{"id":"6fd23fec-86c3-4e22-96ef-477d54f1716b","arxiv_id":"2502.01774","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"Grokking can be triggered systematically by shifting the training distribution through imbalanced subclass sampling, even with dense data and little hyperparameter tuning.","lead":"This paper argues that grokking, the sudden late generalization seen in neural networks, happens when the training data distribution is shifted relative to the test distribution, not just because data are sparse or heavily regularized. It introduces synthetic datasets with class and subclass structure that can be undersampled to trigger grokking on demand, and it reports MNIST experiments designed to show the same effect on real images.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim underdetermined: Fig 3a shows grokking without induced imbalance, while Fig 3b shows stronger imbalance (f=0) prevents grokking; distribution shift is never directly measured, so the causal attribution is confounded.","rationale":"Good-faith reading: the paper proposes that distribution shift is the cause of grokking and provides synthetic benchmarks to induce it. For that claim, the experimental manipulation must isolate shift from confounds. It does not. Three pieces of internal evidence cut against it. (1) Figure 3a: balanced down-sampling—no deliberate subclass imbalance—still groks at small sizes. If finite-sample noise is counted as shift, then the term is doing no predictive work; if not, shift is unnecessary. (2) Figure 3b: f=0, the largest induced imbalance, prevents grokking, while f=0.01 triggers it. This non-monotonicity may be explainable by partial coverage or weak signal, but the paper offers no formalization of shift magnitude to resolve it. (3) The paper's own text attributes the late improvement to weight decay after convergence, yet claims regularization only amplifies grokking; no ablation of weight decay or initialization scale is provided. Missing promised transformer and MNIST result sections further weaken the architecture-independence claim, but the core issue is the confound between distribution shift, sample size, and retained weak signal. A label-corruption experiment at fixed proportions would settle whether the retained correct signal is necessary. If it is, the paper should be rewritten as a claim about partial coverage and weak relational signal, not distribution shift. The reader's REJECT is justified; our stress-test supports that verdict without changing it.","tokens_in":10527,"tokens_out":7516,"duration_ms":71105,"concrete_test":"On the equidistant dataset, set f=0.01 (the condition that groks) and corrupt the labels of the retained examples in the downsampled subclass—assign them to a random other subclass—while keeping the train/test subclass proportions identical. If grokking still occurs, imbalance alone is causal; if it reverts to the f=0 plateau, the retained correct weak signal is doing the work, and the paper's causal story fails. Repeat at init scale 1 with weight decay 0 to check the 'minimal tuning' claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim (Section 3) is that grokking is caused by a distribution shift between training and test data, with sparsity and regularization merely amplifying it. The evidence does not establish this causal direction. In §5.2, Figure 3a, balanced down-sampling of the equidistant dataset—no subclass forced to be sampled more than any other—still produces grokking at small dataset sizes, so the phenomenon appears without deliberately induced shift. In Figure 3b, removing a subclass entirely (f=0) prevents grokking, while retaining 1–20% of that subclass (f=0.01–0.2) triggers it. If larger shift means more cause, f=0 is a larger shift and should grok; the non-monotonic pattern is equally explained by whether any correctly labeled examples of each subclass remain (partial coverage / weak training signal). The paper also never quantifies the shift (e.g., KL divergence or total variation between train and test subclass proportions), so 'distribution shift' is an inferred label for a confounded manipulation. The fixed weight decay (1e-4) and initialization scale (8) are known grokking-promoting hyperparameters in the cited literature, and the paper itself attributes the late improvement to 'weight decay led the network to a more sparse internal representation' (§5.2), undercutting the claim that regularization is merely an amplifier.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that grokking (delayed generalization) is a statistical phenomenon caused by a distribution shift between training and test data, with data sparsity and regularization acting only as amplifiers rather than causes. The authors define grokking informally, introduce two synthetic datasets with class-subclass hierarchies, and induce distribution shifts by downsampling subclasses by a fraction f. Experiments with MLPs and a two-layer transformer show grokking across several settings, including dense datasets, and the authors claim that grokking can be reproduced systematically with minimal hyperparameter tuning. The paper also announces MNIST experiments validating the findings beyond synthetic data, and provides code and datasets for replication.","tokens_in":10749,"tokens_out":5707,"duration_ms":50789,"significance":"If the central hypothesis were convincingly established, the paper would offer a unifying statistical explanation of grokking and provide useful controlled benchmarks for studying delayed generalization. The synthetic dataset construction and the systematic f-sweep are valuable contributions, and the release of code and data is a strength. However, the causal claim is currently under-supported: the operationalization of 'distribution