Scalable Chain of Thoughts via Elastic Reasoning
Authors: Yuhui Xu
Hanze Dong
Lei Wang
Doyen Sahoo
Junnan Li
Caiming Xiong
Venue: ICLR 2026 Poster (2026)
Keywords: Test time scaling
Large language models
Length control Abstract:
Primary area: foundation or frontier models, including LLMs
TL;DR: A novel framework for scalable chain of thoughts that explicitly separates reasoning into two phases—thinking and solution—with independently allocated budgets.

Abstract: Large reasoning models (LRMs) have achieved remarkable progress on complex tasks by generating extended chains of thought (CoT). However, their uncontrolled output lengths pose significant challenges for real-world deployment, where inference-time budgets on tokens, latency, or compute are strictly constrained. We propose Elastic Reasoning, a novel framework for scalable chain of thoughts that explicitly separates reasoning into two phases—thinking and solution—with independently allocated budgets. At test time, Elastic Reasoning prioritizes the completeness of solution segments, significantly improving reliability under tight resource constraints. To train models that are robust to truncated thinking, we introduce a lightweight budget-constrained rollout strategy, integrated into GRPO, which teaches the model to reason adaptively when the thinking process is cut short and generalizes effectively to unseen budget constraints without additional training. Empirical results on mathematical (AIME, MATH500) and programming (LiveCodeBench, Codeforces) benchmarks demonstrate that Elastic Reasoning performs robustly under strict budget constraints, while incurring significantly lower training cost than baseline methods. Remarkably, our approach also produces more concise and efficient reasoning even in unconstrained settings. Elastic Reasoning offers a principled and practical solution to the pressing challenge of controllable reasoning at scale. Code is available in the supplementary material.

Published as a conference paper at ICLR 2026
SCALABLE CHAIN OF THOUGHTS VIA ELASTIC REA- SONING
Yuhui Xu Hanze Dong
Lei Wang
Doyen Sahoo
Junnan Li Caiming Xiong
Salesforce Al Research
ABSTRACT
Large reasoning models (LRMs) have achieved remarkable progress on plex tasks by generating extended chains of thought (CoT). However, their un- controlled output lengths pose significant challenges for real-world deployment. where inference-time budgets on tokens, latency, or compute are strictly con- strained. We propose Elastic Reasoning, novel framework for scalable chain of thoughts that explicitly separates reasoning into two chases-thinking and so- lution-with independently allocated budgets. At test time, Elastic Reasoning pri- oritizes the completeness of solution segments, significantly improving reliability under tight resource constraints. To train models that are robust truncated think- ing, introduce lightweight muddet-constrainner rollout strategy, integrated into GRPO, which teaches the model to reason adaptively when the thinking pro- is short and generalizes effectively unseen budget constraints without additional training. Empirical results on mathematical (AIME, MATH500) and programming (LiveCodeBench, Codeforces) benchmarks demonstrate that Elastic Reasoning performs robustly under strict budget constraints, while incurring sig- nificantly lower training cost than baseline methods. Remarkably, our approach also produces more concise and efficient reasoning even in unconstrained settings. Elastic Reasoning offers principled and practical solution the pressing chal- lenge of controllable reasoning at scale. Our code has been made available at [
INTRODUCTION
Large reasoning models (LRMs) (DeepSeek-Al al. 2025 OpenAl et al. (0224) have demonstrated remarkable performance complex reasoning tasks by producing extended Shain-of-Thoughe (CoT) outputs, which facilitate effective problem-solving in domains such as mathematics and pro- gramming. Reinforcement learning (RL) techniques (Schulman et al. [017 Zelikman et al. 2022 Rafailov et al. 2023 Dong et al. 2023 Shao et al.) 2024) have been employed to optimize these rea- soning trajectories, enabling LRMs generate longer, more informative chains. These RL-driven methods scale effectively across diverse benchmarks (Zhang et al. 2024, Dong et al. 2024 Luo al., 2025c: Xiong al. 2025b Luo al.) 2025b), yielding substantial gains in both solution accuracy and robustness; while they often incur significantly longer inference chains (DeepSeek-
AT et al. 2025 Du et al.
2025 Yu al. 2024 Qin al. 2024 Xiong al., (0255)). Notably, the length of the reasoning trajectory remains uncontrolled, making it difficult to allocate fixed compute budget at inference time while maintaining a desired performance level.
Two primary lines of research have been proposed to address this challenge. The first, known as Long2Short (Team et al. 2025 Kang et al. 2024), seeks reduce reasoning length through rein- forcement learning with trajectory penalties compression-awe aware fine-tuning, where the model is trained on shortened trajectories to preserve performance while minimizing inference cost. The sec- ond line of work focuses on length control (Muennighoff et al., 2025 Aggarwal & Welleck 2025 Yuan et al. 2024). S1 (Muennighoff et al., (025) introduces simple mechanism that prompts the model emit special tokens (e.g.. "Wait", "Final Answer") regulate reasoning length. However, this approach significantly degrades performance, as it overlooks the critical role of the solution seg- ment. L1 (Aggarwal & Welleck 2025 proposes reinforcement learning framework that enforces
https: thu.. www./orceaeaaaaeeeeeeeeaaaaaae Hee.frrcceaeeeerceec///11111
Reasonin ng

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Published as a conference paper at ICLR 2026
explicit length constraints over the entire trajectory. While more flexible, this method demands sub- stantial training resources and still results in noticeable performance degradation compared to the original model.
We propose Elastic Reasoning, a simple yet effective method that enables large reasoning models to achieve scalable and adaptive length control. As illustrated in Figure the S1 aoproach-gennnee the answer by emitting a special token such as "Final Answer" lewver--prffomss better than directly trun- cating the full reasoning trajectory, underscoring the importance of preserving the solution segment. Motivated by this, we propose separate budgeting which explicitly divides the total token budget c into two parts: tokens for the thinking phase and tokens for the solution phase, where c = + s. Once the model consumes t tokens in the thinking phase, we forcibly terminate it by appending the special token </thinks and transition to solution generation. Separate budgeting outperforms under varying generation budgets.
To further improve solution quality under in- complete reasoning, we introduce novel train- ing strategy called budget-constrained rollout, which teaches the model to generate high- quality answers even with partial CoT trajec- tories. This method is integrated into GRPO training and is highly efficient-requiring only 200 training steps on math tasks with max- imum response length of 2K tokens (t" 1K, s* 1K), compared 700 steps for LI- Exact and 820 steps for LI-Max with a 4K re- sponse length. Moreover, models trained with Elastic Reasoning generalize effectively to ar- bitrary reasoning budgets without the need for further fine-tuning.
AIME
SI (Budget Forcing)
Separate-Ralgeeing
Vanilla- Vanillo-Budgeeing
E1 MatM-a15555
Fasig1
4X
Tokens Used
We evaluate Elastic Reasoning on both mathe- matical and programming reasoning tasks, in- troducing two models: E1-Math-1.5B and E1-Code-14B. (1) E1-Math-155B outperforms
Figure I: Separating thinking and solution phases allows precise length control. Each point shows Pass@] I at a given token budget, with the x-axis showing average tokens per problem.
both L1-Exact and S1, and achieves performance comparable to LI-Max, while requiring signifi- cantly fewer training steps. For instance, the AIME2024 dataset, our method achieves 35.0% accuracy, compared to 27.1% for L11Maxx, 24.2% for L1-Exact, and 41.0% for the original model. (2) E1-Code-14B demonstrates strong scaling with varying inference budgets, achieving Code- forces rating of 1987 and placing in the 96.0 percentic--comparale
to 01-2024-12-17 (Low), which scores 1991 and ranks in the 96.1 percentile. (3) A surprising observation is that, after train- ing, the trajectories generated by our models are significantly shorter than those from the original DeepScaleR and DeepCoder models across both math and code tasks. This suggests that budget- constrained rollout not only improves length control but also encourages the model to reason more concisely and generate more efficient solutions.
