FROST: Filtering Reasoning Outliers with Attention for Efficient Reasoning
Authors: Haozheng Luo
Zhuolin Jiang
Md Zahid Hasan
Yan Chen
Soumalya Sarkar
Venue: ICLR 2026 Poster (2026)
Keywords: Efficient Reasoning
Attention Outlier
Reasoning
Primary area: foundation or frontier models, including LLMs
TL;DR: We propose FROST, an attention-aware method that identifies and removes reasoning outliers to prune uncritical reasoning paths, producing shorter and more reliable reasoning trajectories without sacrificing reasoning capacity.

Abstract: We propose **FROST**, an attention-aware method for efficient reasoning.  Unlike traditional approaches, FROST leverages attention weights to prune uncritical reasoning paths, yielding shorter and more reliable reasoning trajectories.  Methodologically, we introduce the concept of *reasoning outliers* and design an attention-based mechanism to remove them.  Theoretically, FROST preserves and enhances the model’s reasoning capacity while eliminating outliers at the sentence level.  Empirically, we validate FROST on four benchmarks using two strong reasoning models (Phi-4-Reasoning and GPT-oss-20B), outperforming state-of-the-art methods such as TALE and ThinkLess.  Notably, FROST achieves an average **69.68%** reduction in token usage and a **26.70%** improvement in accuracy over the base model.  Furthermore, in evaluations of attention outlier metrics, FROST reduces the maximum infinity norm $\||\mathbf{x}\||_{\infty}$ by **15.97%** and the average kurtosis by **91.09%** compared to the base model.

Published as a conference paper at ICLR 2026
FROST: FILTERING REASONING OUTLIERS WITH AT- TENTION FOR EFFICIENT REASONING
Haozheng Luo Zhuolin Jiang"
Md Zahid Hasan
Yan Chen
Soumalya Sarkar2
DDpartment of Computer Science, Northwestem University, Evanston, IL 60208 USA
RRT Technology Research Center (RTRC), East Hartford, CT 061 18 USA
DDeaarmentt of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA
bluoQu. www.weetenrwwweercccco edu zahid@iastate.cm edu
fehuooli..jaangssoomaa soumalya. wtt.:mallyy..aaarrrrrarrrra sarkarrrrrztccoomo com ychendnorthvesterr.cc
ABSTRACT
We propose FROST, attention-aware method for efficient reasoning. Unlike traditional approaches, FROST leverages attention weights to prune uncritical reasoning paths, yielding shorter and more reliable reasoning trajectories. Method- ologically, we introduce the concept of reasoning outliers and design an attention- based mechanism remove them. Theoretically, FROST preserves and enhances the model's reasoning capacity while eliminating outliers at the sentence level. Empirically, we validate FROST on four benchmarks using two strong reasoning models 1hii4-Rcasooinn and (PTT-oss-2BB) outperforming state-of-the-art meth- ods such as TALE and ThinkLess. Notably, FROST achieves an average 69.68 reduction in token usage and %6.70% improvement in accuracy over the base model. Furthermore, in evaluations of attention outlier metrics, FROST reduces the maximum infinity norm ||x|| by 15.97% % and the average kurtosis by 91.09 % compared to the base model.
INTRODUCTION
We observe that large reasoning models (LRMs) often generate numerous irrelevant steps, which we term reasoning outliers. To mitigate this, we introduce FROST, an efficient reasoning method that leverages attention weights to prune uncritical reasoning paths, producing shorter and more reliable trajectories. More specifically. FROST replaces the standard Softmax function with Softmax,, enabling attention to better identify and suppress outliers. This directs LRMs toward critical reasoning steps, thereby enhancing their overall reasoning capacity.
Efficient reasoning is critical for large reasoning models (LRMs), which have shown strong perfor- mance tasks such mathematical problem-solving (Luo al., 2025a; Yang al., 2024; Shao et al., 2024), coding (Ding et al., 2024a;b), and scientific question answering (Comanici et al., 2025; Hurst al., 2024). Yet, these models often generate large amounts of uncritical information- commonly arising from redundant self- verification that introduce inefficiencies and potential inaccuracies. Numerous methods have been proposed to improve reasoning efficiency. Token-level approaches such as TALE (Han et al., 2025) and R2R (Fu et al., 2025) risk pruning essential reasoning steps, as reasoning paths are naturally sentence-basee based. Sentence-level approaches, including DRP (Jiang et al., 2025b) and GRPO-S (Tan & Pan, 2025), perform iterative refinement of reasoning paths, but this often comes at the cost of increased computational cost and latency.