shift' is confounded with partial subclass coverage, the shift itself is never quantified, and the fixed grokking-promoting hyperparameters are not ablated. The reported MNIST validation is missing from the manuscript. These issues currently prevent the paper from supporting its main claim, though the underlying hypothesis remains plausible and testable.","major_comments":[{"comment":"The non-monotonic relationship between f and grokking undercuts the causal attribution to distribution shift magnitude. Removing a subclass entirely (f = 0) prevents grokking, while retaining only 1-20% of that subclass (f = 0.01-0.2) induces it. If a larger distribution shift caused more grokking, f = 0 should grok the most; instead, the pattern is equally or better explained by whether any correctly labeled examples of each subclass remain (partial coverage / weak training signal). The manuscript should quantify the induced shift (e.g., KL divergence or total variation between training and test subclass proportions) and test monotonicity, or reframe the claim as requiring partial subclass coverage rather than shift magnitude.","section":"§5.2, Figure 3b"},{"comment":"Balanced downsampling of the equidistant dataset, with no subclass forced to be sampled less than another, still produces grokking at small sample sizes. If the test set is the full balanced 10,000-sample set, the small training set has the same expected subclass proportions but a different finite-sample realization; whether this constitutes 'distribution shift' is not established. Without a direct measure of the shift, small sample size itself remains a viable candidate cause, which conflicts with the paper's claim that sparsity is merely a mechanism for achieving distribution shift.","section":"§5.2, Figure 3a"},{"comment":"The experiments fix weight decay at 1e-4 and initialization scale at 8, both known grokking-promoting choices from Liu et al. (2022), yet no ablation is reported. Moreover, §5.2 states that 'weight decay led the network to a more sparse internal representation,' which attributes the late improvement to regularization. This is difficult to reconcile with the claim that regularization merely amplifies and does not cause grokking. The authors should either ablate these hyperparameters or clarify the precise role of regularization in their account.","section":"§5.1 and §5.2"},{"comment":"The abstract and introduction claim that MNIST experiments validate the findings beyond synthetic data, but no MNIST results appear in the main text or in Appendix A. This claim is currently unsupported and must either be substantiated with results or removed from the paper.","section":"Introduction and Abstract"}],"minor_comments":[{"comment":"The caption says 'It occurs in every case,' while the text says grokking 'only existing in considerably small sample sizes.' These statements are contradictory and should be reconciled.","section":"Figure 3a caption and §5.2"},{"comment":"The inequality |C(Ci,j) - C(Ci,l)| > |C(Ci,j) - C(Ck,l)| for i≠k appears to contradict the prose that subclasses of the same class are closer than subclasses of different classes; if the intended property is the opposite, the inequality should use '<'.","section":"Appendix A.2"},{"comment":"The text states results are aggregated from 10 independent runs, but the figures show only average curves without error bars or confidence bands; given that grokking detection is sensitive to training dynamics, variance information should be included.","section":"§5, experimental setup"},{"comment":"Definition 3.1 relies on informal thresholds and symbols such as α0, α1, and 'S ≫ (α1 - α0)' without operational criteria; please specify how these quantities are estimated from training curves in practice.","section":"§3, Definition 3.1"},{"comment":"There are minor typographical issues, including 'liner models' instead of 'linear models,' and the reference to Quiñonero-Candela et al. (2022) should cite the original 2008 work on dataset shift.","section":"§2 and References"}],"recommendation":"major_revision","confidential_remarks":"The central hypothesis is plausible and the synthetic benchmarks could be useful to the community, but the evidence as presented does not support the causal claim. I would be open to a revised version that quantifies distribution shift, distinguishes partial coverage from shift magnitude (especially with respect to the f=0 versus f=0.01 results), ablates the fixed hyperparameters, and includes the claimed MNIST experiments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's real contribution is the two synthetic datasets, which let you induce grokking on demand by undersampling subclasses. The equivariant dataset result—grokking even when a subclass is entirely absent—is genuinely interesting and not in the prior literature. If you work on grokking, these benchmarks are worth having.\n\nThe central hypothesis, that grokking is caused by a distribution shift between train and test, is not established by the experiments. The most direct problem is internal: in the equidistant dataset, removing a subclass entirely (f=0) prevents grokking, while keeping 1% of it (f=0.01) triggers late generalization. If bigger shift meant more grokking, f=0 should be the strongest case. The pattern is equally explained by the presence of a weak training signal from a few labeled examples. The paper never measures the shift (e.g., KL or total variation between train and test subclass proportions), so \"distribution shift\" is an inferred label for a confounded manipulation.\n\nThe paper also undercuts its own \"regularization is merely an amplifier\" claim. In Section 5.2 it attributes the late improvement to weight decay leading to a sparse internal representation. With fixed weight decay (1e-4) and initialization scale 8—both values known to promote grokking—and no ablations, the separate roles of shift, sparsity, and regularization remain tangled.