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Fast on the Easy, Deep on the Hard: Efficient Reasoning via Powered Length Penalty

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

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Pith's one-line read Adding a powered length penalty to the reward function teaches language models to be fast on easy questions and deep on hard ones, cutting tokens on easy benchmarks by up to 90 percent while broadly holding accuracy.

desk verdict A plausible reward-shaping knob with a genuinely interesting 'thought disappearance' effect, but the headline numbers are assembled from different alpha values and need honest re-reporting. read the letter →

arxiv 2506.10446 v1 pith:TDDJRNJJ submitted 2025-06-12 cs.CL

Zehui Ling , Deshu Chen , Hongwei Zhang , Yifeng Jiao , Xin Guo , Yuan Cheng This is my paper
classification cs.CL
keywords largelanguagemodelsefficientreasoninglengthpenaltyreinforcementlearningRLOOchain-of-thoughtmathematicaldifficultyadaptation
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

Large reasoning models solve hard problems by writing long chains of thought, but they pay that cost on easy problems too, and this paper tries to teach them instead to be fast on the easy and deep on the hard. The whole method is one reward change: a correct answer earns $1 + \alpha/\mathrm{len}^{\gamma}$, a wrong answer earns 0, and a reinforcement-learning estimator called RLOO (REINFORCE leave-one-out) scores each sampled answer against the other answers to the same question. Because the penalty is steep for short responses and nearly flat for very long ones, the learning signal compresses easy answers while leaving long reasoning alone, so the model itself decides how much to think. Trained on grade-school math (GSM8K) with 1.5B and 7B reasoning models, the method cuts output tokens on easy benchmarks by roughly 40% to 90% while holding or improving accuracy, and it reports better accuracy on the hard AIME2024 contest at fewer tokens (27.7% to 33.7% for the 1.5B model). The authors show analytically that the spread of rewards among long correct answers shrinks as length grows, which is why hard questions are governed by correctness rather than brevity; the paper's stated scope is mathematical reasoning with models up to 7B parameters.

What carries the argument

The carrying mechanism is the Powered Length Penalty (PLP), the reward function $f(\mathrm{len}) = 1 + \alpha/\mathrm{len}^{\gamma}$ with $\alpha \geq 0$ and $\gamma > 0$ (the analysis takes $\gamma = 0.5$), applied to correct responses inside the RLOO advantage estimator, which gives each of $k$ sampled answers the difference between its reward and the mean reward of the other $k-1$ answers to the same question. Adding the penalty to 1 rather than subtracting it from 1 keeps long correct answers far above the zero reward of wrong answers, so on hard questions the model is rewarded for correctness, while on easy questions, whose correct answers are short, the penalty creates large reward gaps that drive compression. The statement that carries the argument is the variance formula $\operatorname{Var}(Z) = (\ln b - \ln a)/(b-a) - 4/(\sqrt{a}+\sqrt{b})^{2}$ for $Z = 1 + 1/\sqrt{\mathrm{len}}$ with $\mathrm{len}\sim U(a,b)$: the variance shrinks as the length scale grows, so the same fixed function penalizes short answers heavily and long answers almost not at all. The paper contrasts this with standardized penalties, whose variance stays fixed at 1 regardless of length, meaning they keep punishing long answers even where reasoning is needed.

