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REVIEW 4 major objections 5 minor 42 references

Beyond Human-Like Processing: Large Language Models Perform Equivalently on Forward and Backward Scientific Text

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Pith's one-line read This paper claims that language models trained on character-reversed neuroscience text predict experimental outcomes as well as models trained on normal text, showing they act as general pattern learners rather than human-like language…

desk verdict A genuinely interesting training experiment, but the equivalence claim is unsupported by the statistics and is confounded by where the outcome appears in reversed text. read the letter →

arxiv 2411.11061 v1 pith:TUZY6BAY submitted 2024-11-17 cs.CL q-bio.NC

classification cs.CLq-bio.NC
keywords largelanguagemodelsmodelevaluationbackwardtextcharacter-levelreversalneurosciencepredictionperplexitypatternlearninghuman-likeprocessing
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

This paper asks whether large language models succeed on language tasks because they model human language or because they are flexible pattern learners. To separate the two, the authors trained GPT-2 models from scratch on twenty years of neuroscience publications in normal order and in character-reversed order, then tested both on BrainBench, a benchmark where models and human experts choose which of two versions of an abstract reports the real experimental outcome. Backward-trained models performed as well as, and at the largest size better than, forward-trained models, and both matched or exceeded human experts. The authors conclude that the forward structure of language is not needed for these models' predictive success, so their linguistic performance should not be treated as evidence for human-like processing.

What carries the argument

The central mechanism is character-level reversal combined with retraining: a fresh byte-pair-encoding tokenizer is learned on the reversed neuroscience corpus, and GPT-2 models of three sizes are trained from scratch on the reversed sequence, preserving the next-token objective. Evaluation uses perplexity: for each BrainBench item the model chooses the version with lower perplexity, and confidence is the perplexity gap. This design isolates whether the forward direction of natural language is necessary for the model's predictive success.

What would settle it

Take the 200 BrainBench items and score both model types on versions with the final outcome sentence removed, or with the outcome sentence moved to the start of an otherwise forward abstract; if backward-trained models lose their parity once the decisive sentence is no longer automatically early, the equivalence is a position effect rather than evidence of format-general learning.

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Extended reading notes

Core claim

The paper reports that GPT-2 models trained from scratch on twenty years of neuroscience text reversed character-by-character perform as well on the BrainBench neuroscience-outcome prediction benchmark as models trained on normal text, with the largest backward model exceeding the published human-expert average. Because the reversed text violates the forward direction of human language, the authors read this as evidence that transformer language models are general pattern-learning machines: they extract useful predictive structure from any sufficiently regular input, and their success on language tasks should not be taken as evidence of human-like language mechanisms.

Load-bearing premise

The load-bearing assumption is that choosing the less surprising abstract measures the same predictive skill in both formats, even though character reversal relocates each abstract's decisive outcome statement to the beginning of the sequence.

Editorial extensions

If this is right

  • Because backward-trained models match forward-trained models, the forward direction of natural language is not necessary for accurate next-token prediction on scientific text.
  • At the largest model size tested, both directions exceed the published human-expert average on BrainBench, so expert-level neuroscience prediction does not require human-like language order.
  • Backward-trained models score lower on human-like difficulty judgments yet remain accurate, meaning human and model errors on BrainBench only weakly overlap.
  • The authors recommend that strong LLM performance on linguistic tasks should not be read as evidence for human-like cognitive mechanisms.

Reading between the lines

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

  • A testable extension: move the outcome sentence to the front of forward-order abstracts; if forward models then match backward models, the equivalence is positional rather than format-invariant.
  • The same forward-versus-reversal comparison could be applied to non-linguistic sequences, such as time series or protein sequences, to see whether the equivalence is a general property of next-token training or particular to text.
  • A caution about the benchmark: because the altered BrainBench abstracts change the outcome, a model that finds the outcome sentence early in the reversed input gets a shortcut; scoring models on result-free portions would test whether they actually track the reasoning of the abstract.
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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 / 5 minor

Summary. The paper trains GPT-2 models of three sizes (124M, 355M, 774M) from scratch on two decades of neuroscience literature, either in normal forward order or with all characters reversed, and evaluates them on the BrainBench benchmark, where models must choose which of two abstracts is the original. The central claim is that forward- and backward-trained models perform equally well on this task, and that this equivalence supports the view that transformer-based LLMs are general pattern-learning machines rather than human-like language processors. The paper also reports that backward-trained models have higher perplexity on validation and test items, that forward-trained models correlate more with human difficulty judgments, and that models at the largest size match or exceed human expert performance.