To address these challenges, we propose FROST, reasoning method that improves efficiency by pruning uncritical reasoning paths through attention weights. We observe that LRMs typically assign low attention to uncritical steps and higher attention to critical ones, consistent with findings that critical steps exhibit higher sentence entropy (Tan & Pan, 2025). We therefore introduce the concept of reasoning outliers- uneritical steps with both low attention weights and low entropy (Wang et al., 2025; Fu et al., 2025) and design FROST to eliminate them, yielding shorter and more reliable reasoning paths. Our approach sharpens the attention distribution of LRMs, suppressing
Thii work carried out the RTX Technology Research Center during a Summer 2025 internship.

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Published as a conference paper at ICLR 2026
Orgiall Mad
seporan, dn perdtaed culiiie
Figure 1: The Example of The GPT-OSS-20B Model.
low-weight steps while preserving high-weight ones. Building on prior work (Luo et al., 2025b; Hu et al., 2024; Xiao et al., 2024), we adopt Softmax; in place of Softmax, which effectively drives low weights to zero while maintaining large weights. Finally, we propose training strategy that integrates Softmax; with supervised fine-tuning on reasoning tasks, producing efficient reasoning models without sacrificing accuracy.
Contributions. We present FROST (as shown in fig. 1), reasoning outlier-free LRM designed to enhance reasoning efficiency. Our main contributions are:
We introduce the concept of reasoning outliers and propose FROST to prune uncritical reasoning steps characterized by low attention.
Theoretically, we analyze Softmax, and show its effectiveness in suppressing low attention weights while preserving high ones, thereby enhancing the reasoning capacity of LRMs.
Methodologically, we design a training strategy that combines Softmax) with supervised fine- tuning, enabling efficient reasoning without sacrificing accuracy.
Empirically, we demonstrate the effectiveness of FROST across multiple benchmarks, achieving up 26.70% accuracy gain while reducing reasoning path length by 69.68% compared with base models. We also measure attention outlier values to verify their impact on efficient reasoning: FROST reduces the maximum infinity norm ||x|| by 15.97 % and the average kurtosis by 91.09% % In addition, FROST cuts inference time by at least 28.6% and reduces training time by 42.2% relative to other SFT baselines.
RELATED WORK
Reasoning Models. In recent years, Large Language Models (LLMs) such as DeepSeek-Ri R1 (Guo al., 2025), OpenAl ol (Jaech al., 2024), and Gemini 2.0 Pro (Team al., 2023) have demon- strated strong reasoning capabilities, particularly on mathematical and logical tasks (Hao et al., 2024). To further improve reasoning performance, numerous methods are proposed, falling into the main paradigms (Ke et al.): inference scaling and learning-to-reason For inference-time scaling, numer- ous methods have been proposed, including few-shot prompting (Brown et al., 2020), in-context learning (Brown et al., 2020), Chain-of-Though (CoT) reasoning (Wei et al., 2022), and Search & Planning (SP) (Besta et al., 2024). Numerous studies focus on improving the LLM reasoning at inference time, with CoT emerging process and generates interpretable reasoning traces, A simple example involves adding a prompt like "Let's think step by step" after a question (Wei al., 2022). Recent research increasingly combines CoT with other inference-time scaling methods, such as ReAct (Yao et al., 2023), Self-Ask (Press al., 2023) and agentic reasoning (Pan et al., 2025; 2024), further enhance reasoning capabilities. For seerming-to-reasse
key technique. CoT strengthens the model's reasoning
approaches. many methods aim to build reasoning ability through alignment, including reinforcement learning (RLHF (Ouyang et al., 2022), DPO (Rafailov et al., 2023), GRPO (Ramesh et al., 2024)), supervised fine-tuning, and energy-based model (EBM) reasoners (Jiang et al., 2025a). However, LLMS with reasoning capabilities- paralities--pptiiulllaryy those with smaller parameter generate excessively detailed reasoning chains, including unnecessary tracebacks and redundant alternative paths (Hou et al., 2026; Chen et al., 2025). This overthinking not only increases computational cost during inference but also negatively impact response quality accuracy (Cuadron et al., 2025) and safety (Kumar et al., 2025). To address this, we propose an iitention-awart adaptation method that optimizes reasoning paths, yielding efficient reasoning models.