RELATED WORKS
2.1 TEST-TIME SCALING IN LARGE LANGUAGE MODELS
Increasing computation during inference, often referred to as test-time scaling (TTS), has been shown to improve the reasoning capabilities of LLMs (Wei et al. 2023 Wang et al. 2023 Snell ct al. 2024; DeepSeck-Al Al et al. 2025 Team et al. 2025 Muennighoof et al.) 2025). Early works, such as chain-of-thouuht prompting ( (Wei et al. 2023), show that producing a series of intermediate reasoning steps significantly improves LLMs* performance on complex reasoning tasks. Building on this, self-consistency (Wang al.) 2023) further boosts performance by sampling a diverse set of reasoning paths and selecting the most consistent answer. Recent studies have formalized these findings into test-time inference scaling laws (Snell et al. 2024, Wu et al. 2024). [wu et al. (2024) explore the trade-offss between model size and inference- timeence-timeeeee computation. Snell et al. (2024) inves- tigated how fixed but non-tri non-trivial inference-time budgets can significantly boost LLM performance.
The remarkable successes of advanced reasoning models, such ol (OpenAl al. 2024 and
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RI (DeepSeek-Al et al. 2025). have further amplified interest in leveraging TTS techniques. While much of the existing works primarily focuses on improving performance by increasing inference- time computation, work takes a different perspective: How can we enable LLMs perform effective long reasoning under strict oulput length constraints?
2.2 LENGTH CONTROL IN LARGE LANGUAGE MODELS
Controlling the generation length of LLM directly affects both latency and monetary cost infer- ence time. Earlier approaches to length control are designed mainly for general text generation (Jie)
et al. 2023 Yuan et al. 2024). Typical methods include (i) manipulating positional encodings to
achieve exact sequence lengths (Butcher et al. 2024). (ii) modifying training objectives to penal-
ize deviations from length targets (Jic et al. [023] Singhal et al. 2024). and (ii) fine-tuning on instructions that explicitly state the desired output length (Yuan et al. 2024). Although effective for tasks such as summarization or constrained writing, these techniques generally aim to verbosity or enforce maximum- length limits, and overlook the intricate, step-bg step-by-step reasoning processes re- quired for many reasoning tasks. Recent works have begun to explore efficiency in reasoning by encouraging shorter chains (Kang et al. 2024; Arroa & Zanette (025); however, they typically lack mechanisms for precise, user-defined length targets that align with explicit compute budgets. One notable attempt. budget forcing (Muennighoff et al. 2025), enforces strict token caps by truncating or padding with special tokens. This can yield incomplete reasoning or unnatural, forced outputs, ul- timately harming both accuracy and interpretability. Additionally, L1 (Aggarwal & Welleck 2025) uses reinforcement learning to let models dynamically allocate inference compute based on con- straints provided in the prompt. Our approach does need to include length instructions in the prompt. Instead, we truncate reasoning trajectories to meet a given budget and train the model under these constraints via reinforcement learning.
2.3 EFFICIENT REASONING IN LARGE LANGUAGE MODELS
Making complex reasoning in LLMs more efficient, particularly by shortening the reasoning pro- cess, is crucial to reducing computational costs and making these models practical for real-world deployment. This has become vibrant research area with several promising directions to encour- age more concise and effective reasoning strategies (Kang et al.) [024 Xu al.) 2024) Hao al.] 2024 Liao et al.] 2025 Luo et al., 2025a). One common strategy involves incorporating explicit re- wards into RL. to encourage the model to find shorter reasoning paths (Team al., 2025: Luo al.) 2025a). Some focus creating datasets with examples of concise reasoning paths and then using SFT to teach models how to generate compact and knowledgeable reasoning steps (Kang et al,, 2024 Yu et al. 2024). Instead of relying solely explicit textual reasoning, methods exploring latent reasoning aim to compress these intermediate steps into more compact, internal representa- tions (Hao al.) 2024 Shen et al.) 2025 Saunshi et al. 2025). Efficiency also be improved during inference, without needing to retrain the model. These training-free techniques dynamically adapt the reasoning strategy based on the specific input or task demands (Liao al. 2025 Fu al., 2025). In this work, we introduce a training approach using reinforcement learning under strict budget constraints to encourage the model to balance reasoning quality with cost efficiency.
3 METHODOLOGY
3.1 PRELIMINARIES: REASONING LANGUAGE MODELS
We consider augmentedd language models that generate outputs consisting of two distinct segments: a thinking part and a solution part. Following prior work, we denote the reasoning phase using special tokens such as <th nk> and </think> to explicitly mark the model's intermediate thoughts.
Formally, given an input prompt x, the model generates an output sequence y = (y think y solution). ), where ylinkk contains the intermediate reasoning steps (enclosed between <think> and </think>) and y solation contains the final solution. Typically, think accounts for most of the total tokens, while y colutioo provides concise summary and final answer. The overall generation structure is:
y == (<thh nk> intermediate reasoning </thinks, solution)
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Thinking segment
Training
Inference
S.
Solution segment
Q
Policy Model
Q
Ell Mott/Code
Inferance budget
Si
A
62
T
f
T:
82
Tx
8x
To
TG
Ac
Budget-Cone Conget--onsttaalee Rollout
Separate Budpeting for Inference
Figure 2:
The framework of Elastic Reasoning. Elastic Reasoning comprises two key components:
(I) GRPO training with budget-constrained rollout, and (2) separate budgeting for inference. Left: During training, the model is optimized using GRPO under a fixed token budget (t', s*). Right: At inference time, the trained E1 model can generalize to arbitrary token budgets c = t, + s", enabling flexible and efficient reasoning. The red squares are visual markers for different budget settings.
3.2 ELASTIC REASONING
3.2.1 BUDGET-CONSTRAING INFERENCE
In many real-world applications, inference cost must be carefully controlled due to constraints on latency, computation, or memory. A common approach is to truncate generation after a fixed number of tokens c, enforcing:
vl S
where |y| denotes the number of generated tokens. However, naively truncating the output often results in incomplete missing y volution leading invalid unusable predictions.
3.2.2 SEPARATE BUDGETING FOR THINKING AND SOLUTION
To address this limitation, we propose Separate Budgeting, a method that explicitly allocates inde- pendent budgets for the reasoning and solution phases. A key observation is that even when the reasoning phase is forcibly terminated (e.g., by inserting </think>), the model is still capable of producing coherent-ann often correct-Solution
Given a total generation budget c, we divide it into two components: a budget t for the thinking phase and budget for the solution phase, such that t
During inference:
The model begins generating within a <think> block.
If the model emits </think> before reaching the budget I, we transition immediately to the solution phase.
If the budget is exhausted before </think> is emitted, we forcibly terminate the rea- soning by appending </think>.
The model then continues generating the solution segment, up to maximum of tokens.
This approach ensures that both the reasoning and solution components explicitly accommo- dated within the total budget c, thereby avoiding unintended truncation of the solution segment. The thinking budget t can be flexibly adjusted at inference time to match different application scenarios, while the solution phase always retains a guaranteed allocation. As shown in Figure [ Separate Budgeting outperforms both vanilla budgeting (naïve truncation) and S1 (budget forcing). By ded-

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accuracy during training
Averge durng trainnn
Pung2
0.20
0.5
0.30
0.5
0.34
0.40
0.60
0.20
Siaca
Slapa
(a) Pass@1 on AIME2024 over steps
(b) Reward over steps
Figure Validation accuracy and reward curves of E1-Math-1.... over training steps.
icating fixed token budget for solution generation, Separate Budgeting significantly improves the reliability and quality of model outputs under tight inference-time constraints.