\n\nCompleteness is an issue: the intro and contribution list promise MNIST and transformer experiments, but neither appears in the main text or appendix provided. The appendix only contains the dataset construction procedure. Error bars are absent despite 10 runs; that is minor but easy to fix.\n\nOn the positive side, the definition attempt is reasonable, the dataset construction is clearly described, code is provided, and the limitations section is honest. The citation of Levi et al. is appropriate; this paper extends their insight to class–subclass structure.\n\nI would send this to peer review rather than desk-reject, because the benchmarks are useful and the hypothesis is testable. But a serious referee should push for a direct quantification of distribution shift, an ablation of weight decay and initialization, and the missing experiments. As it stands, the explanation is overclaimed.","headline":"Useful synthetic benchmarks and a plausible hypothesis, but the causal claim about distribution shift is confounded by partial subclass coverage and the paper's own weight-decay explanation.","tokens_in":11349,"tokens_out":4893,"would_cite":true,"duration_ms":42163,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that grokking—delayed generalization—is fundamentally a statistical consequence of distribution shift between training and test data, not of data sparsity or regularization, and that it can be systematically induced by…","keywords":["grokking","delayed generalization","distribution shift","class hierarchy","subclass sampling","synthetic dataset","equivariant map","late generalization"],"falsifier":"Train the same architecture on the equidistant dataset with $f=0$ (subclass absent) while sweeping weight decay and initialization scale: if grokking appears under some hyperparameters, the effect is not due to the shift alone. Alternatively, compare $f=0.01$ imbalanced sampling with a balanced sample of the same size; if balanced sampling also groks, the distribution shift is not the determining factor.","tokens_in":10255,"feed_emoji":"📉","tokens_out":6268,"duration_ms":50614,"temperature":0.7,"pith_summary":"Grokking is the phenomenon where a model's test performance suddenly jumps long after training loss has converged. This paper argues that the underlying cause is a distribution shift between training and test data, and that data sparsity and regularization merely amplify this shift rather than cause it. To support this, the authors build two synthetic datasets with controlled class–subclass structure and induce grokking by under-sampling specific subclasses, even with thousands of samples and minimal hyperparameter tuning. They also reproduce the effect on MNIST by clustering digit representations, and show the behavior appears in both MLPs and transformers. The payoff is a systematic, testable account of late generalization and a basis for better stopping criteria in training.","feed_headline":"Grokking happens when training data shifts from test data","feed_subtitle":"Controlled subclass sampling reproduces late generalization with dense data and minimal tuning.","key_machinery":"The central machinery is a pair of synthetic datasets built from multivariate normal subclasses organized into classes. In the equidistant dataset, all subclass centroids are equally far apart; in the equivariant dataset, subclasses within a class are closer to each other than to subclasses of other classes. A sampling fraction $f \\in [0,1]$ controls how many examples are kept from selected subclasses, creating a controlled distribution shift. These datasets let the authors isolate whether grokking tracks sample size, regularization, or the shift itself; the class–subclass geometry also lets them test whether relational proximity among subclasses can substitute for missing samples. A formal Definition 3.1 of grokking, based on the transition interval between memorization and generalization, provides the measurement used throughout.","core_discovery":"The paper's central claim is that grokking is a statistical phenomenon: it occurs when the training distribution is shifted away from the test distribution, specifically when some subclasses of a class hierarchy are under-represented. The authors formalize grokking with a definition based on inflection points $\\alpha_0$ and $\\alpha_1$ and the gap $\\Delta S$ between low- and high-performing test regions. Using synthetic data where classes are unions of equidistant or equivariant subclasses, they show that a faint training signal from a down-sampled subclass—even 1% of its samples—can trigger late generalization, while removing the subclass entirely ($f=0$) blocks it in the equidistant setting. In the equivariant setting, where subclasses of a class are close to each other, grokking occurs even when a subclass is completely absent, because nearby subclasses leak information. They conclude that small sample sizes are only a convenient route to the necessary distribution shift, not the cause, and that grokking can be induced with dense data and minimal tuning.","pith_inferences":["If shift magnitude rather than mere presence of samples were the driver, grokking probability should vary monotonically with $f$; the $f=0$ versus $f=0.01$ discontinuity suggests a threshold experiment worth running: sweep $f$ finely and check whether grokking appears abruptly or gradually.","The equivariant result implies that datasets with hierarchical or geometric label structure may be prone to grokking under mild imbalance; an untested extension is that curriculum or active-learning