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Efficient Reasoning Methods. To address overthinking, current approaches to optimizing reasoning paths fall into three categories (Sui et al., 2025): comptt-bseed methods, supervised tine-tuning, and reinforcement learning. Prompt-bare commpt-based methods (Liu et al., 2025; Xu et al., 2025a; Han et al., 2025) introduce token-budget constraints to shorten reasoning paths. For instance, TALE (Han et al., 2025) limits the token budget per instance to reduce reasoning length while maintaining task accuracy. Supervised fine-tuning (SFT) methods (Ma et al., 2025; Xia et al., 2025a) improve reasoning conciseness by training models on compressed reasoning paths. For example, DRP (Jiang et al., 2025b) fine-tunes models on distilled reasoning data by pruning unrelated reasoning steps. Reinforcement learning (RL) methods (Hou et al., 2026; Li et al., 2025; Yi et al., 2025) guide concise reasoning by introducing reward functions that penalize overly long reasoning paths. For example, Chia al. (2024) introduce reward score based on reference loss and exploration loss from diverse paths, encouraging favorable reasoning branches and penalizing unfavorable ones to improve overall problem-sole solving performance. However, prompt- methods rely on handcrafted prompts and often perform unreliably on complex problems. In contrast, SFT and RL approaches require substantial computational resources for fine-tuning, limiting accessibility for users without adequate hardware. To address these challenges, eliminates reasoning outliers through attention analysis. Recent studies (Choi et al., 2025; Cai et al.) also analyze internal attention patterns in reasoning models, particularly the sentence level, but their objectives differ substantially from ours and focus on KV-cache -y-eeche-bbaed inference efficiency. Think Clearly (Choi al., 2025) examines sentence-level attention spikes near the end-of- of-thinking token and uses these patterns to prune redundant sentences for faster decoding. In contrast, our Figure analyzes sentence-level contributions the final-answer token, enabling attribution of which specific reasoning sentences actually affect the model's prediction, rather than identifying redundancy for pruning. R-KV (Cai et al.) likewise detects redundant attention interactions to compress the KV cache, but does not study how individual reasoning steps functionally influence final-answer formation. Our work therefore provides a finer- aner-grained, component-lewe attribution analysis of the reasoning trace going beyond redundancy detection to clarify how different reasoning segments vary in contribution, which constitutes the key novelty relative to these approaches.
propose new reasoning outlier-removal strategy that
REASONING OUTLIER
In this section, we analyze the attention distribution of reasoning traces generated by LRMs. We then examine the impact of different components of the trace on final answer prediction, followed by our definition and characterization of reasoning outliers.
3.1 ATTENTION DISTRIBUTION OF REASONING TRACES
We consider representative LRMs, including DeepSeek-R RG et al., 2025), Phi-4 (Abdin et al., 2025), and GPT-4o (Hurst et al., 2024), which generate text in autoregressive manner by predicting the next token given the preceding context. To study the attention distribution, we visualize the attention heatmap of each token in the reasoning trace when predicting the final answer.
Let the reasoning process be a sequence of tokens T f2......... tn]. where each t, denotes a token in the process. The attention weight matrix A is defined as:
A = [ayy] where aij Attention (z).
Here, az represents the attention weight from token t; to token tz.
As an illustrative example, we use a sample question from GSM8K (Cobbe et al., 2021) and generate the reasoning trace with the Phi-4-Reasoning model (Abdin et al., 2025). fig. 2 shows the corresponding attention heatmap. The results indicate that in the shallow layers, the attention distribution is relatively uniform across all tokens. However, as we move to deeper layers and later heads, the model begins to focus more on specific tokens, particularly those the reasoning steps and the final answer. This suggests that the model progressively refines its focus towards the most relevant parts of the reasoning trace as processes the information.