Significance. If the central claim were established, this would be an important result: it would demonstrate format-invariant predictive learning in transformers and strengthen the case that LLM success on linguistic tasks should not be interpreted as evidence for human-like mechanisms. The paper has strengths: it retrains models from scratch with domain-specific tokenizers, uses a challenging expert-level benchmark, and provides a direct forward/backward comparison. However, the main inference is currently undercut by the weak statistics and by a positional confound in the backward condition, so the significance of the result in its present form is limited.

major comments (4)
  1. [Section 2.1, Fig. 3] The claim that forward- and backward-trained models 'performed equally well' rests on a non-significant ANOVA (F(1,2)=2.77, p=.238). A non-significant difference does not establish equivalence, especially when only three model sizes are available and there are no repeated training runs. The authors should report an equivalence test (e.g., TOST or a Bayesian analysis) or a sensitivity power analysis showing what effect sizes the design could detect. Without this, the central conclusion is unsupported.
  2. [Methods 4.2, 4.8 and Section 2.1] The character-level reversal confounds the forward/backward comparison. BrainBench abstracts differ primarily in the outcome statement, which is typically located near the end of the forward abstract. Reversing the characters moves these discriminating tokens to the beginning of the backward input, while the decision rule (Eq. 1) averages log-likelihood over the entire sequence. Thus backward-trained models receive early access to the exact tokens that determine the correct choice, whereas forward-trained models must process the shared context first. The paper does not control for token position, so the observed equivalence may reflect a positional artifact rather than format-invariant pattern learning. An analysis that conditions on the position of the altered tokens, or that truncates the input to remove the early-outcome advantage, is needed.
  3. [Methods 4.8, Section 5.1] The backward-trained models use a separately trained tokenizer on reversed text, which produces a different vocabulary (27% neuroscience-related tokens versus 25.4% for the forward tokenizer). This tokenizer difference is a confound: the similar accuracy could be due to the tokenizer rather than to format invariance. The authors themselves hypothesize this possibility in Section 2.1 but do not test it. A control experiment using the same tokenizer for both directions, or evaluating a forward model with the backward tokenizer, would be needed to separate these factors.
  4. [Methods 4.9] The statistical model is underspecified. The text states that model size was coded as a continuous variable and that model was treated as a within-subjects factor, but does not give the actual repeated-measures error term or the model formula used in aov(). The reported degrees of freedom (1,2) for the direction effect suggest that the comparison is effectively across only three model sizes, which provides very low power. Please clarify the design and consider a more appropriate analysis that treats items as the random factor and models the nested structure explicitly.
minor comments (5)
  1. [Abstract] The abstract states that models performed equally well 'eclipsing human expert performance for both forward and backward orders,' but Section 2.1 and Fig. 3 show that only the 774M models surpass human experts, while the 124M models match them. Please soften this wording to avoid overstatement.
  2. [Methods 4.2 vs. 4.8] It is unclear whether the prompt 'You are a neuroscientist...' is also reversed in the backward condition or whether it is prepended after character-level reversal. Please clarify, as this affects the input distribution seen by the backward models.
  3. [Section 2.3] The reported t(2)=20.848, p=0.002 appears implausible for a t-distribution with 2 degrees of freedom; please verify the statistic and the degrees of freedom.
  4. [Methods 4.9] There is a typo in the text: 'ANOV A' should be 'ANOVA'. Additionally, the description of the error term in aov() should be explicit (e.g., Error(item/(direction*model_size))).
  5. [Discussion] There is a typo in 'Moreover. our study' in the Discussion; it should be 'Moreover, our study'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the forward-backward equivalence is an empirical benchmark result with no by-construction reduction.

