-RSPG (a) feyman_ll_6_11 0 100 200 300 400 500 600 Epoch 0.6 0.7 0.8 0.9 1.0 Reward DDSR-GRPO DDSR-RSPG (b) strogatz_predpre…
Figure 5
Figure 5. Figure 5: Learning curves examples for DDSR with GPRO and with RSPG. Error bars denote one standard deviation. First, our random mask-based diffusion improves solution accuracy rate — by 2.2% on the Feynman dataset and 35.7% on the Strogatz dataset — compared to the standard D3P…
Figure 6
Figure 6. Figure 6: Breadth first search ordering of an expression compared to the pre-order traversal [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: The architecture of the diffusion model in DDSR. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Learning curves of 20 problems for DDSR with GPRO and with RSPG. These problems are randomly selected from the Feynman and Strogatz dataset. Colored regions denote one standard deviation [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

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Cited by 1 Pith paper

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