3.2 IMPACT OF REASONING TRACE COMPONENTS ON ANSWER PREDICTION
To quantify the impact of different components of the reasoning trace on final answer prediction, we conduct additional experiment analyzing the summed attention weight distribution to the final answer token </think>, which allows us to measure how strongly each reasoning step contributes to the model's ultimate decision and provides insights into whether the model grounds its prediction
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Alteadion Weaght Layer
Attentien Wetght
Takes Prettinn
Atteadio Weights Layer
Welghtt Layer
Attcatk Welghts Lager
Takon Panttee
Tokea Position
Tokan Pestice
Figure 2: Attention Heatmap of Reasoning Tokens. We use the Phi-4-Reasoning model (Abdin et al., 2025) to generate reasoning trace for sample GSM8K question (Cobbe et al., 2021). The figure shows attention heatmaps from transformer layers O, and 39, with the first head (top row) and last head (bottom row). Yellow indicates higher attention weights and blue indicates lower ones. In shallow layers, contributions to the final answer are nearly uniform, while deeper layers and later heads highlight specific tokens with stronger influence.
in meaningful intermediate reasoning or relies on superficial correlations. We divide the reasoning process into four components: the question Q, the reasoning steps R1, R2...., .. Rms and the final answer A. For each component, we compute the total attention weight contributing to the first token of the final answer: W uraee Σ,, Exi aid, where Tince is the set of tokens in given component, and 0;A4 denotes the attention weight from token t to the </think> token.
Hall
121
17
Figure 3: Total attention weight distribution to the final answer token </think> from different components of the reasoning trace. We visualize the total attention weight distribution of the Phi-4-Reasoning model on sample GSM8K question, using transformer layers 1, 30, and 40. The results show that few reasoning traces contribute strongly to the final token </think>, while many traces have negligible influence, particularly the layers and 40.
As shown in fig. 3, different reasoning traces contribute unequally to final answer generation. While a few traces show strong influence, most contribute weakly, and some exhibit almost no contribution all.
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sepdett saquimay unng
Number
Number
Vanilla
FROST
Figure 4: Theoretical Analysis of Reasoning Outlier Removal. We conduct a theoretical analysis with Phi-4-Reasoning model to demonstrate that removing reasoning outliers using the Softmax function (FROST) can preserve or even enhance the model's reasoning capacity. As shown in the figure, the attention weight distribution before and after outlier removal indicates that the model's focus critical reasoning traces maintained or improved, while the influence of outliers is significantly reduced.
3.3 DEFINING AND CHARACTERIZING REASONING OUTLIERS
As observed in section 3.1, many reasoning traces contribute negligibly to the final answer. These traces often correspond to verification, self-checking, or repetition of prior reasoning steps. Their presence forces LRMs to generate more tokens than necessary, substantially reducing reasoning efficiency. A potential cause (Sui et al., 2025) is that model developers often encourage extended reasoning steps maximize accuracy. In the meantime, the model may generate redundant or irrelevant information, leading to inefficient and incorrect reasoning. As a result, we define reasoning traces with low attention weight and negligible contribution to the final answer as reasoning outliers.
To identify and remove reasoning outliers, we observe that they share similar characteristics with attention outliers (Luo al., 2025b; Hu al., 2024). Motivated by this, we adopt Softmax; (eq. (1)) to detect and climinate reasoning outliers during the reasoning process, and provide a comprehensive proof of its efficiency in section 5.
exp(x1:) Softmax (xttmaax(x;))).... = Eg exp(xx) -1'
where represents the attention weight of token 44-
Theoretical Analysis. We conduct a theoretical analysis to show that removing reasoning outliers with the Softmax, function preserves, and can even enhance, the reasoning capacity of I.RMs. In our experiments, we use the Phi-4-Reasoning (Abdin et al., 2025) to generate reasoning traces for sample GSM8K question (Cobbe et al., 2021). Specifically, we compare the last layer's attention distribution in head 15 under vanilla attention and Softmax attention (FROST), As shown in fig, 4, Softmaxx reduces the influence of outliers while maintaining or strengthening focus on critical reasoning traces. This analysis supports approach of using Softmax, effectively identify and eliminate reasoning outliers, thereby improving the efficiency and reliability LRMs. For more details of the theoretical proof, please refer to section 5.