3.2.3 BUDGET-CONSTRAINED ROLLOUT
While Separate Budgeting ensures dedicated budgets for both reasoning and solution phases, we observe that naively truncating the thinking part-especially on complex tasks such as code gen- eration-can lead to significant performance degradation. To mitigate this issue, propose reinforcement learning (RL) fine-tuning procedure that explicitly trains the model under reason- ing budget constraints, allowing it to produce more effective and concise reasoning within limited budgets.
We adopt GRPO RL algorithm. Let ne denote the policy of language model parameterized by 0, which generates response y (y hink yf solution) for given input a, subject to total budget constraint t" s c" In standard GRPO, the policy generates full trajectory without any structural constraints. contrast, our budget-constrained rollout conditions the policy on a fixed thinking/solution budget pair s*. During training, simulate the Separate Budgeting procedure used at inference time: the policy rolls out a reasoning segment y tlink up to a maximum of t tokens. the model emits the </think> token before reaching this limit, it proceeds to generate the solution segment as usual. Otherwise, we forcibly append </think> once the budget t" is reached. The model then generates the solution segment solution using the remaining tokens.
Let rly) denote task-specific reward function. The training objective is to maximize the expected reward:
J(0)= EgwD gomnel-la (* [r())]
We optimize ((0) using GRPO with the following gradient estimator:
VoJ Ezz,y [A(x, y) log To xx s"),")), A(z. v) r(y) Ey ~m lm (r((())
(12) (((')
where the conditioning in TT(- x t², s*) simply indicates that trajectories are sampled under these rollout constraints, rather than from unconstrained generation. In our training setup, we fix the budget pair to (t", (1K, 1K) for simplicity and efficiency. Surprisingly, we find that the learned policy generalizes well to a wide range of unseen budget configurations at test time, without requiring any additional fine-tuning. As shown in Figure [ the E1-Math-1.5B model achieves sub- stantial improvements while generalizing robustly across various generation budgets. This indicates that Elastic Reasoning encourages the model to intermalize a flexible reasoning strategy that adapts to different resouree constraints. This RL-based adaptation helps the model prioritize informative reasoning content earlier in the generation process, thereby improving both robustness and solution quality under test-time truncation.
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AIME
AMC
Olympind-Bench
45%
70%
57%
405
Pawgl
2355
2065
40%
3006
2056
855
10%
10%6
MATH
Minerva
40%
(Budget Forcing)
LI-Exact
L. Max
25%
E1-Mathh-155 (Budge
DeepScaleR- 1.5 1.5D-24K 24K
4855
20%
L1-Esact-24K
L1-Max-2a 34K
10%
EI M.--Mabb-----44 24K
10%
256
256
1K
Tokeas Esed
Figure 4: Comparison of EI-Math-L.5B with LI and S1 baselines under varying generation budgets. Each point shows Pass@1 at a given token budget, with the x-axis showing average tokens per problem. 24K / 64K settings correspond the maximum token lengths permitted during inference, with no explicit budget constraints imposed.
4 EXPERIMENT RESULTS
4.1 MODELS AND DATASETS
(Luo et al. 2025c and DeepCoder-145 14B- Preview (Luo et al. 20256), which are fine-tuned from DistseeeRR--isttll---eeeen 14B (DeepSeek-Al et al.) (2025) through iterative context lengthening. For training data, we fol- low same datasets used in (Luo et al. 2025c b). In the math domain, the training set con- sists of AIME (1984-2023). AMC, Omni-Math (Gao et al. (022)) and STILL (Min et al.) [024]) For code training, we use TACO (Li et al. 2023), SYNTHETIC-1 (Mattern et al.) (0025), and LiveCodeBench 2003300/01-2224/0777
Our base models are DeepScaleR-L.5B-Peee
and
(Jain et al. 2024). For evaluation, MATH500 (Hendrycks et al. 2021). AMC, Olympiad- -ennh (Gao et al., 2024), and Minerva Math (Lewkowycz al. 2022) for mathematical reasoning. For code-related tasks, evaluate on LiveCodeBench (2024 08/01 18/01-2025/0/00 (Jain et al. 2024), Codeforces, and HumanEval+ (Liu al. 2023). For mathematical reasoning tasks, report averages over 16 runs, whereas for code- related tasks the results are averaged over runs. 24K / 64K settings correspond to the maximum token lengths permitted during inference, with no explicit budget constraints imposed. More training details are in Appendix
AIME 2024,
4.2 MATHEMATICAL REASONING RESULTS
We visualize the reward and validation Pass@) performance on AIME2024 every 10 steps during training in Figure ing phase and begins to converge after approximately the 1h50th step. Meanwhile, the validation accuracy (Pass@1 1) improves rapidly, rising from around 0.07 to 0.20 over the course of training- This demonstrates that, through budget-constrained rollout, the model can quickly learn to reason effectively when the thinking phase is incomplete.
can be observed that the reward steadily increases during the initial train-
We report Pass @ accuracy versus the number of tokens used across five math benchmarks: AIME, AMC, Olympiad-Bennt Bench, MATH500, and Minerva Math in Figure Our proposed method, E1- Math-1.5B 1.5B, under both budget-constrain
and 24K-token settings (red stars), consistently outper- forms (Budget Forcing) and LI LI-Exact, and performs competitively with L1-Max, while requir- ing significantly fewer training steps. On MATH500, E1-Math-1.55 1.5h achieves P accuracy

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Table 1: Comparison of models across LiveCodeBench, Codeforces, HvmanEval+, and AIME benchmarks. El-code-14B variants trained exclusively code data; their AIME scores, obtained on math problems unseen during training, demonstrate that E1-code- 14B retains strong math per- formance.
LiveCudeBench
Codeforces Rating
Codeforces Percentile
HumanEval+
01-2024-12-17 (Low)
59.5
1991
96.1
90.8
74.4
03-Mini 2025-1-33 (Low)
60,9
1918
94.9
92.6
60.0
O1-Preview
42.7
1658
88.5
89.0
40.0
DeepSeck-RI
62.8
1948
95.4
92.6
79.8
DeepSeek-BR RI-Distil-Qwen.
DeepCoder- Prev
El-code 141l (f = 1k, 1k)
37.3
1457
78.1
88.1
17.9
El-code-14P
2h,
El-code-14B (1 3k, 12)
44.1
1711
90.6
90.8
35.4
El-code-14B (1 4k, a 1k)
47.0
1771
92.3
92.0
41.9
El El-cole-114 14B
58.4
1987
96.0.
91.4.
70.6
of 83.6% using only 1619 tokens per question, whereas L1-Exact and LI-Max yield lower or comparable performance with more tokens (L1-Exact: 79.9% with 1959 tokens; L1-Max: 83.6% with 1796 tokens). Notably, when evaluated without inference-time budget constraints, E1-Math- 1.5B achieves higher accuracy than all baseline methods across all benchmarks. For example, AIME2024, E1-Math-1.55 1.5B exhibits a performance degradation of only 6.0% relative to the original model, compared 12.9% for LI-Max and 16.8% for L1-Exact. These results demonstrate that our method is not only effective in enforcing inference- serrence-time budget constraints but also preserves most of the original model's performance. When compared with the original DeepScaleR- 1-55B, E1- Math-1.5B reduces the average number of tokens used across datasets by more than 30%, including a 32. 1% reduction on AIME2024 (see further analysis in Appendix
Furthermore, similar to L1, $1, and 01, we observe a clear log-linear scaling pattern in El: per- formance improves approximately linearly with respect the logarithm of the number of generated reasoning tokens. We provide additional results for other model variants in Appendix
4.3 CODE REASONING RESULTS
As shown in Figure we visualize the Pass@1 accuracy on LiveCodeBench under varying generation budgets, comparing our method to simple separate budgeting strategy for think- ing and solution. We observe that the orig- inal BeCCooder--4B-Peviiie
LiveCodeBench
Depladar-
DespCraan Br-roinnnt EL-Cole
fails to gener- ate correct outputs when reasoning is incom- plete, consistently achieving less than 10% ac- curacy when inference budget is less than 4K even using separate budgeting. In contrast, E1-Code- 14B model demonstrates impressive scalability: its performance improves steadily as the inference budget increases, highlighting the effectiveness of our training strategy in en- abling the model to reason adaptively under constrained thinking. Notably, E1-Code-14D also achieves performance improvement of 0.3% on LiveCodeBench even in the uncon- strained setting, while simultaneously reduc- ing the average number of generated tokens by 37.4%-from 17,815 to 11,145 tokens. This indicates that method only scales well with inference budgets but also promotes more concise and efficient reasoning.
