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REVIEW 4 major objections 5 minor 1 cited by

Scalable Complexity Control Facilitates Reasoning Ability of LLMs

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that controlling model complexity during pretraining, through the initialization rate and weight decay, improves scaling laws and reasoning benchmarks across model and data sizes.

desk verdict Plausible and valuable direction, but the scaling-law claim is not backed by the data as presented—send to review, demand better evidence. read the letter →

arxiv 2505.23013 v1 pith:SL3RUARY submitted 2025-05-29 cs.LG

Liangkai Hang , Junjie Yao , Zhiwei Bai , Tianyi Chen , Yang Chen , Rongjie Diao , Hezhou Li , Pengxiao Lin
show 12 more authors
Zhiwei Wang Cheng Xu Zhongwang Zhang Zhangchen Zhou Zhiyu Li Zehao Lin Kai Chen Feiyu Xiong Yaoyu Zhang Weinan E Hongkang Yang Zhi-Qin John Xu
This is my paper
classification cs.LG
keywords complexitycontrolinitializationrateweightdecayscalinglawLLMreasoningpretrainingcondensationgeneralization
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

The paper tries to establish that how much complexity a language model is allowed during pretraining is a primary lever on its reasoning ability. Concretely, initializing weights with a larger initialization rate (so smaller initial scales) and applying stronger weight decay makes test loss fall faster as either model size or data size grows. The authors back this with scaling-law curves on 0.8B models and with large-scale runs: a 0.9B model trained on 600B tokens gains 4.6 points on average over 15 benchmarks against a high-complexity baseline, and a 2.4B model trained on 1T tokens gains 3.4 points, with the largest jumps on math and commonsense reasoning. If true, these are essentially free gains: no new data, no new architecture, only a different choice of two standard hyperparameters.

What carries the argument

The mechanism is a pair of hyperparameters: the initialization rate, defined by sampling each weight from a normal distribution with scale $d_{\mathrm{in}}^{-\gamma}$, and the weight decay coefficient $\lambda$ in the optimizer update. A rate above $\gamma = 0.5$ counts as small-complexity initialization, while $\gamma = 0.5$ is the standard He/LeCun-style choice. The theoretical engine is an interpolated function norm $\|f\|_\gamma$ that connects the kernel-regime norm at $\gamma = -1/2$ with the Barron norm at $\gamma = -1$; under the paper's idealization, minimizing this norm makes training prefer sparse deep circuits over dense shallow circuits. In the trained network this preference shows up as condensation (neurons collapsing into few groups), higher cosine similarity among embeddings, and larger condensation and low-rank metrics in attention projections.

What would settle it

Measure the test-loss gap between the small- and large-complexity configurations on a 2.4B model trained on 10B or more tokens: if the gap closes as data grows, the leftward shift in the paper's scaling-law figure is not a durable scaling-law improvement.

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Extended reading notes

Core claim

The paper's central claim is that model complexity, set by the initialization rate and weight decay, is a controllable proxy for whether a pretrained LLM learns compressed, generalizable rules or memorizes surface patterns. In the language of the paper, small-complexity pretraining steers training toward a circuit ensemble with few, deep circuits rather than many shallow ones, which is why it improves reasoning while also improving the scaling-law exponent. The empirical evidence has three parts: test-loss curves for 0.8B models shift left when complexity is reduced, in both the data-size and model-size directions; at scale, the small-complexity 0.9B/600B model beats the large-complexity model on 14 of 15 benchmarks and the 2.4B/1T model on 13 of 15, with average gains of +4.6 and +3.4 points; and mechanistic analyses show small complexity increases embedding similarity and condensation/low-rank structure in attention projections. The paper also argues, via a norm-interpolation calculation, that lower initialization rate favors exactly the sparse-deep solutions that generalize.

Load-bearing premise

The argument depends on the assumption that test-loss rankings seen on small 0.8B models trained on at most 1.4B tokens still hold at 600B and 1T token scales, and that the 2.4B small-complexity gain is not caused by the extra normalization used only in that run.