4 FROST
To enhance the reasoning efficiency of LRMs, we propose supervised fine-tuning (SFT) with reasoning outlier removal, as illustrated in fig. 5.
In the SFT stage, we train on math problems with detailed reasoning steps and answers. During training, we replace the vanilla Softmax with Softmax; (eq. (I)), enabling the model to focus on critical reasoning traces while suppressing outlers. Unlike prior methods that employ Softmax, either training from scratch (Hu et al., 2024) multi-step continual learning (Luo et al., 2025b)-20000 our approach achieves effective outlier removal with only a few steps of fine-tuning from existing pretrained checkpoints, making it more efficient and practical. We optimize model parameters using cross-entropy loss and apply LoRA (Hu et al., 2021) to further reduce training cost.
for outlier
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Outlier-wwe aware normilization (Softmaxs)
suppresses extreme attention spikes,
Softmax
FFN Layer
K
Traditional Reasoning Model
Suppressed
under Softmaki
How could ignore the uncritical reasoning steps?
LRM
PRN Layer
Efficient Reasoning Model Figure 5: Overview of the FROST workflow We replace the vanilla Softmax layer with an outlier- removal layer based on Softmax,, followed by SFT to adapt model parameters the new activation function. We observe that our method significantly reduces the number of low-attention sentences.
Outter spiicc (Seftmax)
Outller removed (Soliman)
THEORETICAL ANALYSIS
In this section, provide brief theoretical analysis showing that Softmax, operate at the sentence level to remove reasoning outliers in LRMs. We provide a theoretical proof that our method achieves deployment-tima suppression in efficient reasoning, consistent with our findings in fig. 4.
Setup. Let a token sequence be partitioned into sentences (5.)= For a query E R& and keys {kr} IRd define token compatibilities
Softmax, (g. )oy, where is the token index in S; v
and U/ Rd denotes the token value for each token in sentence S. Let Φ R [t R be monotone pooling operator (e.g., Sun//nean/oossumepp/mxx))) Define sentence scores $; = o (f22)s)) and
($1,..., 8m) E IR", Define the probability simplex mm1
E Rm a; 2 0, Σi= =
Assumption 5.1 (Softmax, operator). There exists Softmax, mapping 01 R A m-1 such
that:
Order preservation: If r; 2 x, then on (x); 2 o) Shift invariance: 01(x + cl) = a1 for all e e R.
Tail contraction: There exists н E (0, 1) such that for all r R", [[] I K (r) (median(11(2)))
|
Smoothness and positivity: σ₁ continuously differentiable on RM and 0r (x); > 0 for all finite
We write the sentence-level attention a (s) m-1 and the layer output as y Disi 0g4 which and v; are attention probabilities and token values corresponding to sentence 8;. Assume ||o|]ooo < B, and that all linear maps used below have finite operator norms B,, which is a constant.
Lemma 5.1 (Monotone pooling preserves sentence dominance). then for any i,j. (Vi S,, Et Sj x ze) tion 5PI a; 0n (s), σ1 (5),
é is monotone coordinatewise, S; > 8j. Consequently, by Assump-
aj-
Proof. See appendix C. for detailed proof.
Theorem 5.1 (Softmaxi suppresses sentence-level attention outliers). Let = (Si,..., 9m) be the sentence scores built via a monotone pooling If heavy-tailed (e.g., ||s||oo/ (alo//mediass)) 1), then for a σ1 (s)
|alloc median(a)
|
K
for some x E (0, 1),
(2)
median(s)
so the relative dominance outliers contracts at the sentence level.

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Proof. See appendix C.2 for detailed proof.