Pasegi
30%
20%
10%
0%
2k
Tokens Used
Figure Pass@1 accuracy on LiveCodeBench under varying reasoning budgets with the x-axis showing average tokens per problem. Both of the models inference with separate budgeting.
Results reproduced using the authors' official code and model with the evaluation protocol.

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Published as a conference paper at ICLR 2026
In Table report the results of EL-CCde-1443 on four benchmarks: LiveCodeBench, Codeforces, HumanEval Plus, and AIME2024. We observe consistent test-time scaling behavior across all benchmarks under constrained inference budgets. Beyond scalability, model also demonstrates strong performance in the unconstrained setting. Specifically, we observe performance improve- ments LiveCodeBench, Codeforces, and HumanEval Plus, and only slight performance drop on AIME2024. On Codeforces, E1-Code-Iee 1efooeeeBBBBB. achieves 42-point improvement in rating and a 0.6 percentile gain, outperforming 03-Mini-2025-1131 (Low) and performing comparably O1-2024- 12-17 (Low). These results highlight that our method not only enables efficient, budget-constrained reasoning but also enhances overall reasoning capability, even in unconstrained scenarios.
4.4 ANALYSIS AND DISCUSSIONS
In this section, provide ablation studies examining which components of the model enhanced after training, the effect of the training budget t", and the allocation of tokens between the thinking and solution phases. Additional ablations varying the solution-token budget and the zero-thinking setting are provided in Appendix and Appendix K
4.4.1 WIICH PART ENTIANCED AFTER TRAINING?
Table 2:
Ablation of enhanced thinking and solution on DeepScaleR-1.5B 1.55--rreeieww and E1-Math- 1.5B. Budget format 'thinking+solution" (in thousands of tokens).
DeepScaleR-1.59 E1-Math-1.5B
Pass @1 (%)
Thinking Solution Thinking
Solution
0.5K+1K
1K+1K
2K+1K
3K+1K
2.10
4.80
12.5
20.0
3.5041.4
7.90+3.1
20.6+8.1
24.0+4.0
/
10.848.7
14.249.4
21.9+9.4
26.446.4
13.5411.4
17.5+12.7
24.8
27.947.9
To better understand which components the reasoning process are enhanced through train- ing, conduct ablation experiments ALME2024 benchmark. Specifically, we separately generate the thinking and solution segments us- ing both models under varying generation budgets. For example, to generate the thinking part, and then use El-Math-1.5B to generate the corresponding solution based that reasoning. This setup allows us to isolate the contributions of each model to the reasoning pipeline and assess how training improves each component.
DeepScalcR-1.5B-Previie and E1-Math-1.5B using the
1...caleR--..--Peiiiii
As shown in Table 2 we observe that both the thinking and solution are enhanced after training. Notably, the improvement in the solution component is more substantial, particularly under con- strained thinking budgets. For instance, using the E1 model to generate only the solution segment yields an 8.7% gain in accuracy compared to using the original DeepScaleR model, under genera- tion budget of (0.5K + 1K) tokens. This highlights the effectiveness of budget-constrained rollout in strengthening the model's ability to produce high-quality solutions based on incomplete reasoning.
This observation also helps explain why training with a fixed budget constraint (e.g., (1K, IK)) enables the model to generalize effectively wide range of budget configurations. We hypothesize that the improvement in solution generation plays central role in this generalization, allowing the model to adapt even when the available thinking tokens reduced.
4.4.2 ABLATION OF TRAINING BUDGET t"
To further investigate the role of the thinking budget t* in our proposed budget-constraieed roll- out, we conduct experiments to evaluate the model's performance under four settings: e (0.5K, 1K, 2K, 3K). while keeping the solution budget fixed a* 1K. We evaluate on five math benchmarks: AIME, AMC, Olympiad- -encch, MATH500, and Minerva Math (Figure
Across all configurations, the model demonstrates strong generalization varying inference budgets on all benchmarks. Among the tested values, t* = 1K consistently achieves the best performance, while also maintaining low maximum generation length of 2K tokens, making it highly efficient
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AIME
AMC
Olympiad-Bench
68%
45%
0055
Paw
15%
1.55
38%
46%
3056
Exed
MATH
Minerva
95%
0 E1-Math-i.sh (0.5K*IK)
E1-Math-1.55 (2K-1K)
131-Matb-1L.. (3K-1K)
Paxvi1
1065
22%
1K
Tokeas Esed
Figure 6: Ablation study on training reasoning budget t*, We compare four settings: t" {0.5K, 1K, 2K, 3K), while keeping the solution budget fixed at 1K. Each point shows Pass @1 I at given token budget, with the x-axis showing average tokens per problem. LiveCodeBench
AIME
12009
Thinking
20000
Thinidng Solution
Solutton
Teaal 17845
10000
Number Tokenss
32.1%
Number Tokeuss
15000
Trtal 0R230
Tedal: 11149
30000
D905
610
5000
2000
Totah 2310
020
Tetah 1791
1960
27137
1uuah 1781
1013
1957
Tetak 1378
3618
E1-Math (0.5k+1K)
EI-Math (1++1K))
Et-Math (2k+1K)
EL-Math (3k+
EI-Math (244)
El-Coder (ti++1))
El El-Coder (2k+118)
E1-Coder Gk++10
E1-Cnder (4k+1K)
El- El-Codor (64K)
DeapCoder (64K)
Figure 7: Distribution of tokens for thinking and solution across different generation budgets.
and effective setting. Based on the trade-off between performance and computational cost, we adopt (t" 1K, 5 1K) as our default configuration (Appendix H
4.4.3 TOKEN ALLOCATION BETWEEN THINKING AND SOLUTION
Figure visualizes the distribution of thinking and solution tokens within generated trajectories under different generation budget constraints. We select AIME2024 for the math task and Live- CodeBench for the coding task.
For AIME2024, the inference budget decreases, the number of tokens used in the thinking seg- ment decreases accordingly, while the number of tokens the solution segment slightly increases. A similar trend is observed LiveCodeBench, where the thinking tokens decrease with tighter budgets, while the number of solution tokens remains relatively stable.
Notably, even when evaluated without budget constraints, trained E1 models demonstrate sub- stantial token efficiency: they reduce total token usage by 32.1% on AIME2024 and 37.4% on Live- CodeBench, while maintaining strong performance (even slightly better than the baseline model). This suggests that the model has learned to reason more concisely and generate efficient solutions post training. Qualitative analysis is provided in Appendix G

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Published as a conference paper at ICLR 2026
5 CONCLUSION
We introduce Elastic Reasoning, unified framework for enabling large reasoning models to gen- erate accurate and efficient chain-of-thought outputs under strict inference-time constraints. By explicitly separating the reasoning process into thinking and solution phases, and training with novel budget-constrained rollout strategy, our approach ensures robustmess truncated reasoning while preserving or even improving overall performance. Elastic Reasoning significantly reduces token usage during inference, generalizes across unseen budget configurations, and outperforms prior length control baselines in both mathematical and programming domains. Our findings offer scalable and principled solution for real-world deployment of reasoning LLMs where computation budgets are limited. We believe this framework opens new directions for budget-aware reasoning.