full rationale

The central claim that forward- and backward-trained GPT-2 models perform equivalently on BrainBench is an empirical comparison, not a derivation from fitted parameters or self-referential definitions. Models are trained from scratch on a fixed corpus and evaluated with the fixed perplexity rule in Eq. 1; no parameter is fitted to BrainBench outcomes, and no equation is defined in terms of the conclusion. BrainBench and the human expert data come from the same group's prior work (ref 38), but those are real external measurements from 171 neuroscientists and are not produced by the present models or fitted values, so citing them is independent evidence rather than circular. The tokenizer citation (ref 37) is methodological and not load-bearing for the equivalence claim. The skeptic concern that character-level reversal moves discriminating tokens earlier in the input is a potential confound or correctness risk, not a circular step: the paper's Eq. 1 and Methods 4.8 do not reduce the forward-backward comparison to its inputs by construction, and no specific reduction can be exhibited. The paper is therefore self-contained against an external benchmark, with only minor non-load-bearing self-citations.

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

No free parameters are fitted to BrainBench outcomes; the training hyperparameters are standard. The paper relies on several domain assumptions about the validity of perplexity-based selection, the construct validity of BrainBench, and the comparability of forward-only human data to backward-trained models. No new entities are introduced.

assumptions (5)
  • domain assumption Lower perplexity is a valid decision rule for selecting the correct BrainBench abstract.
    Methods 4.2 defines the model's choice as the abstract with lower perplexity; this assumes perplexity differences track correctness equally in forward and backward text.
  • domain assumption Character-level reversal removes human-like language structure while retaining sufficient structure for learning.
    Section 1 and Methods 4.8 assume reversed text is an impossible language that still contains predictive patterns; the paper does not formally characterize what sufficiently structured means.
  • domain assumption BrainBench accuracy measures the ability to predict neuroscience experimental outcomes.
    The benchmark's construct validity is inherited from [38]; the paper does not independently validate it.
  • domain assumption The human expert data from [38] are directly comparable to models that read either forward or backward text.
    Section 2.1 and Figure 3 compare models to human accuracy, but humans were only tested on forward text; this assumption is partly violated.
  • domain assumption GPT-2's architecture is representative of transformer-based LLMs generally.
    Section 1 justifies GPT-2 as foundation of GPT-4, Claude, and Gemini, but the results are not shown to generalize to other architectures or pretraining regimes.

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Cite this review

Pith. "Pith review of Beyond Human-Like Processing: Large Language Models Perform Equivalently on Forward and Backward Scientific Text." pith.science (2026). https://pith.science/paper/TUZY6BAY

@misc{pith2026241111061,
  author       = {Pith},
  title        = {Pith review of: Beyond Human-Like Processing: Large Language Models Perform Equivalently on Forward and Backward Scientific Text},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUZY6BAY}},
  note         = {Machine review of arXiv:2411.11061}
}
read the original abstract

The impressive performance of large language models (LLMs) has led to their consideration as models of human language processing. Instead, we suggest that the success of LLMs arises from the flexibility of the transformer learning architecture. To evaluate this conjecture, we trained LLMs on scientific texts that were either in a forward or backward format. Despite backward text being inconsistent with the structure of human languages, we found that LLMs performed equally well in either format on a neuroscience benchmark, eclipsing human expert performance for both forward and backward orders. Our results are consistent with the success of transformers across diverse domains, such as weather prediction and protein design. This widespread success is attributable to LLM's ability to extract predictive patterns from any sufficiently structured input. Given their generality, we suggest caution in interpreting LLM's success in linguistic tasks as evidence for human-like mechanisms.

Figures

Figures reproduced from arXiv: 2411.11061 by the authors.