Editorial extensions

If this is right

  • Higher initialization rate and weight decay improve sample efficiency: the same test loss is reached with fewer tokens or fewer parameters.
  • A constant initialization rate, rather than a fixed initialization standard deviation, is the right scaling prescription, so the recipe can transfer across model widths.
  • Complexity control stacks with supervised fine-tuning: the reported SFT gains are larger on small-complexity base models.
  • The gains concentrate on reasoning-heavy tasks such as GSM8K, MATH, and CommonsenseQA, so complexity control can be treated as a reasoning-oriented pretraining intervention.
  • Realizing the gains at scale requires stabilizing training with a moderate rate and additional normalization, so the recipe must be adapted as model size grows.

Reading between the lines

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

  • If the scaling-law ordering from 0.8B models holds beyond 10B parameters, choosing the initialization rate and weight decay may become standard pretraining practice, but that extrapolation goes beyond what the paper's single large runs can confirm.
  • The sparse-deep-circuits account predicts that small-complexity models should show measurable phase-transition or grokking-like behavior on small algorithmic tasks; this is testable with existing circuit-probing experiments.
  • Because the 2.4B small run used extra embedding and sandwich normalization, the cleanest reading is that complexity control plus stability engineering jointly produced the gain; the paper does not isolate the two components.
  • The authors point to post-training complexity control as future work; if it transfers, the same reasoning gains could be obtained after pretraining at even lower cost.
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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. This paper proposes that controlling model complexity during pretraining, implemented through the initialization rate γ and the weight decay coefficient λ, improves the scaling laws of large language models and enhances their reasoning ability. The authors train Llama-style models at several scales: 180M models on 40B tokens for analysis, 0.8B models on 0.2–1.4B tokens for scaling-law curves, 0.9B models on 600B tokens, and 2.4B models on 1T tokens. They report that small-complexity configurations (larger γ, larger λ) shift test-loss curves leftward and produce higher benchmark scores, with average gains of +4.6% and +3.4% over 15 tasks for the 0.9B and 2.4B comparisons, respectively. The paper also includes mechanistic analyses of embedding similarity and attention-matrix condensation, and a heuristic theoretical argument based on interpolated function norms and circuit-ensemble representations.

Significance. If the claims are substantiated, the paper offers a simple and computationally free intervention for improving LLM reasoning by tuning two hyperparameters, with no additional data or architecture changes. The reported gains on math and commonsense benchmarks are practically meaningful, and the focus on complexity control as a design principle is a useful complement to data- and scale-centric approaches. The authors have provided substantial experimental effort, including training a 2.4B model on 1T tokens, and the auxiliary analyses (embedding similarity, attention condensation) suggest a mechanism consistent with the condensation literature. However, as detailed below, the headline scaling-law conclusion currently rests on evidence that is too weak to fully support the paper's claims.