What would settle it

Build a training set with the same questions and answers as GSM8K but with the length-difficulty correlation removed, for example rewriting easy problems so their correct solutions are long and hard problems so their correct solutions are short, then train with the same PLP reward and measure accuracy by difficulty tier. If the model then shortens the long easy answers while accuracy on the short hard answers falls as tokens shrink, the penalty is tracking length statistics rather than true difficulty, and the reported fast-on-the-easy behavior is specific to the length-difficulty correlation in GSM8K. A cheaper check: Table 3's prediction that the thinking-stage fraction tracks difficulty can be tested question by question against human difficulty ratings.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that a powered, absolute length penalty produces difficulty-adaptive reasoning without any difficulty estimator, extra training signal, or inference-time prompt. With $\gamma$ fixed at 0.5 in the analysis, the reward $Z = 1 + 1/\sqrt{\mathrm{len}}$ for correct answers makes the advantage differences among short responses large, so easy answers are pushed to be shorter, while the differences among long responses are small, so hard answers are judged almost purely on correctness; the variance of $Z$ when length is uniform on $(a,b)$ is $(\ln b - \ln a)/(b-a) - 4/(\sqrt{a}+\sqrt{b})^{2}$, which decreases as the length scale grows. Empirically, training on just 3,200 GSM8K questions transfers to harder benchmarks: the thinking stage nearly disappears on GSM8K itself (26 of 1,319 sampled responses keep the thinking label, versus 1,319 of 1,319 for the original model), drops to 162 of 500 on MATH500, and remains on all 30 AIME2024 responses, while AIME2024 accuracy rises from 27.7% to 33.7% (1.5B model) and from 54.7% to 55.7% (7B model) at fewer tokens. The claim, as the authors would state it, is that the same reward that teaches conciseness on easy problems preserves, and in these runs slightly improves, correctness on hard ones because the penalty field is automatically tuned by the length distribution of correct answers.

Load-bearing premise

The load-bearing premise, stated in the paper's introduction, is that response length is a reliable indicator of question difficulty, so that penalizing length hits easy questions hard and spares hard questions, and this premise gives way if easy questions regularly draw verbose correct answers or hard questions draw terse ones.

Editorial extensions

If this is right

  • Training on a simple dataset can transfer efficiency to harder benchmarks: with 3,200 GSM8K questions as the only training data, MATH500 tokens drop by about a third on the 1.5B model with accuracy preserved, and AIME2024 accuracy improves at fewer tokens on both model sizes.
  • The model learns on its own to skip the explicit thinking stage on easy questions, with the thinking label appearing in 26 of 1,319 GSM8K responses versus 1,319 of 1,319 originally, so inference cost adapts to the model's own judgment of difficulty rather than a fixed reasoning budget.
  • Accuracy and efficiency are not a fixed trade-off on hard problems: on AIME2024 the 1.5B model goes from 27.7% to 33.7% accuracy with tokens reduced from 15,643 to 13,327, and the 7B model from 54.7% to 55.7% with tokens reduced from 12,816 to 9,056.
  • The strength parameter $\alpha$ traces a controllable effort-accuracy frontier: larger $\alpha$ compresses outputs further, but past a model-specific point accuracy collapses, so practitioners can dial the trade-off rather than accept a single compression level.
  • Because the method needs no critic or reward model, correctness on a verifiable answer being the reward itself, it adds no inference-time prompts and keeps the training pipeline memory-light on top of RLOO.

Reading between the lines

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

  • My inference: because the model spontaneously drops the thinking stage on easy questions, the fraction of responses that keep a thinking stage could serve as a per-question measure of perceived difficulty, and comparing that fraction with human difficulty ratings would show whether the learned economy tracks genuine difficulty or only length statistics.
  • My inference: the big-reward design, adding the penalty to 1 instead of subtracting it from 1, suggests that the safety margin of correct long answers above zero is what preserves accuracy, so other length-control schemes could reproduce the effect by keeping correct long answers far from zero rather than by copying the exact functional form.
  • My inference: the method's transferability is likely weakest in non-mathematical domains such as legal or medical reasoning, where difficulty and response length correlate less reliably, so running the same recipe on such a domain and comparing the accuracy-efficiency frontier with the math results would be the natural test of the paper's implicit claim that the mechanism generalizes.
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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 / 4 minor