Figure 1
Figure 1. Forward and backward tokenization and training. Both forward and backward trained models were optimized to predict the next token in the training data sequence. (A) The forward tokenizer and models were trained on 20 years of neuroscience literature. (B) In contrast, the backward tokenizer and models were trained on the same data with text reversed at the character level. on neuroscience experiment prediction tasks … view at source ↗
Figure 2
Figure 2. BrainBench is a benchmark for neuroscience. (A) BrainBench evaluates test￾takers’ ability to predict neuroscience results. Test-takers chose between the original abstract and one altered to significantly change the result while maintaining coherency. (B) Human experts and Language Models (LLMs) were tasked with selecting the correct (i.e., original) version from the two options. Human experts made choices, and provi… view at source ↗
Figure 3
Figure 3. BrainBench performance of GPT-2 models trained forward and backward. GPT￾2 models, trained from scratch on two decades of neuroscience literature, rival or exceed human expert performance, demarcated by the blue dashed line. Models trained on the same data re￾versed at the character level performed non-significantly better than their forward-trained coun￾terparts. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Backward-trained models exhibit higher perplexities on both validation and BrainBench items. (A) Perplexity of the validation set items; (B) Perplexity of the correct options in BrainBench items. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Comparison of model and human judgments on BrainBench difficulty. Model judgments (both forward and backward-trained) correlate more strongly with each other than with human expert judgments. Backward-trained models show significantly lower correlation to human judgmen…

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Works this paper leans on

42 extracted references · 21 canonical work pages

  1. [1]

    author Vaswani, A. et al. title Attention Is All You Need ( year 2023 ). ://arxiv.org/abs/1706.03762. note ArXiv:1706.03762 [cs]

  2. [2]

    title Language Models as Models of Language ( year 2024 )

    author Millière, R. title Language Models as Models of Language ( year 2024 ). ://arxiv.org/abs/2408.07144. note ArXiv:2408.07144 [cs]

  3. [3]

    & author Blything, L

    author Ambridge, B. & author Blything, L. title Large language models are better than theoretical linguists at theoretical linguistics . journal Theoretical Linguistics volume 50 , pages 33--48 ( year 2024 ). ://www.degruyter.com/document/doi/10.1515/tl-2024-2002/html

  4. [4]

    , author Dupoux, E

    author Linzen, T. , author Dupoux, E. & author Goldberg, Y. title Assessing the Ability of LSTMs to Learn Syntax - Sensitive Dependencies ( year 2016 ). ://arxiv.org/abs/1611.01368. note ArXiv:1611.01368 [cs]

  5. [5]

    title Assessing BERT 's Syntactic Abilities ( year 2019 )

    author Goldberg, Y. title Assessing BERT 's Syntactic Abilities ( year 2019 ). ://arxiv.org/abs/1901.05287. note ArXiv:1901.05287 [cs]

  6. [6]

    author Lakretz, Y. et al. title Mechanisms for handling nested dependencies in neural-network language models and humans . journal Cognition volume 213 , pages 104699 ( year 2021 ). ://linkinghub.elsevier.com/retrieve/pii/S0010027721001189

  7. [7]

    author Lampinen, A. K. title Can language models handle recursively nested grammatical structures? A case study on comparing models and humans ( year 2023 ). ://arxiv.org/abs/2210.15303. note ArXiv:2210.15303 [cs]

  8. [8]

    , author Kovaleva, O

    author Rogers, A. , author Kovaleva, O. & author Rumshisky, A. title A Primer in BERTology : What we know about how BERT works ( year 2020 ). ://arxiv.org/abs/2002.12327. note ArXiv:2002.12327 [cs]

Show all 42 references
  1. [9]

    , author Tan, Y

    author Lin, Y. , author Tan, Y. C. & author Frank, R. title Open Sesame : Getting Inside BERT 's Linguistic Knowledge ( year 2019 ). ://arxiv.org/abs/1906.01698. note ArXiv:1906.01698 [cs]

  2. [10]

    author Schrimpf, M. et al. title The neural architecture of language: Integrative modeling converges on predictive processing . journal Proceedings of the National Academy of Sciences volume 118 , pages e2105646118 ( year 2021 ). ://pnas.org/doi/full/10.1073/pnas.2105646118

  3. [11]

    & author Ramscar, M

    author Arnon, I. & author Ramscar, M. title Granularity and the acquisition of grammatical gender: How order-of-acquisition affects what gets learned . journal Cognition volume 122 , pages 292--305 ( year 2012 )

  4. [12]