major comments (4)
  1. [4.1, Figure 2] The scaling-law conclusion is extrapolated from a heavily undertrained regime: the 0.8B models are trained on at most 1.4B tokens, which gives a token-to-parameter ratio below 2:1, while the headline benchmark comparisons use roughly 600B tokens per 0.9B model and 1T tokens per 2.4B model. The figure shows single runs with no error bars, and the claimed difference in scaling slopes is assessed visually rather than fitted. Without intermediate-scale points or multiple seeds, the data do not establish that the ordering of test losses persists across two orders of magnitude in data, which is the basis for the abstract's claim of consistent improvement in scaling laws.
  2. [4.2, Table 1, and Appendix B] The 2.4B small-complexity model is not a clean comparison for complexity control: it uses gamma equal to 0.58 instead of the gamma equal to 1 used in Figure 2, and it additionally employs embedding normalization and sandwich normalization (stated in Appendix B and Section 6.1). These architectural changes were introduced to mitigate loss spikes, but they confound the attribution of the plus 3.4 average gain to complexity control alone. The authors should either isolate the effect of gamma while holding normalization fixed, or present a matched 2.4B model without the extra normalization.
  3. [Table 1 and Appendix C] All reported benchmark differences are based on a single training run per configuration, with no standard errors or significance tests. Several deltas are small (for example, MMLU-Pro minus 0.5, OpenBookQA plus 0.8, GPQA plus 0.9, and IFEval plus 0.7 for the 2.4B comparison) and could be within evaluation noise. The assertion that complexity control improves performance consistently across tasks needs either multiple seeds or a paired statistical analysis across the 15 benchmarks.
  4. [6.3] The theoretical analysis rests on two very strong assumptions: convergence to the minimum-gamma-norm minimizer, and decomposition of the parameter distribution into product measures with per-layer masses epsilon or 0.9. These assumptions are not validated empirically and are not shown to be necessary for the reported benchmark gains. As a heuristic the discussion is illustrative, but as written it may be misread as a mechanistic explanation. Please clarify that this is a plausibility argument rather than a derivation of the scaling-law results.
minor comments (5)
  1. [3, Eq. (1)] The displayed weight-decay update rule appears to have a sign typo: it reads theta_{t+1} = -hat_theta_t - lambda * C * theta_t, but the standard AdamW update would be theta_{t+1} = hat_theta_t - lambda * C * theta_t. Please correct the sign.
  2. [6.3] The derivation of the minimized function norm is hard to follow because several intermediate displays are truncated or ambiguous, including the final expression for the norm with the factor 0.9 and epsilon. Please expand the calculation and define the norm notation precisely.
  3. [Table 1 caption] The table caption does not state that the 2.4B Small model uses gamma equal to 0.58 while the 0.9B Small model uses gamma equal to 1. Adding this information would prevent conflating the two configurations.
  4. [Figure 2 right] In the right panel, all models are trained on 1B tokens, so the token-to-parameter ratio varies from 20 for the 50M model to 1.25 for the 0.8B model. This confounds model-size scaling with data-to-model ratio and should be discussed.
  5. [Abstract and Section 4.1] The phrase that the initialization rate turns out to be the right invariant for the scaling laws is stronger than the evidence provided, since only a few discrete gamma values are tested. Please consider qualifying this claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the scaling-law and benchmark gains are direct controlled comparisons against external benchmarks; self-citations are background, and the heuristic circuit-depth analysis is explicitly assumption-based rather than a fitted prediction.

full rationale

The central empirical claim—smaller initialization rate and larger weight decay improve test loss and downstream benchmarks—is established by direct controlled experiments (Figures 2, 4, 5; Tables 1, 4, 6) using external benchmarks (MMLU, GSM8K, etc.). No parameter is fitted to the target benchmark gains: the small-complexity configuration (γ=1, λ=1) is defined before the comparison, and the 2.4B γ=0.58 choice is a stability-motivated hyperparameter, not selected by the reported benchmark deltas. The scaling-law extrapolation from 0.2–1.4B-token runs and single-run large-scale comparisons is a substantive correctness risk, but it is not circularity because the curve ordering is observed, not derived from the conclusion. The paper's self-citations to condensation work [39, 69, 88] are heavy but background; the experiments in this paper stand independently of those citations. The theoretical analysis in Section 6.3 is explicitly labeled 'heuristic calculations'; it constructs an interpolation between RKHS and Barron norms and assumes (Assumption 1) that training with rate γ selects the minimum of that norm. The circuit-depth conclusion follows from that stated assumption and the constructed norm, but the paper does not use this calculation to generate the benchmark numbers, so it is an explanatory model rather than a circular prediction. No equation is shown to be equivalent to its own inputs by construction. Overall, the empirical contribution is self-contained against external benchmarks, so circularity is minimal.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The empirical claim does not depend on the theory, but the mechanistic explanation rests on two unproved assumptions and a free epsilon. The benchmark comparison itself is a single-seed experiment with no released code or data.

