{"id":"7dd1dc36-6a65-4df6-b7e3-a9f8f395efa7","arxiv_id":"2505.24776","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A masked discrete diffusion model trained with token-wise GRPO and a long short-term risk-seeking replay pool improves symbolic regression solution rates and expression simplicity on SRBench.","lead":"DDSR is a new symbolic regression method that generates math equations using a masked diffusion model and trains it with the GRPO reinforcement learning algorithm. On the SRBench benchmark it recovers more ground-truth expressions than the DSR baseline and produces simpler formulas than most genetic programming methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reverse generation in §3.1 is internally inconsistent: sampling all masked tokens contradicts one-token-per-step, and applying Eq. (5) would re-mask rather than unmask; without code, the actual sampler is unclear.","rationale":"The reader identified the same process inconsistency in the reverse generation, and I find it is indeed the most load-bearing issue. The paper otherwise presents substantial SRBench experiments, ablations, and complexity comparisons, and I do not see a separate flaw that would overturn the empirical claim if the sampling procedure is clarified. However, without code or a coherent algorithm, the central claim cannot be fully verified; the existing CONDITIONAL verdict is appropriate.","tokens_in":14533,"tokens_out":4689,"duration_ms":53677,"concrete_test":"Ask the authors for the exact reverse-sampling code or a corrected pseudocode that specifies (i) whether one masked token or all masked tokens are sampled at each step, (ii) how Eq. (5) maps the intermediate X_0 to X_{t-1}, and (iii) how already reconstructed tokens are preserved. Then re-run the SRBench experiments with a standard one-token-per-step masked-diffusion reverse loop and compare Table 1; if the reported results require Algorithm 2's autoregressive sampler, the paper's diffusion formulation and its claimed advantage over DSR are not supported by the experiments as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the reverse (generation) process in Section 3.1. The paper says that at each backward step t we input X_t, predict q(X_0), 'sample the masked tokens in X_t' to form an intermediate X_0, and then 'apply (5) to obtain X_{t-1}'. This is not executable as written. First, sampling all masked tokens at once would fill M positions in one step, contradicting the claim that tokens are reconstructed one at a time. Second, Eq. (5) is the forward masking operator: applying it to an intermediate X_0 masks tokens, so it cannot reduce the number of masked tokens from t to t-1. Third, Appendix Algorithm 2, the only concrete sampling procedure given, is an autoregressive token-by-token sampler with valid-token constraints; it does not invoke X_t, the predicted q(X_0), or Eq. (5) at all. If Algorithm 2 is what the code actually runs, then the generated expressions are not produced by the proposed masked-diffusion reverse process, and the central comparison in Table 1 does not validate the paper's claimed method. Since no code or data are released, a reader cannot determine which procedure was used. This ambiguity is load-bearing because the reverse process is the heart of the contribution; if it works differently than described, the novelty and the interpretation of the empirical gains change materially.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14950,"tokens_out":7553,"duration_ms":80985,"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":[{"comment":"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.","section":"Section 3.1 and Algorithm 2"},{"comment":"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.","section":"Eq. (5) in Section 3.1"},{"comment":"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.","section":"Section 3.2 and Algorithm 1"},{"comment":"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.","section":"Section 5.2 and Figure 4"}],"minor_comments":[{"comment":"In the second contribution bullet, 'GPRO' should be spelled 'GRPO'.","section":"Section 1"},{"comment":"The entropy-gradient term is mentioned but its exact form is not specified; please provide the formula.","section":"Section 3.2"},{"comment":"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.","section":"Section 3.3"},{"comment":"Table 1 reports no confidence intervals; consider adding them to support the claim of statistical significance.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The main text and the appendix describe two different generation procedures; if the reviewer's reading is correct, the authors need to determine which one was actually used and either fix the text or the appendix. It would be advisable to release the code to allow verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe paper is a genuine attempt to bring masked discrete diffusion to symbolic regression, and on its face it works: DDSR beats DSR on most SRBench settings, matches or beats GP methods on solution rate under noise, and produces simpler expressions than TPSR or GP-GOMEA. The ablations are real—D3PM is worse, GRPO helps over plain risk-seeking, and the long short-term replay pool adds a point or two. That is a solid empirical package. The new pieces—random single-token masking, token-wise GRPO with a trust region, and the long short-term risk-seeking policy—are sensible combinations of existing ideas, and the paper does not oversell them.