methods that deliberately expose a few examples from each subclass could trigger or suppress grokking on demand.","A practical consequence the paper leaves implicit: train–test distribution divergence measured during training (e.g., by two-sample tests on hidden representations) could serve as an early-warning signal for imminent grokking, enabling adaptive stopping rather than waiting for the late jump.","The definition of grokking via $\\Delta S$ and the support interval $S$ could be turned into an automated detector, though the paper does not propose thresholds for these quantities."],"forward_implications":["Grokking can be induced systematically in class-hierarchical data by under-sampling subclasses, without extreme sparsity or heavy regularization.","A very small number of examples from an under-represented subclass ($f=0.01$) is enough to produce late generalization, while zero examples blocks it in equidistant structure.","When subclasses are close within a class, grokking can occur even with a subclass completely absent, showing that relational structure in the data carries part of the training signal.","The phenomenon is architecture-independent: similar late-generalization curves appear for MLPs and transformers.","MNIST experiments with clustered latent representations show induced distribution shifts can reproduce grokking outside synthetic data, supporting new stopping criteria based on the likelihood of late generalization."],"supporting_citations":[{"why":"First reports the grokking phenomenon on small algorithmic datasets; the baseline result this paper reinterprets as distribution-shift-driven.","marker":"Power et al., 2022"},{"why":"Studies hyperparameters, including the initialization scale factor 8 used here, and MLP grokking on MNIST with 1,000 training examples; provides the tuning context the paper claims is not the cause.","marker":"Liu et al., 2022"},{"why":"Shows grokking in linear estimators can arise from data distribution rather than model complexity, directly motivating the paper's hypothesis.","marker":"Levi et al., 2023"},{"why":"Provides the mechanistic-interpretability account of grokking via progress measures on modular arithmetic; the paper contrasts its statistical explanation with this mechanism-focused view.","marker":"Nanda et al., 2023"},{"why":"Proposes grokking as a lazy-to-rich transition and argues weight decay alone cannot explain it; supplies the alternative explanation the paper's experiments weigh against.","marker":"Kumar et al., 2024"},{"why":"Supplies two-sample testing methods for detecting dataset shift, which the paper identifies as applicable to detecting the shifts that drive grokking.","marker":"Rabanser et al., 2019"},{"why":"Defines dataset shift, the concept the paper uses as the causal mechanism for grokking.","marker":"Quiñonero-Candela et al., 2022"}],"fun_headline_variants":["Grokking is a distribution shift, not a small sample size","Late generalization? It's about train-test shift, not sparse data","Grokking explained: it's the shift, not the sample count","Dense data can grok too—just shift train from test","Grokking's real trigger: train-test distribution gap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that under-sampling a subclass counts as a distribution shift and that the delayed generalization is caused by that shift, rather than by simply having a few examples of each subclass; the paper's own result that $f=0$ blocks grokking while $f=0.01$ triggers it makes this distinction the crux, and the fixed weight decay and initialization scale are not ablated.","fun_headline_variants_meta":{"raw":{"variants":["Grokking is a distribution shift, not a small sample size","Late generalization? It's about train-test shift, not sparse data","Grokking explained: it's the shift, not the sample count","Dense data can grok too—just shift train from test","Grokking's real trigger: train-test distribution gap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000767,"raw_usage":{"total_tokens":3414,"prompt_tokens":975,"completion_tokens":2439,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":2363}},"tokens_in":591,"tokens_out":2439,"duration_ms":17818,"temperature":1.0,"reasoning_tokens":2363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:31:27.348835+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the same architecture on the equidistant dataset with $f=0$ (subclass absent) while sweeping weight decay and initialization scale: if grokking appears under some hyperparameters, the effect is not due to the shift alone. Alternatively, compare $f=0.01$ imbalanced sampling with a balanced sample of the same size; if balanced sampling also groks, the distribution shift is not the determining factor.","supporting_citations":[{"cited_title":"Progress measures for grokking via mechanistic interpretability","cited_arxiv_id":null,"evidence_quote":"Provides the mechanistic-interpretability account of grokking via progress measures on modular arithmetic; the paper contrasts its statistical explanation with this mechanism-focused view."},{"cited_title":"J., and Pehlevan, C","cited_arxiv_id":null,"evidence_quote":"Proposes grokking as a lazy-to-rich transition and argues weight decay alone cannot explain it; supplies the alternative explanation the paper's experiments weigh against."},{"cited_title":"Failing loudly: An empirical study of methods for detecting dataset shift","cited_arxiv_id":null,"evidence_quote":"Supplies two-sample testing methods for detecting dataset shift, which the paper identifies as applicable to detecting the shifts that drive grokking."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines dataset shift, the concept the paper uses as the causal mechanism for grokking."}],"review_version":1}