Summary. This paper proposes a powered length penalty (PLP) for RLOO-based reinforcement learning of reasoning models. For a correct response of length len(y), the reward is R(y,x)=1+α/len(y)^γ, and incorrect responses receive 0. The intent is to penalize verbosity strongly on easy questions while leaving long reasoning chains on hard questions nearly unpenalized. The authors train DeepSeek-R1-Distill-Qwen-1.5B, DeepSeek-R1-Distill-Qwen-7B, and Qwen2.5-7B-Instruct on 3,200 GSM8K training questions, then evaluate on GSM8K, MATH500, and AIME2024. They report token reductions with preserved or improved accuracy on the easier sets and improved accuracy on AIME2024. Section 4 gives a variance analysis of the penalty under a uniform length distribution, and Section 5.4 reports ablations over α.

Significance. If the headline claim held—that a single PLP-trained model is fast on easy, deep on hard, preserving or improving accuracy across three difficulty levels while cutting tokens—this would be a simple and practical contribution to efficient reasoning. The idea of making the length penalty decay with response length is intuitive, and the choice to train on one dataset and test across difficulty levels is a clean experimental design. I credit the paper for reporting ablations over α and for stating that three training runs were performed per setting. However, the central empirical claim is not currently established: the 1.5B results in Table 1 are assembled from different values of α for different datasets, the AIME2024 differences are within the noise of a 30-question benchmark, and no error bars or significance tests are reported despite the stated three-run protocol. These issues are fixable within the manuscript's scope, so the contribution remains potentially valuable.

major comments (4)
  1. [§5.2, Table 1; Appendix Tables 1–3] The 1.5B 'Ours' row in Table 1 does not correspond to a single trained model. GSM8K (α=5, 411 tokens, 86.3%) and AIME2024 (α=5, 13327 tokens, 33.7%) use α=5, while MATH500 (α=2, 3606 tokens, 85.1%) uses α=2. Under α=5, the 1.5B MATH500 accuracy is 81.9%, which is below the original model's 82.9%. Therefore, the claim that PLP preserves or improves accuracy on all three benchmarks for a single configuration is not supported by the reported data. Please present results for one hyperparameter setting per model, or explicitly reframe the contribution as per-dataset hyperparameter selection.
  2. [§5.1, Table 1, AIME2024] AIME2024 contains only 30 questions. With the reported evaluation protocol (N=10 conditional samples per question), the 1.5B improvement of 6.0 points corresponds to roughly two questions, and the 7B improvement of 1.0 point corresponds to 0.3 questions. The implementation details say three independent training runs were performed, but no per-seed variance, confidence interval, or significance test is reported anywhere. Without this information, the claimed AIME2024 accuracy gains are not distinguishable from sampling noise.
  3. [§5.4, Figures 5 and 6] The hyperparameter selection rule for α is unclear and appears post-hoc. The text states that for the 1.5B model 'α=2 yielded the best results,' yet Table 1 uses α=5 for GSM8K and AIME2024 and α=2 for MATH500. For the 7B model, 'the optimal performance was achieved when α=4,' which does match the Table 1 row. Since α is a free parameter and no validation-based selection criterion is given, the reader cannot tell whether the reported numbers were chosen after looking at the test benchmarks. Please specify a single selection rule (e.g., choose α on a validation split and fix it for all evaluation datasets), or report the full per-α table as the main result and avoid cherry-picked rows.
  4. [§3.4, §5.1] The reward function depends on both α and γ, but the experimental section never states which value of γ was used. Section 4 restricts its analysis to γ=0.5, and Figure 2 uses γ=0.5, yet the implementation details in Section 5.1 mention only α in the ablations. If γ is fixed to 0.5 throughout, say so explicitly in the experimental setup; if γ varies, report its values. This is needed for reproducibility of Table 1.
minor comments (4)
  1. [§3.3] The heading 'Length Cliping' contains a typo; it should be 'Length Clipping.'
  2. [§4] The sentence 'The variances of the above two different methods also exhibit the same trend in general distributions' asserts a generalization from the uniform-distribution calculation without proof or citation. Please either provide a supporting argument or soften the claim.
  3. [Appendix B] The appendix introduction states 'The same parameter was tested on GSM8K, MATH500, and AIME2024,' but the tables show that the reported 'Ours' row uses different α values across datasets for the 1.5B model. This wording is misleading and should be corrected.
  4. [Table 2] The TokenSkip result is taken from the reference paper rather than measured under the same evaluation protocol. Please state this in the caption or table note so the comparison is not over-interpreted.