\n\nThe soft spot is not the experiments; it's the description of the generation process. Section 3.1 says that at each backward step you sample the masked tokens from the predicted q(X0), then \"apply (5)\" to get X_{t-1}. Equation (5) is the forward masking operator, so applying it to a fully reconstructed X0 masks tokens rather than unmasking them; it cannot produce the X_{t-1} with one fewer masked token. The appendix's Algorithm 2, the only concrete sampling procedure given, is an autoregressive token-by-token sampler with valid-token constraints—it never uses X_t, q(X0), or Eq. (5). So the paper's own appendix contradicts its method section. Without code, a reader cannot tell whether the reported results come from the diffusion process as described or from a simpler masked-language-model-style generator. That ambiguity is load-bearing because the novelty claim is specifically about diffusion-based generation.\n\nThe paper also ships no code or data, and the authors themselves list limitations—longer runtimes on complex problems and reduced robustness at high noise—which is honest but does not fix the reproducibility gap.\n\nMy take: the empirical results are worth taking seriously, and the framework is a reasonable engineering contribution, but the paper needs a major revision to specify the actual reverse sampling procedure, reconcile it with Algorithm 2, and release code. As it stands, the central method is not yet a \"diffusion\" process in a way the reader can verify.\n\nI'd send it to a serious referee, but I would not cite it in my own work until the discrepancy is resolved and code is available. If the authors can clarify that generation really is the diffusion reverse process, this could be a useful addition to the DSR line—maybe reading-group material.","headline":"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.","tokens_in":15392,"tokens_out":5804,"would_cite":false,"duration_ms":62408,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["symbolic regression","discrete diffusion","token masking","reinforcement learning","GRPO","risk-seeking policy","SRBench","expression generation"],"falsifier":"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.","tokens_in":14381,"feed_emoji":"🧮","tokens_out":13723,"duration_ms":130200,"temperature":0.7,"pith_summary":"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.","feed_headline":"Masked diffusion recovers true equations at almost double DSR's rate","feed_subtitle":"Beats deep-learning and most genetic-programming baselines on known-formula benchmarks, writing far shorter equations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines DSR, the deep-learning baseline DDSR is compared against and must beat; supplies the risk-seeking policy, token library, and reward scheme that DDSR extends.","marker":"(Petersen et al., 2019)"},{"why":"Introduces D3PM, the standard discrete diffusion model whose all-token perturbation the random mask schedule replaces and outperforms in ablations.","marker":"(Austin et al., 2021)"},{"why":"Supplies the masked-token diffusion idea for discrete sequences that DDSR adapts to one-mask-per-step expression generation.","marker":"(Nie et al., 2025)"},{"why":"Defines GRPO, the group-relative policy optimization that DDSR converts into token-wise updates with trust-region clipping.","marker":"(Shao et al., 2024a)"},{"why":"Provides SRBench, the benchmark whose 133 ground-truth and 120 black-box problems generate all reported comparison numbers.","marker":"(La Cava et al., 2021)"},{"why":"Establishes the denoising diffusion framework that the discrete masking process is built upon as its categorical counterpart.","marker":"(Ho et al., 2020)"},{"why":"Introduces the priority training queue that the long short-term candidate pool in DDSR is explicitly modeled on.","marker":"(Mundhenk et al., 2021)"}],"fun_headline_variants":["Masked diffusion symbolic regression nearly doubles DSR's rate","Diffusion model recovers 46.5% vs DSR's 24.8% in symbolic regression","RL-trained masked diffusion beats DSR with simpler equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Masked diffusion symbolic regression nearly doubles DSR's rate","Diffusion model recovers 46.5% vs DSR's 24.8% in symbolic regression","RL-trained masked diffusion beats DSR with simpler equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000539,"raw_usage":{"total_tokens":2563,"prompt_tokens":897,"completion_tokens":1666,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":1603}},"tokens_in":513,"tokens_out":1666,"duration_ms":14849,"temperature":1.0,"reasoning_tokens":1603,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:14:21.023372+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"K., Landajuela, M., Mundhenk, T","cited_arxiv_id":null,"evidence_quote":"Defines DSR, the deep-learning baseline DDSR is compared against and must beat; supplies the risk-seeking policy, token library, and reward scheme that DDSR extends."},{"cited_title":"D., Ho, J., Tarlow, D., and Van Den Berg, R","cited_arxiv_id":null,"evidence_quote":"Introduces D3PM, the standard discrete diffusion model whose all-token perturbation the random mask schedule replaces and outperforms in ablations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the masked-token diffusion idea for discrete sequences that DDSR adapts to one-mask-per-step expression generation."},{"cited_title":"P., faissol, D., and Petersen, B","cited_arxiv_id":null,"evidence_quote":"Introduces the priority training queue that the long short-term candidate pool in DDSR is explicitly modeled on."}],"review_version":1}