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Transformers learn state-tracking solutions separately for each sequence length, while recurrent networks share a single step-by-step mechanism across lengths — even when training and test lengths match.

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2026-08-02 21:56 UTC pith:6PYY7K2X

load-bearing objection Large, careful study whose sharing-factor metric is a real contribution, but the abstract overstates the data-efficiency gap because N* is computed under a fixed optimization budget and the headline comparison ignores each model's preferred supervision format. the 3 major comments →

arxiv 2602.18333 v2 pith:6PYY7K2X submitted 2026-02-20 cs.LG cs.CL

On the "Induction Bias" in Sequence Models

M.Reza Ebrahimi , Micha\"el Defferrard , Sunny Panchal , Roland Memisevic This is my paper
classification cs.LG cs.CL
keywords induction biasstate trackingsample complexitylength generalizationtransformersrecurrent neural networksweight sharing across lengthschain-of-thought supervision
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 argues that transformers' well-known trouble with state tracking is not just an out-of-distribution failure: even when training and evaluation use the same sequence lengths, transformers need dramatically more data as the state space or sequence length grows, and they fail to reuse what they learn across lengths. By measuring the minimal training set size needed to learn modular addition and permutation composition, the authors find that recurrent networks (an LSTM and a dense state-space model) amortize the computation through a shared transition operation, while a 6-layer GPT-2-style transformer learns near-independent solutions for each length, sometimes with destructive interference. The authors attribute this to an 'induction bias': recurrent networks update a hidden state one step at a time, factorizing the next-token distribution through that state, whereas transformers recompute state from the whole context. If correct, the result reframes state-tracking limitations as an in-distribution data-efficiency problem and points to weight sharing across lengths as the mechanism behind length generalization. A sympathetic reader would care because it says scaling up data alone is an inefficient fix for transformers on compositional, stateful tasks.

Core claim

The central claim is that the presence or absence of an induction bias — the property that the next-token distribution conditioned on history factorizes through a hidden state, p(x_{t+1} | x_1,...,x_t, h_t) = p(x_{t+1} | x_t, h_t) — determines how efficiently a sequence model learns state tracking in distribution. The paper demonstrates this with a large-scale measurement of N*, the minimal sample size needed to reach a fixed validation-loss threshold, on modular addition over Zm and permutation composition over S5, under three supervision formats (outcome, chain-of-thought, aligned chain-of-thought) and three length distributions. Transformers only learn reliably with chain-of-thought super

What carries the argument

The load-bearing objects are (1) the induction bias, a formalized property of step-by-step state updates: p(x_{t+1}|x_1,...,x_t,h_t)=p(x_{t+1}|x_t,h_t), which makes state updates single-step and repeatable, and therefore shareable across lengths; and (2) the sharing factor κ, defined as the ratio of the sum of minimal sample sizes N*_n for models trained separately on each fixed length to the minimal sample size N*_joint for one model trained jointly on all lengths. κ > 1 means amortized learning across lengths, κ ≈ 1 length-isolated solutions, and κ < 1 destructive interference. The paper also uses a single-layer LSTM and a dense bilinear state-space model (hidden-to-hidden transition matri

Load-bearing premise

The comparison assumes the specific transformer, LSTM, and dense state-space models are comparable enough in capacity and optimization that the measured sample-efficiency gaps reflect architectural inductive bias rather than mismatched size, training budget, or hyperparameters.

What would settle it

Re-run the minimal-sample-size search for transformers on modular addition with matched per-step compute and hyperparameter tuning; the paper predicts κ ≤ 1 for every supervision format, so finding any transformer configuration with κ > 1, or with N* decreasing as maximum length L grows under aligned chain-of-thought, would falsify the central claim.

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

If this is right

  • State-tracking failures of transformers cannot be dismissed as an out-of-distribution artifact: the paper measures steep in-distribution sample-complexity growth with sequence length and state-space size.
  • A transformer that trains on multiple lengths at once can be less data-efficient than training separate models per length (κ < 1 in CoT), implying length-specific circuits actively compete for capacity.
  • Recurrent models in their preferred supervision format (aligned chain-of-thought) need orders of magnitude fewer samples and get more efficient as maximum length grows, because longer sequences provide reusable step-by-step supervision.
  • The sharing factor κ is a predictor of length generalization: configurations with κ ≫ 1 generalize to twice the training length; configurations with κ ≤ 1 do not.
  • For tasks that require sequential state updates, the data hunger of transformers is a function of the task's state space and length, not just of model scale.