Circularity Check

1 steps flagged · score 6.0 of 10

The 1.5B 'Ours' results are a composite of per-benchmark α selections, so the headline accuracy-efficiency tradeoff partly reduces to test-set hyperparameter selection; the powered-length-penalty method itself is not circularly derived.

  1. fitted input called prediction [Table 1 (Section 5.2); Appendix Tables 1-3; Section 5.4 Ablation Studies]
    ""Ours86.3% 411 85.1% 3606 33.7% 13327"; "R1-1.5B 2 3606 85.1%"; "R1-1.5B 5 13327 33.7%"; "In the 1.5B model, we found that selecting α=2 yielded the best results.""

    The Table 1 'Ours' row for the 1.5B model is not produced by a single trained model: GSM8K (86.3%, 411 tokens) and AIME2024 (33.7%, 13,327 tokens) match the α=5 runs, while MATH500 (85.1%, 3,606 tokens) matches the α=2 run. Appendix Table 2 shows that with α=5 the same 1.5B model obtains 81.9% on MATH500, below the original 82.9%. Thus the reported 'accuracy preserved on MATH500' is the result of selecting α=2 after seeing MATH500 test results, and the α used for the headline GSM8K/AIME numbers would make MATH500 accuracy worse.

full rationale

The paper's core mechanism—training with RLOO and reward f(len)=1+α/len^γ—is empirical and externally anchored to original-model baselines; it is not a derived-from-first-principles claim that reduces to its own definition. Section 4's variance formula Var(Z)=ln(b/a)/(b-a)-4/(√a+√b)^2 is an internal mathematical property of the chosen penalty under a uniform-length assumption; it is not used to produce the empirical accuracy numbers and so is not circular. The paper does not rely on load-bearing self-citations. The main circularity is the 'fitted input called prediction' pattern at the level of hyperparameter selection: for the 1.5B model, the Table 1 'Ours' row assembles results from α=5 (GSM8K and AIME2024) and α=2 (MATH500), and under the α=5 configuration MATH500 accuracy falls below the original baseline. Since α is selected on the same test sets whose results are reported, the headline accuracy-preservation claim for MATH500 is not out-of-sample for the 1.5B model. The 7B row is a single α=4 configuration across all three datasets, so the method retains independent empirical content; the circularity is partial, not total. The small AIME sample size and absence of per-seed variance are additional statistical robustness concerns, but they are not themselves circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central empirical claim depends on two fitted hyperparameters (α, γ) and on the domain assumption that output length tracks difficulty. The variance analysis in Section 4 is a property of the chosen reward function, not an independent check of the central claim.

free parameters (2)
  • Alpha (α) = 1.5B: 5 for GSM8K/AIME2024 and 2 for MATH500 in Table 1; 7B: 4; swept values 1-30
    Penalty strength coefficient. Selected per dataset after evaluating on the test benchmarks, so the main results are not from one fixed configuration.
  • Gamma (γ) = 0.5
    Power in the denominator, fixed without ablation 'for analytical convenience' (Section 4), making the variance formula tractable.
assumptions (3)
  • domain assumption Response length is an indicator of question difficulty.
    Section 1: 'we propose using model response length as an indicator of question difficulty.' If easy problems can produce long outputs or hard problems produce short outputs, the penalty misaligns.
  • ad hoc to paper The length distribution of model outputs is approximately uniform for the variance analysis, and the variance trend generalizes to real distributions.
    Section 4 assumes len(y) ~ U(a,b) and γ=0.5 to derive Var(Z); the paper then asserts the same trend 'in general distributions' without proof.
  • standard math RLOO with k=8 samples gives unbiased advantage estimates and stable training.
    Standard result from Ahmadian et al. (2024), used as the RL framework.