ETHICS STATEMENT
Our study focuses on improving the efficiency and scalability of reasoning in large language mod- els by introducing budget-constrainet training and inference. All experiments were conducted on publicly available datasets mathematics (e.g., AIME, AMC, MATH500, Minerva Math) and pro- gramming (e.g.. LiveCodeBench,, Codeforces, HumanEEal+)), without the use of private or sensitive user data. No personally identifiable information (PII) or human subject data was collected, and therefore IRB approval not required.
We acknowledge the broader ethical considerations surrounding large language models, including potential risks of misuse, fairness, and environmental impact. Our method, Elastic Reasoning, is designed to reduce inference costs by making models more token-efficient. This can mitigate envi- ronmental impact by lowering computational and energy requirements.
REPRODUCIBILITY STATEMENT
We have taken several steps to facilitate reproducibility of our work. An anonymous implementation of our proposed Elastic Reasoning framework is included in the supplementary materials. The main text and Appendix B describe the methodology and training procedure detail, including the budget-constraineed rollout strategy and evaluation protocol. We provide complete information about the datasets used in both the math and code domains Dection[4.II, Hyperparameters, training configurations, and additional ablation studies are documented in the Appendix to enable faithful reproduction of our results. Together, these resources ensure that both the reported results and future extensions of this work can be reproduced by the community.
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A USE OF LARGE LANGUAGE MODELS (LLMs)
We used large language models (LLMs) solely as a writing assist tool to improve grammar, clarity, and style. The research ideas, methodology, experiments, analysis, and conclusions were developed entirely by the authors without reliance on LLMs. No part of the scientific contribution, experimental design, or result interpretation involved LLM usage.
B TRAINING DETAILS
For GRPO training, we adopt the same hyperparameters as those used in 1eepScaleR-115B-PP Preview and DeepCode- 14B-Preview For 11-MMth-1.5BB we use learning rate of 1 x 10 and batch size of 128. The maximum context length is set to IK tokens for the thinking segment and IK tokens for the solution segment. Training is performed for 200 steps using the VeRL (Sheng et al.) 2024) framework. For E1-Code-14P 14B, we use the same learning rate and batch size. The context length configuration mirrors that of the math model: 1K tokens for thinking and 1K tokens for solution. Training is conducted for only 30 steps.
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Model
AMC
Olympiad-Benchs
MATH500
Minerva Math
DepSScaleR-I.BB224K
72.06 (5571)
49.59 (5956)
87.63 (3124)
30.56 (4881)
LI-Exact-24K
66.42 (4024)
45.41 (4111)
83.45 (3937)
29.16 (4221)
L1-Max-244 24K
67.70 (3615)
45.69 (3547)
83.66 (3375)
28.81 (3440)
E1-Math- M-MMaahhhBBBB 1.5B-245 24K
71.23 (3653)
46.50 (3865)
85.05 (2011)
30.17 (2631)
Table C.I:
Pass @ I accuracy (%) under 24K-token unconstrained inference across math benchmarks. Numbers in parentheses indicate average tokens used.
Reward design. We follow the reward function used in DeepScaleR-1.5B, which is standard in recent Long-Cole CoT RL works. The reward binary:
if the final answer correct according automatic LaTeX/Sympy checker. otherwise (incorrect answer, malformed output, or missing <think> </think> delimiters).
Importantly, since our rollout is budget-constrained, the reward is computed based on the trajectory generated under the IK+1K budget. This ensures that the reward signal is aligned with the intended inference-time behavior.
Hyperparameter settings. We follow exactly the same hyperparameters as DeepScaleR-1..3 for fair comparison, except for our modified rollout procedure:
Batch size: 128
Learning rate: 1e-6
Rollout sampling temperature: 0.6
Number of rollouts per prompt:
Budget-constrained rollout: 1K thinking tokens, 1K solution tokens
Total RL steps: 200 (math), 30 (code)
C COMPARISON UNDER 24K-TOKEN UNCONSTRAINED INFERENCE
To highlight the effectiveness of our method, we compare 11-Math--1.5B with L1-Exact, L1-Max. and the vanilla DeepScaleR-1.55B under the 24K-token unconstrained inference setting across five math benchmarks (Table C.I). Note that E1-Math-1..5 1.5B, L1-Exact, and LI-Max are all fine-tuned from DeepScaleR-1.52
Compared to LI-Exact and LI-Max, our method achieves the best overall trade-off between perfor- mance and length control. For example, on Minerva Math, E1 E1-Math-1.5B uses only 2,631 tokens on average (vs. 3,440 for L1-Max and 4,221 for LI-Exact), while achieving 17% accuracy 1.36% higher than LI-Max and 1.01% higher than LI-Exact.
Relative to the vanilla D our accuracy remains comparable while requiring substan- tially fewer tokens. For instance, on AMC, E1-Math- 1.5B achieves 71. 71.23% accuracy with 3,653 to- kens, whereas DeepSScalRR-1.5 -1.5a requires 5,571 tokens to reach 72.06%. These results demonstrate that our approach preserves accuracy while significantly improving length efficiency.
D REASONING COMPLETENESS RATIO
We measure the reasoning completeness ratio tte fraction of cases where the full reasoning fits within the thinking budget-and and the corresponding Pass@ accuracy under different budget settings AIME, averaged over 16 runs.
and
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Inference Budget
0.5K+1K
IK+1K
2K+1K
3K+1K
Complete Ratio (vanilla)
0.0%
0.0%
2.3%
5.4%
Accuracy (vanilla)
5.2%
9.6%
15.8%
22.7%
Complete Ratio (E1)
0.0%
1.3%
7.5%
9.2%
Accuracy (EI)
13.5%
17.5%
24.8%
27.9%
Table D.I: Reasoning completeness ratio and @ accuracy on AIME.
At low budgets (0.5K+1K and IK+IK), the completeness ratio remains near zero, even after train- ing. Nevertheless, E1 achieves substantially higher accuracy, indicating that it learns to be robust truncated reasoning. E1 becomes more token-efficient over time, which slightly increases the completeness ratio byproduct- but this is not the main source of its gains.
Additional evidence comes from Table 2 when comparing the first and third rows, the thinking segment is held fixed, so the completeness ratio remains the same. Yet, El significantly outperforms vanilla model in accuracy. This confirms that El's improvements arise not from completing reasoning more often, but from better of limited budgets.
E
ITERATIVE TRAINING
Table E.1: Pass 1(%) AIME 2024 across different budget configurations in two iterations.
Iteration
0.5K+1K
1K+1K
2K+1K
3K+1K
1st (IK+IK)
13.5
17.5
24.8
27.9
2d (3K+1K)
11.5
17.1
22.9
26.7
1 (3K+IK)
9.2
14.8
21.7
25.2
2dd (IK+IK)
11.7
15.0
22.9
25.5
We investigate whether the model benefits from iterative training, i.e., performing a second round of training with different compute budget. Concretely, consider two schedules: (i) train with a budget of (t* = 1K, a = 1K) and continue from the checkpoint with (t" = 3K, a* = 1K); and (ii) the reverse order. Results AIME2024 (Table E. show noticeable improvement from the second round. This suggests that once the model has learned to reason effectively under a given budget, further training with different budget provides little additional benefit.