free parameters (3)
  • initialization rate gamma = 0.1, 0.3, 0.5, 0.8, 1 for 180M; 0.5 vs 1 and 0.58 for 0.9B/2.4B
    Chosen by hand as the complexity-control knob; not fitted to data, but the central comparison depends on these values. The small-complexity configurations use gamma=1 or 0.58, not a value derived from a criterion.
  • weight decay coefficient lambda = 0, 0.1, 1
    Chosen by hand and paired with gamma to define complexity configurations. The large-complexity and small-complexity setups differ in both gamma and lambda simultaneously, so individual contributions are not isolated.
  • epsilon in Assumption 2 = unspecified, stated as 0 < epsilon << 1, with 0.9 used in the norm formula
    Free scale in the toy calculation of circuit norms. The conclusion that small gamma favors deep circuits depends on the ratio 0.9/epsilon being large, and epsilon is not determined from data or theory.
assumptions (4)
  • ad hoc to paper Training initialized with rate gamma converges to a global minimizer with minimum ||f||_gamma norm (Section 6.3, Assumption 1).
    The paper cites the 2-layer kernel regime and says it expects similar behavior in general settings. No proof is given for deep networks or Transformers.
  • ad hoc to paper Every minimizer's parameter distribution decomposes into weighted product measures with per-layer masses epsilon or 0.9 (Section 6.3, Assumption 2).
    Introduced to make the norm calculation tractable. The paper states the purpose is to simplify computation and expects the result to hold more generally.
  • domain assumption Deep residual networks with 2-layer blocks are a valid proxy for Transformer circuits (Section 6.3).
    The heuristic calculation is performed for residual nets, and the transfer to Transformers is asserted rather than demonstrated.
  • standard math RKHS and Barron norm characterizations of 2-layer networks are standard background (Section 6.3).
    Background results from cited references 14, 15, 16, 17, and 48. These are accepted mathematical facts, not new assumptions.
invented entities (1)
  • circuit ensemble representation {(c_i, L_i)}
    purpose: Models the trained network as a weighted sum of product-measure circuits to explain why small initialization selects sparse deep solutions.
    This is a representational assumption (Assumption 2) used in the heuristic calculation. The paper does not directly measure these circuits in the trained LLMs.

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Cite this review

Pith. "Pith review of Scalable Complexity Control Facilitates Reasoning Ability of LLMs." pith.science (2026). https://pith.science/paper/SL3RUARY

@misc{pith2026250523013,
  author       = {Pith},
  title        = {Pith review of: Scalable Complexity Control Facilitates Reasoning Ability of LLMs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SL3RUARY}},
  note         = {Machine review of arXiv:2505.23013}
}
read the original abstract

The reasoning ability of large language models (LLMs) has been rapidly advancing in recent years, attracting interest in more fundamental approaches that can reliably enhance their generalizability. This work demonstrates that model complexity control, conveniently implementable by adjusting the initialization rate and weight decay coefficient, improves the scaling law of LLMs consistently over varying model sizes and data sizes. This gain is further illustrated by comparing the benchmark performance of 2.4B models pretrained on 1T tokens with different complexity hyperparameters. Instead of fixing the initialization std, we found that a constant initialization rate (the exponent of std) enables the scaling law to descend faster in both model and data sizes. These results indicate that complexity control is a promising direction for the continual advancement of LLMs.

Figures

Figures reproduced from arXiv: 2505.23013 by the authors.

Figure 1
Figure 1. Next-token prediction accuracy with varying model complexity. The colors indicate the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Test loss across varying data and model scales under different complexity configurations. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Performance improvement via SFT across complexity configurations(0.9B model). It is [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Parameter norm evolution across complexity configurations. Left to right: [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: (A) Evaluation scores (average, GSM8K, HellaSwag) under varying complexities. Top: [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Cosine similarity among 350 embedding vectors which occur most frequently in training [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Dc and Ds of WQ and WK in each layer under different model complexity configurations. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: The evolution of parameter norm and average score with two distinct [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: The dynamics of parameter norm (left) and loss (right) of 2.4B model with [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: Performance landscape of all tasks across [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Score-complexity relationship with dashed lines denoting baseline performance levels. [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Cosine similarity among 350 embedding vectors which occur most frequently in training [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Dc of WQ and WK in 0.9B model’s each layer under different model complexity configurations [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Ds of WQ and WK in 0.9B model’s each layer under different model complexity configurations. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 15
Figure 15. Figure 15: Dc of WQ and WK in 2.4B model’s each layer under different model complexity configurations [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: Ds of WQ and WK in 2.4B model’s each layer under different model complexity configurations. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]

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Cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.