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Cite this review

Pith. "Pith review of Fast on the Easy, Deep on the Hard: Efficient Reasoning via Powered Length Penalty." pith.science (2026). https://pith.science/paper/TDDJRNJJ

@misc{pith2026250610446,
  author       = {Pith},
  title        = {Pith review of: Fast on the Easy, Deep on the Hard: Efficient Reasoning via Powered Length Penalty},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TDDJRNJJ}},
  note         = {Machine review of arXiv:2506.10446}
}
read the original abstract

Large language models (LLMs) have demonstrated significant advancements in reasoning capabilities, performing well on various challenging benchmarks. Techniques like Chain-of-Thought prompting have been introduced to further improve reasoning. However, these approaches frequently generate longer outputs, which in turn increase computational latency. Although some methods use reinforcement learning to shorten reasoning, they often apply uniform penalties without considering the problem's complexity, leading to suboptimal outcomes. In this study, we seek to enhance the efficiency of LLM reasoning by promoting conciseness for simpler problems while preserving sufficient reasoning for more complex ones for accuracy, thus improving the model's overall performance. Specifically, we manage the model's reasoning efficiency by dividing the reward function and including a novel penalty for output length. Our approach has yielded impressive outcomes in benchmark evaluations across three datasets: GSM8K, MATH500, and AIME2024. For the comparatively simpler datasets GSM8K and MATH500, our method has effectively shortened output lengths while preserving or enhancing accuracy. On the more demanding AIME2024 dataset, our approach has resulted in improved accuracy.

Figures

Figures reproduced from arXiv: 2506.10446 by the authors.

Figure 1
Figure 1. Comparison of model outputs under the same [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Difference between original RL and ours. The same color represents the question corresponding to the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Comparison between standardized and absolute length penalty methods across two example ranges: [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Difference between big reward and small reward when the last sample is incorrect. 5.3 Empirical Analysis Differences in model parameter scale We have observed that, in both the 1.5B and 7B models, the reinforcement learning with length penalty reduced the number of tok…
Figure 5
Figure 5. Figure 5: Difference between our method and the efficient method. For our method, the coefficients are 1, 2, 3, 4, 5, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Difference between our method and the efficient method. For our method, the coefficients are 1, 2, 3, 4, 5, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Forward citations

Cited by 6 Pith papers

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

  1. A First-Principles Theory of Slow Thinking and Active Perception

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  2. Compress the Easy, Explore the Hard: Difficulty-Aware Entropy Regularization for Efficient LLM Reasoning

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    CEEH selectively applies entropy regularization to hard questions and a shortest-correct-length penalty to easy ones, reducing reasoning length while preserving accuracy across six math benchmarks.

  3. Learning to Reason Efficiently with Discounted Reinforcement Learning

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  4. Baichuan-M2: Scaling Medical Capability with Large Verifier System

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  5. Reward Modeling for Reinforcement Learning-Based LLM Reasoning: Design, Challenges, and Evaluation

    cs.LG 2026-02 conditional novelty 4.0 of 10

    A taxonomy-driven survey arguing that reward design is the central mechanism shaping reliable LLM reasoning, with maps of reward paradigms, reward-hacking failure modes, and benchmark pitfalls.

  6. Towards Concise and Adaptive Thinking in Large Reasoning Models: A Survey

    cs.AI 2025-07 conditional novelty 3.0 of 10

    A comprehensive review that categorizes methods for shortening and adaptively triggering chain-of-thought reasoning in large language models.

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Pith tools

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