Where Pith is reading between the lines

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

  • If the mechanism is right, 'context rot' in very long contexts could be a symptom of length-isolated circuits; a direct test would be to estimate κ on natural-language state-tracking tasks with pretrained models.
  • The paper's small, task-specific models leave open whether large pretrained transformers have the same isolation; a testable extension is to measure transfer of fine-tuning data across context-length buckets in an LLM.
  • The results suggest architectural remedies: forcing recurrence or a recurrent bottleneck into transformer layers might restore amortized sharing and improve data efficiency, not just length extrapolation.
  • κ could be used as a pre-deployment diagnostic: measure whether a model shares solution mechanisms across input-length ranges before trusting it on variable-length compositional workloads.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper reports a large-scale empirical study (≈190k training runs) of in-distribution data efficiency for transformers, LSTMs, and Dense-SSMs on modular addition and permutation composition. Across three supervision formats (outcome supervision, chain-of-thought, aligned chain-of-thought) and three length distributions, it measures the minimal sample size N* required to reach a validation threshold within a fixed optimization budget, and defines a sharing factor κ comparing a single model trained jointly over lengths with the sum of models trained per length. The main findings are that transformers require more data as sequence length and state-space size grow, that RNNs are more efficient under their preferred supervision formats, and that transformers exhibit low or even negative cross-length sharing (κ≈1 or κ<1) while RNNs show high κ under their preferred formats. The paper connects these in-distribution results to length generalization, arguing that the absence of an "induction bias" explains transformers' data inefficiency.

Significance. If the central claims hold, the paper makes a valuable contribution by moving the state-tracking debate from out-of-distribution failures to in-distribution sample efficiency, and by proposing a concrete, measurable quantity (κ) that links data amortization across lengths to length generalization. The scale of the study, the systematic binary-geometric search procedure, the inclusion of two tasks and three supervision regimes, and the appendix checks with smaller models are clear strengths. The finding that transformers learn length-specific solutions in isolation, if confirmed, has direct implications for architecture design and for understanding the data needs of transformer-based agents. However, the interpretation of κ and the abstract's broad claim rest on assumptions that are not fully checked, most notably the treatment of N* as a pure sample-complexity measure independent of the fixed optimization budget.

major comments (3)
  1. [§2, Eq. (2); Appendix A.1; §4, Eq. (4)] N* is defined as the smallest training set for which at least one configuration reaches validation loss ≤ 1e-4 within a fixed budget of 250k optimization steps (Appendix A.1). It is therefore not a pure sample-complexity measure; it conflates data efficiency with optimizer convergence at a fixed compute budget. For the joint-length model in Eq. (4), the optimization problem is harder than for a fixed-length model, so N*_joint may be inflated by the step limit. The paper never varies the step budget, so the claim (Observations 4.1–4.2) that transformers exhibit κ=0.28 under CoT because they learn "length-specific solutions in isolation" cannot be distinguished from the possibility that 250k steps is insufficient to resolve interference between length-specific shortcuts, and that more optimization would raise κ. Please provide a budget sweep (e.g., N* as a function of training steps) or co
  2. [Abstract; Table 1 (CoT rows)] The abstract states that "the amount of training data required by transformers grows much more rapidly with state-space size and sequence length than for RNNs." This is contradicted by Table 1 when the comparison is made under the same supervision format. For example, under the uniform length distribution and Chain-of-Thought, m=5: transformer N* is 148 at L=5 and 1.7K at L=20, while LSTM N* is 2K and 68.1K, respectively; LSTM grows roughly 34× versus 11× for the transformer. Similar patterns hold for m=3. The claim may be true only when each model is evaluated in its preferred format (transformer+CoT vs LSTM/Dense-SSM+ACoT), as in Observations 3.1–3.2. The abstract and introduction should qualify this, or the headline claim should be revised to avoid overgeneralization.
  3. [§2, Models; Appendix B.3] The three architectures are not matched in parameter count, depth, or per-step FLOPs: a 6-layer GPT-2 with d=256, a single-layer LSTM with d=768, and a single-layer Dense-SSM with d=256. The appendix's smaller-model checks (Tables 3–4) show similar qualitative trends, which is reassuring, but the main text does not report parameter counts or FLOPs, and the comparisons that drive the central conclusions use these specific configurations. Since the paper attributes differences to "induction bias," the lack of capacity matching leaves open whether some of the N* differences reflect model size or optimization landscape rather than architectural inductive bias. Please report parameter/FLOP counts and, if feasible, include a truly matched-size comparison for the key κ results.
minor comments (5)
  1. [§1, formal definition] The formal definition p(x_{t+1}|x_1,...,x_t,h_t)=p(x_{t+1}|x_t,h_t) uses a hidden state h_t that is not defined for transformers; the connection to the empirical notion of "induction bias" is loose. Clarify how this formal condition applies to the models studied.
  2. [Algorithm 1] The pseudocode is slightly unclear: the Success flag is set inside the for-loop but used after it; also, if the first candidate at Nmax fails, the algorithm returns -1 without setting L. Please make the control flow explicit.
  3. [Figure 6] The caption does not specify which length distribution is used for N*_joint. The text implies the joint model is trained on a distribution over lengths, but the figure would be clearer if the distribution (e.g., uniform) and the number of independent models (L-1) are stated directly.
  4. [Table 1] Many entries are marked with a dash; it would be helpful to distinguish "not attempted" from "not learned at the maximum sample size." Currently the reader must infer this from the text.
  5. [References] The reference list contains two separate entries for "Terzić" with different author lists (Terzić et al., 2025a and 2025b); the formatting is inconsistent. Also verify the year of Ebrahimi & Memisevic (2025) matches the cited NeurIPS 2025 paper.