F MMLU RESULTS
To further assess generalization, we evaluate both models on MMLU (Hendrycks et al. 2020). We adopt the same budget-constrained settings and additionally report results under unconstrained inference (24K). All results are averaged over 16 runs:
Inference Budget
0.5K+1K
IK+1K
2K+1K 3K+1K
24K
Base Model
48.3%
1st Iteration (1K+1K)
47.4% 47.4%
47.7% 47.6%
48.1 19.1%
48.6%
48.4%
2nd Iteration (3K+1K)
47.9%
48.2%
48.2%
Table F.1:: Performance on MMLU under different inference budgets, averaged over runs.
The second-iteration model exhibits no degradation in general capability. This demonstrates that E1 retains its broad reasoning ability even after adapting to a larger budget, confirming that it learns budget handling transferable skill without forgetting prior competencies.
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G QUALITATIVE ANALYSIS
To examine how reasoning adapts under budget constraints, we compare outputs from the original model and our E1 model (trained with 0.5K thinking IK solution budget) on an AIME2024 example.
Original model (1537 tokens). The reasoning path is verbose, with many tokens consumed within individual steps.
E1 model (1091 tokens). After training with constrained budget, the reasoning steps become denser and more structured, reflecting more purposeful use of limited space. The model transi- tions more smoothly and directly into the solution, while maintaining accuracy. The final answer is reached correctly and with fewer tokens.
<think>
Okay, I have this problem about Aya's s walk. Let me try to understand what it's asking.
Aya walks 9 kilometers every morning, when she walks at a speed of s 7 kilometers per hour, it takes her 4 hours, including t minutes spent in the coffee shop. Then, when she increases her speed to s+2 km/h, the tctal time becomes 2 hours and 24 ninutes, including the the coffee shop. Now, we need to find the time it takes if she walks at + 1/2 km/h, again including t minutes in
same t minutes
the coffee shop.
Alright, let's s break this down step by step.
First, when she walks at speed s km/h, the total time including the coffee shop is 4 hours. That total time includes both walking and
the time in the coffee shop. Similarly, when her speed increases minutes, which is s+2 km/h, the total time becomes 2 hours
4 2 + 24/60 2 + 0.4 2.4 hours.
So, let's dencte:
let t be the time spent in the coffee shop in minutes.
Let á be the distance she walks before entering the coffee shop. Wait, but actually, the total distance is km. So, perhaps it's better consider the walking time and the coffee shop time separately.
Wait, perhaps another approach: the total time is walking time plus
+ coffec shep time. So, when she walks at km/h, the total time is hours, which is walking time + minutes. But is in minutes, convert everything to hours or minutes.
perhaps we
Let me clarify the units.
The walking speed is in km/h, the walking time would be distance/speed The coffee shop time in minutes, so we need to convert it to
+ hours or keep it as minutes. It's probably easier to convert
+ everything to hours.
so, let's define:
When she walks at speed km/h, total time is 4 hours, which includes t 4 minutes in the coffee shop. So, the walking tine is 4 (t/60) hours.
similarly, when she walks at s+2 km/b, the total time is hours and 24 4 minutes, which is 2 24/60 2 0.4 2.4 hours</think>
let's dencte:

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Published as a conference paper at ICLR 2026
( s V) her walking speed in km/h. time spent in the coffee shop in minutes.
(( t M)
The total time when walking at \ s A) km/h is hours, including \ + V minutes in the coffec shop.
Therefore, the walking time is 1< \frac(t)(60) hours.
Since she walks 9 km, the walking time can also be expressed as M \frac + 19)(s) V) hours.
So, we have: NI \frac(9) 4 (frac(t) Aquad (e)xt((1) MI
Similarly, when she walks at \ 2 V km/h, the total time is 2 hours and minutes, which is \ \2rac(24) (60) 2.4 V) hours.
Again, the walking time is (( 2.4 \frac(t) A) hours.
Expressing walking time ( (f)ac(9) 2) \) hours, get: \frac(9) (s 2) 2.4 \frac(t) (60) \quad \text( (2)) A]
Now, we have two equations:
1. ( (9 4 \frac(t)[00 V
2. ( 19rac(91s + 2 2.4 \fracit]160) V
Let me dencte \ \frac(t) \) as A x \) for simplicity. Then,
4 equation (1) becomes:
AI \frac(9) 4 \quad ( (1a)) A]
Equation (2) becomes:
A \frac(9)[s (s 2 = 2.4 - x \quad \text (2a) AJ
express / V) Al (9rac19) \quad \text (1b)) V
From equation (1a),
Substitute equation (1b) into equation (2a) Simplify:
AI (s)ac(9) 2) 2.4 (1eft (s))(() M
Al (frac(9) 21 2.4
\frac|91 \J
Al (frac(9) 21 -1.6 \frac(9]]s) \I
Bring all terms to one side:
M (f)ac(9) 2) - (frac(9) -1.6 A]
Factor out 9:
AI 9\1eft (frac(1) 2) (s) \right) -1.6 \
Simplify the expression inside the parentheses:
A (s 2) (fr) \fracls - (s (s + 2) \frac
(s 2))
Thus:
Al \times \ (2 2) \right) = -1.6 AI
Simplity:
AI (sa 2)) - -1.6 \I
Multiply both sides by -1:
AI (f) + 2) 1.6 A]
Solve for \ s(s + 2) \):
Al ss 2) = (1).6 = \frac (16) \frac(45) = 11.25 AJ
So:
11.25
Solve this quadratic equation: Using quadratic formula:
A = \fract -2 Apm (4 + 45) (2) = -Arrac -2 Apm \aqrt(49) 112
\frac( -2 \pm (2) \]
We discard the negative solution since speed can't be negative:
AI = \rrac(b/121 2.5 \text( km/hl AI
Now, substitute \/ 2.5 \) back into equation (la):
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| (9).2(9)
A[ 3.6 A]
Thus:
M[ xx== -
Listing Response from original model with 1537 tokens.
<think> Alright, I've got this problem em about Aya's walk and coffee shop time. let try break down step by step.
First, the problem says Aya walks kilometers every morning. When she + walks at speed of km/h, the total time taken, including the
coffee shop time, is hours. Then, when her speed increases to s+2
km/h, the tctal time becomes 2 hours and minutes. both cases the time spent in the coffee shop is the sane, denoted as t minutes.
The question is asking, if Aya walks
0.5 km/h, how many minutes 4 will her tctal walk time be, including the coffee shop time.
Okay, let's parse this informatien.
First scenaric: Speed: a km/h
Distance: 9 km
Total time including coffee: 4 hours
Therefore, time spent walking time in coffee 4 hours
second scenario:
Speed: s 2 km/h
Distance: 9 km
Total time including coffee: hours 24 minutes, which is 24/60
0.4 2.4 hours
Third scenario:
Speea: 0.5 km/h
Distance:
Total time: let's denote as hours, which includes t minutes the
+ coffee shop.
We need to find T in hours, then convert it minutes.
Let's s start by setting up equaticns.
In the first case, the time spent walking is distance divided by speed, so 9/s hours. Then, the time in the coffec shop is minutes, which is t/60 hours. Therefore, total time is 9/s t/60 hours.
In the second case, speed is 2, time walking is 9/ (s 2) hours. Total time is 9/ 21 t/60 2.4 hours.
So, we have two equations:
1) 9/s t/60
2) (/ 2) + t/60 2.4
We can subtract the second equation from the first to eliminate t/60.
So, (9/s t/60) (9/ (s </think>
Now, let's s solve these equations step by step.
From equation
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9/s t/60 --> t/60 9/s
60x 9/s) 240 540/s.