    , author Roads, B

    author Aho, K. , author Roads, B. D. & author Love, B. C. title Signatures of cross-modal alignment in children’s early concepts . journal Proceedings of the National Academy of Sciences volume 120 , pages e2309688120 ( year 2023 ). ://pnas.org/doi/10.1073/pnas.2309688120

  5. [13]

    author Alayrac, J.-B. et al. title Flamingo: a Visual Language Model for Few - Shot Learning ( year 2022 ). ://arxiv.org/abs/2204.14198. note ArXiv:2204.14198 [cs]

  6. [14]

    author Clerkin, E. M. & author Smith, L. B. title Real-world statistics at two timescales and a mechanism for infant learning of object names . journal Proceedings of the National Academy of Sciences volume 119 , pages e2123239119 ( year 2022 ). ://pnas.org/doi/full/10.1073/pn...

  7. [15]

    author Bodnar, C. et al. title Aurora: A Foundation Model of the Atmosphere ( year 2024 ). ://arxiv.org/abs/2405.13063. note ArXiv:2405.13063 [physics]

  8. [16]

    , author Kong, W

    author Das, A. , author Kong, W. , author Sen, R. & author Zhou, Y. title A decoder-only foundation model for time-series forecasting ( year 2024 ). ://arxiv.org/abs/2310.10688. note ArXiv:2310.10688 [cs]

  9. [17]

    author Song, C. H. et al. title LLM - Planner : Few - Shot Grounded Planning for Embodied Agents with Large Language Models . In booktitle 2023 IEEE / CVF International Conference on Computer Vision ( ICCV ) , pages 2986--2997 ( publisher IEEE , address Paris, France , year 20...

  10. [18]

    , author Aggarwal, R

    author Bagal, V. , author Aggarwal, R. , author Vinod, P. K. & author Priyakumar, U. D. title MolGPT : Molecular Generation Using a Transformer - Decoder Model . journal Journal of Chemical Information and Modeling volume 62 , pages 2064--2076 ( year 2022 ). ://pubs.acs.org/do...

  11. [19]

    , author Schmidt, S

    author Ferruz, N. , author Schmidt, S. & author Höcker, B. title ProtGPT2 is a deep unsupervised language model for protein design . journal Nature Communications volume 13 , pages 4348 ( year 2022 ). ://www.nature.com/articles/s41467-022-32007-7

  12. [20]

    author Chen, M. et al. title Evaluating Large Language Models Trained on Code ( year 2021 ). ://arxiv.org/abs/2107.03374. note ArXiv:2107.03374 [cs]

  13. [21]

    title How children learn to communicate discriminatively

    author Ramscar, M. title How children learn to communicate discriminatively . journal Journal of Child Language volume 48 , pages 984--1022 ( year 2021 )

  14. [22]

    & author Matthews, D

    author Bannard, C. & author Matthews, D. title Stored word sequences in language learning: The effect of familiarity on children's repetition of four-word combinations . journal Psychological science volume 19 , pages 241--248 ( year 2008 )

  15. [23]

    title Expectation-based syntactic comprehension

    author Levy, R. title Expectation-based syntactic comprehension . journal Cognition volume 106 , pages 1126--1177 ( year 2008 ). ://linkinghub.elsevier.com/retrieve/pii/S0010027707001436

  16. [24]

    title A Maximum - Entropy - Inspired Parser

    author Charniak, E. title A Maximum - Entropy - Inspired Parser . In booktitle 1st Meeting of the North American Chapter of the Association for Computational Linguistics ( year 2000 )

  17. [25]

    title Suffixing, prefixing, and the functional order of regularities in meaningful strings

    author Ramscar, M. title Suffixing, prefixing, and the functional order of regularities in meaningful strings . journal Psihologija volume 46 , pages 377--396 ( year 2013 )

  18. [26]

    , author Milin, P

    author Dye, M. , author Milin, P. , author Futrell, R. & author Ramscar, M. title A functional theory of gender paradigms . In booktitle Perspectives on morphological organization , pages 212--239 ( publisher Brill , year 2017 )

  19. [27]

    , author Milin, P

    author Dye, M. , author Milin, P. , author Futrell, R. & author Ramscar, M. title Alternative solutions to a language design problem: The role of adjectives and gender marking in efficient communication . journal Topics in cognitive science volume 10 , pages 209--224 ( year 2018 )