Circularity Check

0 steps flagged

No significant circularity: all load-bearing quantities (N*, κ, OOD accuracy) are measured and compared, not derived from the proposed 'induction bias' definition.

full rationale

The paper's central quantities are empirical measurements, not predictions from a fitted model. N* is defined in Eq. (2) and estimated via the binary-geometric search in Algorithm 1; the sharing factor κ is defined in Eq. (4) using those measured N* values. The 'induction bias' factorization in Section 1 is presented as a conceptual definition and explanatory hypothesis, not as a theorem from which N*, κ, or the OOD results are derived. The main conclusions are framed as empirical observations (e.g., Observation 3.1, 4.1, 4.5) based on comparisons across models, formats, and lengths, and the claimed correlation between κ and length generalization is supported by separate OOD evaluations in Table 2. The paper does adopt the Dense-SSM variant from the authors' prior work (Ebrahimi & Memisevic, 2025), but that citation supplies a model architecture under test, not evidence for the paper's conclusions; the central claim does not reduce to that prior work or to any other self-citation. No fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is smuggled in via self-citation to force the result. The fixed 250k-step training budget may confound N* with optimization convergence, but that is a validity threat rather than a circularity because the measured values are not re-used as predictions. Hence, no specific reduction by construction was found.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

No new physical or mathematical entities are postulated. The main hand-chosen numbers are the success threshold and training budget; the key interpretive assumption is that kappa measures mechanism sharing.

free parameters (2)
  • convergence threshold epsilon = 1e-4
    Success criterion in N* definition (Section 2, Sample Efficiency); chosen by hand. The paper states that using perfect accuracy gives similar results, so this is a threshold, not a fitted constant, but it directly determines N*.
  • optimization budget = 250,000 steps
    Fixed training budget (Appendix A.1); affects whether models converge on large datasets, thereby influencing N*. If the budget were larger, some N* values might decrease.
axioms (4)
  • domain assumption Modular addition over Zm is a canonical state-tracking task for commutative operations (every finite abelian group is a product of cyclic groups); S5 is canonical for non-commutative operations (Cayley's theorem).
    Section 2, Methodology; used to justify task selection and to generalize findings to all finite groups.
  • domain assumption A validation loss below epsilon (1e-4) indicates successful learning of the task.
    Section 2, Sample Efficiency; defines the success criterion for N*.
  • domain assumption RNNs satisfy the factorization p(x_{t+1} | x_1,...,x_t,h_t) = p(x_{t+1} | x_t,h_t), and this 'induction bias' is absent in transformers.
    Section 1, formal definition; used to explain observed differences in data efficiency and sharing factor. For RNNs this follows from h_t being a sufficient statistic; the absence in transformers is an architectural claim.
  • ad hoc to paper The sharing factor kappa, as defined in Eq. (4), measures mechanism sharing across lengths rather than confounds such as distribution shift or task difficulty.
    Section 4; the interpretation of kappa > 1 as amortized learning is asserted, not derived, and could be influenced by the length distribution used for the joint task.

pith-pipeline@v1.3.0-alltime-deepseek · 20717 in / 17673 out tokens · 142171 ms · 2026-08-02T21:56:21.176755+00:00 · methodology

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read the original abstract

Despite the remarkable practical success of transformer-based language models, recent work has raised concerns about their ability to perform state tracking. In particular, a growing body of literature has shown this limitation primarily through failures in out-of-distribution (OOD) generalization, such as length extrapolation. In this work, we shift attention to the in-distribution implications of these limitations. We conduct a large-scale experimental study of the data efficiency of transformers and recurrent neural networks (RNNs) across multiple supervision regimes. We find that the amount of training data required by transformers grows much more rapidly with state-space size and sequence length than for RNNs. Furthermore, we analyze the extent to which learned state-tracking mechanisms are shared across different sequence lengths. We show that transformers exhibit negligible or even detrimental weight sharing across lengths, indicating that they learn length-specific solutions in isolation. In contrast, recurrent models exhibit effective amortized learning by sharing weights across lengths, allowing data from one sequence length to improve performance on others. Together, these results demonstrate that state tracking remains a fundamental challenge for transformers, even when training and evaluation distributions match.