From equation 2:
9/ (s + 2) + t/60 = 2.4 - t/60 = 2.4 9/ 2) --> t 60. (2.4 - 9/(s (s + + 2)) 144 540/(s (s 2)
Now, set the two expressions for equal: 240 540/s = 144 540/ (s 2).
Simplity:
240 144 540/s 540/ (s 2)
96 540+ (1/s 1s + 2))
Simplify the right side:
1/s 1/
s) (s 21) 2/ (s 2)).
Thus:
96 540+ (2) (s 2) - 96 = 1080/ (s + 2)
Multiply beth sides by ss 2): 96*8 2) 1080.
Divide both sides by 24:
4+9 2) 45 --> $4s 83 45 0.
solve the quadratic equation: - [-8 s\mms sqrt (64 + 720) 1/ (2+4) - [-8 $1pms sqrt (78 [-8 s s
28)/8.
Since speed can't be negative, s (20) /8 2.5 km/h.
Now, find from equation 1: 240 540/2.5 240 216 24 minutes.
Now, for the third scenario, speed is + 0.5 km/h.
Time walking: 9/3 3 hours.
Total time including coffee: t/60 24/60 0.4 3.4 hours
+ hours 24 minutes.
Thus, the total time is 3 hours 24 minutes.
The answer is (boxed(204) minutes.
Listing 2: Response from El model with 1091 tokens.
H CHOICE OF TRAINING BUDGETS
We fix the solution budget to 1K tokens because, in practice, observe that when the </think> token manually inserted, the model naturally transitions to concise solutions that typically require fewer than IK tokens, even for complex problems. For the thinking budget, consider four settings 1K achieves the lowest training cost while yielding
E {0.5K, IK, 2K, 3K). Among these, validation loss comparable to that of larger budgets (2K and 3K).
The ablation study in Section 4.4.2 further supports this choice: the 1K + 1K configuration con- sistently provides the best trade-off between performance and training efficiency. Based on this one-time selection, we adopt t* 1K and 1K as the default setting across all math and code experiments.
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I MODEL VARIANTS
In this section, we present the results of method on Skywork-ORI-Math- using our budget-constrained rollout method for 200 RL steps with the same hyperparameters E1-Math-1..BB The results AIME24 (Avg@) 16) under multiple budget settings are shown in Table [.
We fine-tuned
Unconstrained performance increases by +1.3% while requiring fewer tokens. Under strict token budgets, the improvements are both substantial and consistent. The trend mirrors the results ob- served E1-Math- 1.5B, demonstrating that our method generalizes well across model architec- tures and training pipelines. E1-Skywork consistently outperforms L1-Skywork across all budget settings.
Model
Full
1.5K
2K
3K
4K
Skywork-ORI-Math
68.3 (13803)
1.0 (1534)
2.1 (2047)
7.7 (3051)
14.0 (4023)
E1 El-Skywork
69.6 (11768)
16.9 (1381)
21.3 (1841)
26.0 (2799)
32.9 (3742)
L1 S---kkooorkaxxs
59.6 (8277)
8.1 (1481)
14.8 (1948)
24.4 (2805)
31.7 (3497)
Table I.1: Performance of Skywork-OR 1-Math-7B and E1 S1-Skkyork-ORRI-aate AIME under different inference budgets, averaged over 16 runs.
J VARYING THE SOLUTION-TOKEN BUDGET
Model
IK + 0.25K
1K + 0.5K
IK + 0.75K
1K + 1K
1.75K + 0.25K
E1-Math-1.5B
4.8%
12.9%
15.8%
17.5%
5.0%
Table J.1: Effect of varying the solution budget under a fixed IK thinking budget.
Reducing the solution budget causes a substantial performance drop, even when the thinking phase preserved. Shrinking the solution phase 0.25K tokens severely limits performance. Even when increasing the thinking budget from IK to 1.75K, the gain is negligible (4.8% - 5.0%). This indi- cates that minimum solution budget necessary reliably articulate the final answer, regardless of how much reasoning is performed. This supports the core motivation of Elastic Reasoning: Reliable performance under strict budgets requires explicitly reserving solution tokens, not just maximizing reasoning length.
K ZERO-THINKING (OK + 1K)
Budget
AIME24
MATH500
1K + 1K OK 1K
17.5%
79.5%
0.4%
55.9%
Table K.1: Performance under zero-thinking (OK 1K) compared to the standard (1K IK) budget.
On hard problems (AIME24), the model is essentially unable to solve tasks without a reasoning phase (17.5% - 0.4%). On easier tasks (MATH500), the model retains some accuracy (55.9%), consistent with prior observations that certain math items can be solved via pattern recognition or shallow heuristics. Overall, this shows that the thinking phase is crucial for deep reasoning, while a sufficient solution budget is necessary to reliably express the final answer.
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L REPORTING MEAN AND STANDARD DEVIATION
We report results as mean + standard deviation on two datasets: AIME2024 (math reasoning) and LiveCodeBench (coding). For AIME2024, compare against both L1-Exact and L1-Max. For LiveCodeBench, we report only El-Code, as the publicly released LI implementation does not support code fine-tuning.
Model
2K 1K
3K 1K
E1-Math-11.BB
24.8% 3.7%
27.9% 3.3%
L1-Exact
14.7% + 3.0%
20.4% + 3.7%
L1-Max
24.0% 4.6%
25.8% 5.7%
Table L.1: AIME performance under different thinking+solution token budgets.
Model
2K 1K
3K 1K
E1-Code-1.5B 41.6% 0.98%
44.1% + 1.3%
Table L.2: LiveCodeBench performance under different thinking+solution token budgets.
M BUDGET-ALIGNED COMPARISON
We provide budget-aligned comparison in Table M. These results evaluate all models under the same inference-tine budget. Under matched budgets, E1-Math consistently outperforms LI-Exact, and surpasses L1-Max medium and larger budgets (3K and 4K). Notably, El retains most of the original model's 24K-token performance while requiring far fewer RL training steps (200 vs. 820).
Inference Budget
1,5K
2K
3K
4K
24K
L1-Exact
4.3%
10.2%
14.7%
20.4%
24.2%
L1-Max
17.5%
21.3%
24.0%
25.8%
27.1%
E1-Math
13.5%
17.5%
24.8%
27.9%
35.0%
Table M.1: Pass@1 performance on AIME2024 under matched inference budgets.
At small budgets (up to 2K), L1-Max is slightly stronger. At medium and large budgets (3K and 4K). E1-Math surpasses LI-Max, consistent with its superior test-time scaling behavior. In the unconstrained setting, E1-Math preserves most of the original model's full-budget accuracy (35.0%), while I.1 -ax shows degradation from overfitting to short reasoning (27.1% (2).1%). Overall, these results indicate that E1 offers more stable and scalable approach as reasoning depth increases.
N SCALING THINKING VS. SCALING SOLUTION
Below report both thinking scaling andsolution scaling results under equal total token budgets. All numbers correspond to @ on AIME24 (Avg@ 16).
Model
IK + 1K
OK + 2K
2K + 1K
OK + 3K
E1-Math-1.5B
17.5%
15.0% 2.9%
24.8% 15.8%
22.5%
1eeS
9.6%
11.4%
Table N.I: Separate-budgel inference under equal total budgets.
Separate budgeting (our inference mechanism). Across these equal- comparisons, sev- eral clear patterns emerge. First, separate budgeting consistently outperforms vanilla truncation,
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demonstrating that enforcing a dedicated reasoning phase is crucial for solving challenging prob- lems. Second, explicit reasoning remains beneficial even when the total budget fixed. For ex- ample, 2K total tokens, the IK+1K configuration (17.5%) surpasses the OK+2K configuration (15.0%), and a similar advantage holds at 3K total tokens, where 2K+1K (24.8%) outperforms OK+3K (22.5%) This indicates that allocating part of the budget structured thinking is consis- tently superior to spending all tokens on the final solution.