  20. [28]

    author Gold, E. M. title Language identification in the limit . journal Information and Control volume 10 , pages 447--474 ( year 1967 ). ://linkinghub.elsevier.com/retrieve/pii/S0019995867911655

  21. [29]

    , author Greco, M

    author Moro, A. , author Greco, M. & author Cappa, S. F. title Large languages, impossible languages and human brains . journal Cortex volume 167 , pages 82--85 ( year 2023 ). ://linkinghub.elsevier.com/retrieve/pii/S0010945223001752

  22. [30]

    & author Bowers, J

    author Mitchell, J. & author Bowers, J. title Priorless Recurrent Networks Learn Curiously . In booktitle Proceedings of the 28th International Conference on Computational Linguistics , pages 5147--5158 ( publisher International Committee on Computational Linguistics , address...

  23. [31]

    , author Papadimitriou, I

    author Kallini, J. , author Papadimitriou, I. , author Futrell, R. , author Mahowald, K. & author Potts, C. title Mission: Impossible Language Models ( year 2024 ). ://arxiv.org/abs/2401.06416. note ArXiv:2401.06416 [cs]

  24. [32]

    , author Wenger, J

    author Papadopoulos, V. , author Wenger, J. & author Hongler, C. title Arrows of Time for Large Language Models ( year 2024 ). ://arxiv.org/abs/2401.17505. note ArXiv:2401.17505 [cs]

  25. [33]

    author Radford, A. et al. title Language Models are Unsupervised Multitask Learners ( year 2019 )

  26. [34]

    title GPT -4 Technical Report ( year 2024 )

    author OpenAI et al. title GPT -4 Technical Report ( year 2024 ). ://arxiv.org/abs/2303.08774. note \_eprint: 2303.08774

  27. [35]

    title The Claude 3 Model Family : Opus , Sonnet , Haiku

    author Anthropic . title The Claude 3 Model Family : Opus , Sonnet , Haiku . type Technical Report , institution Anthropic ( year 2024 ). ://www-cdn.anthropic.com//de8ba9b01c9ab7cbabf5c33b80b7bbc618857627/Model_Card_Claude_3.pdf

  28. [36]

    author Team, G. et al. title Gemini: A Family of Highly Capable Multimodal Models ( year 2024 ). ://arxiv.org/abs/2312.11805. note \_eprint: 2312.11805

  29. [37]

    , author Sun, G

    author Luo, X. , author Sun, G. & author Love, B. C. title Matching domain experts by training from scratch on domain knowledge ( year 2024 ). ://arxiv.org/abs/2405.09395. note ArXiv:2405.09395 [cs, q-bio]

  30. [38]

    author Luo, X. et al. title Large language models surpass human experts in predicting neuroscience results ( year 2024 ). ://arxiv.org/abs/2403.03230. note ArXiv:2403.03230 [cs, q-bio]

  31. [39]

    author Bender, E. M. , author Gebru, T. , author McMillan-Major, A. & author Shmitchell, S. title On the Dangers of Stochastic Parrots : Can Language Models Be Too Big ? In booktitle Proceedings of the 2021 ACM Conference on Fairness , Accountability , and Transparency , pages...

  32. [40]

    title A new algorithm for data compression

    author Gage, P. title A new algorithm for data compression . journal The C Users Journal archive volume 12 , pages 23--38 ( year 1994 ). ://api.semanticscholar.org/CorpusID:59804030

  33. [41]

    , author Haddow, B

    author Sennrich, R. , author Haddow, B. & author Birch, A. title Neural Machine Translation of Rare Words with Subword Units ( year 2016 ). ://arxiv.org/abs/1508.07909. note ArXiv:1508.07909 [cs]

  34. [42]

    & author Hutter, F

    author Loshchilov, I. & author Hutter, F. title Decoupled Weight Decay Regularization ( year 2019 ). ://arxiv.org/abs/1711.05101. note ArXiv:1711.05101 [cs, math]

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Reviewed August 12, 2026 · model on record in the stance chip above.