Figures

Figures reproduced from arXiv: 2602.18333 by Micha\"el Defferrard, M.Reza Ebrahimi, Roland Memisevic, Sunny Panchal.

Figure 1
Figure 1. Figure 1: Example of the three task formats for the addition modulo 5 task applied to the sequence 2 1 0 3 4. Length Distributions: For each generated sample, we first determine the sequence length n ∈ {2, . . . , L}, where L denotes the maximum sequence length. We then sample a sequence x ∈ Z n m without replacement to ensure that every sample in the dataset is unique. We use three distinct strategies for length se… view at source ↗
Figure 2
Figure 2. Figure 2: further illustrates this gap in sample requirements for the case m = 2 (parity) across the two formats. It has been hypothesized that by outputting intermediate steps autoregressively, the model can attend to its own pre￾vious outputs, effectively simulating a larger depth circuit (Li et al., 2024), and the results confirm this hypothesis. In contrast, Aligned Chain-of-Thought forces the model to compress … view at source ↗
Figure 4
Figure 4. Figure 4: Sample complexity (log scale) for transformers trained with CoT and RNNs with ACoT on the parity task. RNNs exhibit the expected improvement in sample efficiency with increasing sequence length, while transformers fail to leverage the additional supervision. Observation 3.5 With outcome supervision, short sequences are more valuable for learning than long sequences in recurrent models. 5 10 15 20 25 30 Seq… view at source ↗
Figure 3
Figure 3. Figure 3: N ∗ for the outcome supervision format with a uniform length distribution and m = 2 (parity). In the absence of interme￾diate supervision, single-layer RNNs significantly outperform the 6-layer transformer. Observation 3.4 With intermediate supervision, longer sequences im￾prove the data efficiency of recurrent models but not transformers. Intuitively, under formats with intermediate supervision (CoT or AC… view at source ↗
Figure 5
Figure 5. Figure 5: Sample complexity (log scale) in the Outcome Supervi￾sion format for the uniform and short-to-long length setting, with m = 2 (parity). Recurrent models require fewer training samples under the short-to-long setting, indicating that shorter sequences provides a stronger learning signal. In the Outcome Supervision setting, we compare the data requirements under the uniform and short-to-long length 5 [PITH_… view at source ↗
Figure 6
Figure 6. Figure 6: Sample complexity comparison between training a single model jointly across all sequence lengths and the cumulative sample complexity of independently trained models for each sequence length, together with the corresponding sharing factor, for the task of addition modulo 5. The results suggest that transformers learn largely isolated solutions for each sequence length. 8 [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 7
Figure 7. Figure 7: Similar to [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Held-Out Transition-Pair Falsifier for Long-Horizon Non-Abelian State Tracking

    cs.LG 2026-06 unverdicted novelty 7.0

    Held-out transition-pair falsifier on S3 x S3 shows projected recurrent state model achieves perfect final-state prediction up to 1M+ tokens while matched baselines fail.

  2. Continuity Laws for Sequential Models

    cs.LG 2026-05 unverdicted novelty 6.0

    S4 models exhibit stable time-continuity unlike sensitive S6 models, with task continuity predicting performance and enabling temporal subsampling for better efficiency.

Reference graph

Works this paper leans on

4 extracted references · 1 linked inside Pith · cited by 2 Pith papers

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    Induction Bias

    The algorithm conducts a search over sample sizes, combining an initial exponential reduction phase with a subsequent binary search phase. The search begins at a predefined maximum sample size Nmax. For any candidate size N, the algorithm trains models using multiple configurations drawn from a fixed hyperparameter grid Φ. In our implementation, each eval...

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    Induction Bias

    Other architectural parameters, including an MLP expansion factor of 4, follow the default GPT-2 (small) settings. Both the LSTM and Dense-SSM use a single-layer recurrent cell followed by a linear classification head to map the hidden state to token logits. We use an input and hidden dimension of768 for the LSTM, and 256 for the Dense-SSM. See Ebrahimi &...

    2025

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    Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization.CoRR, abs/1412.6980,

    Pith/arXiv arXiv

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    On the expressiveness and length generalization of selective state-space mod- els on regular languages

    Terzi´c, A., Hersche, M., Camposampiero, G., Hofmann, T., Sebastian, A., and Rahimi, A. On the expressiveness and length generalization of selective state-space mod- els on regular languages. InProceedings of the AAAI Conference on Artificial Intelligence, 2025a. Terzi´c, M., Rahimi, A., and et al. Structured sparse transition matrices to enable state tra...

    2025