Third, removing the thinking phase entirely leads substantial accuracy degradation, especially before training, showing that clear reasoning segment is essential for deeper mathematical tasks. Fourth, both the thinking and solution budgets influence performance, but increases in the thinking budget lead to noticeably larger gains, suggesting that deeper reasoning is the primary driver of improvement. Finally, the budget-constrained rollout used during training strengthens the scaling behavior of both phases: after training. E1 models exhibit robust improvements in both thinking and solution effectiveness under matched budgets.
DISCUSSIONS
O.1 WHY IS E1-MATH- 1.5B SLIGHTLY BELOW ITS ORIGINAL BASELINE?
We believe the small Math model (1.5B) is more sensitive to perturbations during RL fine-tuning:
Limited capacity, Smaller models have less representational capacity and are more prone to catas- trophic forgetting when optimized with RL signals. Achieving both strong reasoning and strict budget robustness is more challenging in this size regime.
Trade-off' between robustness and maximal performance. Our method explicitly trains the model to produce correct answers even when reasoning is prematurely truncated. This robustness objective can slightly reduce maximum unconstrained performance smaller models, although it improves practical performance under constraints.
O.2 WHY WE CHOOSE IK SOLUTION BUDGET?
Before designing the training and evaluation setup, we examined the token distribution of long- reasoning models. As shown in Figure 7 the solution segment averages approximately 445 tokens on AIME24 (math) and 733 tokens on LiveCodeBench (code). Thus, the natural solution length for both tasks is well below IK tokens.
For this reason, we fixed the solution budget to IK tokens: it comfortably accommodates the typical solution length across math and code datasets, and it provides stable experimental configuration that isolates the effect of varying the thinking budget. In contrast, the thinking budget naturally grows much larger and is the primary factor driving test-time scaling behavior, which is why vary it extensively (1.5K- + 4K + 24K).
23


Reviews:
[{"review_id": "FH13jZryPg", "reviewer": "Reviewer_8Dnp", "summary": "This paper propose a method to control the reasoning sequence length of large language models. The key idea is to have a separate \"token budget\" for the c-o-t and for the answer, so that when the budget is exhausted for the c-o-t, the model can still produce an answer. Training with GRPO uses rollouts produced with that process, which leads the model to learn to deal with limited budget. The method outperforms alternatives that either limit the complete sequence or train the model to generate tokens that \"terminate\" its reasoning.", "questions": "- How is the model informed of the budget during inference? simply because </think> is generated? Hence the model has no information about the budget before it actually exhausts it? Fig 2 gives the impression that additional tokens specify it (red squares)?\n\n- This is not my direct domain of expertise, so are the baselines considered in the experimental part the best available?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Token budget is a key issue for reasoning, the method is very simple and sensical, performance are great.", "weaknesses": "Part 3.2.3 could probably be clarified, in particular the authors should provide a clearer description of the quantities involved and in particular the meaning of the conditioning in the policy, and the differences with a vanilla GRPO procedure.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "l9PSFHQg6q", "reviewer": "Reviewer_apFU", "summary": "The paper proposes Elastic Reasoning, a method that enables large reasoning models to achieve scalable and adaptive length control. To further improve solution quality under incomplete reasoning, we introduce a novel training strategy called budget-constrained rollout, which teaches the model to generate high quality answers even with partial CoT trajectories. This method is integrated into GRPO training. This method produces E1-Math-1.5B and E1-Code-14B.", "questions": "- Just to confirm that I am understanding the novelty correctly: Two phases of thinking and solution seems to be exactly what reasoning models like DeepSeek is doing, right? Am I missing a certain novelty claim by the authors here? Is it simply that prior works were enforcing the overall tokens count, but now budget-constrained rollout limits the thinking tokens count separately from the solution token?\n- Are there qualitative difference between the reasoning patterns of naive truncation vs Elastic Reasoning?\n- Appendix B mentioned that the training is conducted for only 30 steps, but that does not seem to match Figure 3.\n- Any intuition why the performances of the trained models are still lower than the original model’s?\n- Have you tried varying the solution budget instead? or perhaps a control experiment with 0 thinking token and 1K solution tokens may be interesting.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Very simple method that solves the problem of truncated solution in long reasoning.\n- The thinking-solution ablation (4.4.1) is interesting and is good evidence to understand what the proposed training method improves (i.e., generating a solution under an incomplete thinking process).", "weaknesses": "## Major\n\n- Figure 1, 4, 5, 6 are a bit unclear (This may be a minor weakness, but I assigned this as a major weakness for now because it is a crucial experimental setup):\n    - What are the points? Do they correspond to the whole AIME questions across different budgets?\n    - What is the x-axis? Is it the average tokens used?\n    - Could you include the error bars (x- and y-axis)? This is particularly important as there are cases where the Pass@1 and tokens used are not significantly different\n- Lack of model variations\n    - The authors only experimented with one model variant per setup (one for math, and one for coding).\n    - I understand that such experiments can be costly. However, I cannot confidently argue for the generalizability of this finding.\n- Lack of further analyses\n    - I am curious about the qualitative difference between the thinking process before and after the training (e.g., do the models after fine-tuning commit less backtracking? less circular reasoning?)\n\n## Minor\n\n- Missing details of the GRPO training\n    - What is the reward design? It is unclear whether when the training “converges” around 0.5 reward score is a good thing or not.\n    - What about the other hyperparameters?\n    - Please generally be thorough in describing the experiment\n\n## Additional Suggestions\n\n- Typo Section 4.2 title", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6T9Z4mMaPV", "reviewer": "Reviewer_9D8U", "summary": "The paper proposes Elastic Reasoning (ER), a framework that makes large reasoning models more efficient and controllable under limited inference budgets. By separating reasoning into thinking and solution phases with independent token budgets and training via budget-constrained reinforcement learning, ER achieves concise yet reliable reasoning—reducing token use by over 30% while maintaining or improving accuracy on math and coding benchmarks.", "questions": "- Can the authors evaluate Elastic Reasoning on out-of-domain reasoning benchmarks (e.g., MMLU, GPQA) to verify whether the method generalizes beyond math and code tasks?\n- Can the authors test this by removing the thinking phase entirely (i.e., prompting the model to output only the solution) and reporting the resulting accuracy?\n- Could the authors explore asymmetric budgets (e.g., 1.75K for thinking and 0.25K for solution) to test whether the model still maintains performance with a shorter solution phase? Most solutions require far fewer tokens than reasoning.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Novel framework for budget-aware reasoning – The proposed Elastic Reasoning introduces a clear separation between thinking and solution phases, enabling fine-grained control over inference cost without sacrificing performance.\n- Strong empirical efficiency and robustness – The method achieves reduction in token usage while maintaining or even improving accuracy on diverse math and coding benchmarks.\n- Excellent generalization under unseen budgets – Models trained with a single budget configuration generalize effectively to new inference constraints, demonstrating strong adaptability and practical scalability.", "weaknesses": "- The method is only tested on strong reasoning models (DeepScaleR, DeepCoder); it’s unclear whether it generalizes to weaker models like Qwen2.5-Math, which lack explicit CoT structure or strong reasoning priors.\n- The paper shows that most improvement comes from the solution phase, while increasing the thinking budget (e.g., 2K–3K tokens) brings little additional gain. This suggests that the model may not truly improve its reasoning efficiency—instead, it might rely on memorized solutions rather than performing deeper thinking.\n- The evaluation lacks out-of-domain (OOD) reasoning benchmarks such as MMLU or GPQA. As all experiments focus on math and code, it remains unclear whether Elastic Reasoning generalizes to broader reasoning domains or tasks requiring factual and conceptual knowledge.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)