Theorem 5.2 (Deployment time suppression low-attention sentences). Let the output logits be f Woy with WWoooop =: B, and ||e:]] I B,, For sentence with a; S its one-layer contribution to logits is bounded by
| = ||Wo(avii]l & B, e ||lll| 2 B. B,
(3)
For L stacked layers with Jacobians (Je)é- and ||Jellop S Be.
1 | I II
Be B.
(4)
Since the Softmax, map IRV A V is 1-Lipschitz in the lo & norm,
|sm(E - || Bo B, II Be
(5)
Let B m) Bg be the largest operator norm over T. layers. Then
smie + Ac(")) sm(O]], & B,B,B'e = O(B.B,BLe)
(6)
In practice Bo, Be, B are approximately constant, so the bound reduces to O(s). Therefore, low- attention sentences are effectively skipped inference.
Proof. See appendix C.3 for detailed proof.
EXPERIMENTAL STUDIES
We conduct series of experiments to evaluate FROST in providing efficient reasoning, benchmarking its performance on GPT-oss (Agarwal et al., 2025), Magistral-Small (Rastogi al., 2025) and Rhass-nRason (Abdin et al., 2025). Each evaluation is conducted three times with different random seeds, and we report the average and standard deviation for each metric.
Models. In experiments, (Rastogi et al., 2025) and GPT-oss (Agarwal et al., 2025) as backbone models for efficient reason- ing. Specifically, we adopt the Phi-4-Reasoning" Magistral-Small--1 and GPTT-ss--0B-finenln checkpoints, both finetuned on mathematical datasets with detailed reasoning steps and answers using SFT under the FROST method.
Phi-4-Roee Reasoning (Abdin al., 2025), MiagistralSSmall 1.1
Datasets. Following the setup in (Zhao et al., 2025a), we use OpenR1 (Hugging Face, 2025) as the training corpus. To evaluate reasoning efficiency and generalization on complex mathematical prob- lems, adopt four out-of-domain benchmarks: GSM8K (Cobbe et al., 2021). MATH500 (Lightman et al., 2024), AIME24 (of America, 2024), and Minerva (Dyer & Gur-Ari, 2022). All datasets are designed for mathematical question answering.
Metrics. To evaluate the effectiveness of our efficient reasoning strategy, we report pass@1 as the accuracy metric and use the number of tokens in the reasoning response measure token efficiency.
Baselines. We select five representative methods covering key paradigms of efficient reasoning: (1) TALE (Han et al., 2025): a prompt-based approach that uses soft token budget to generate concise reasoning responses.
(2) DRP (Jiang et al., 2025b): an SFT-based method that distills reasoning paths from a teacher model and applies step-level pruning to produce concise, skill-aware reasoning traces. (3) SelfBudgeter (Li et al., 2025): reinforcement lcarning- based method that
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Table 1: Comparison of FROST with Efficient Reasoning Methods. We evaluate reasoning path efficiency by comparing FROST against four baselines across four mathematical datasets (GSM8K, MATH5000, AIME24, and Minerva). Pass@) and token usage (#Tk) are reported as evaluation metrics, with variance omitted since it is consistently S 2%. Best results are shown in bold, and second-best are underlined, In most settings, FROST achieves the best performance among all methods. Specifically, it improves accuracy by 26.70% while reducing token usage by 69.68% compared to the base model.
Type
Method
GSM8K
MATH500
AIME24
Minerva
Apwso
AsT.
@ass@1
#Tk
@ass@1
#TK
Pass@1
#Tk
@ass@1
#TK
Base
0.9242
1017.70
0,5480
1721.95
0.0667
1017.70
0.2500
1898.86
0.000
0.00
-Reusoning
TALE
0.9500
1716.60
0,5800
1874.43
0.2900
2069.97
0.2627
2093.17
+0.074
+524.49
Phi-4
DRP
0,8340
721.00
0.6200
2122.00
0.3333
6135.00
0.2701
1289.50
+0.067
+1152.69
SelfBudgeter
0.9189
1507.14
0,5347
1195.18
0.1342
1372.83
0.2357
2618.27
+0.009
+259.30
ThinkLess
0,9279
1421.90
0,5414
1101.21 21
0. 0616
1405
0.2575
1708.70 70
+0.025
-.75
Ours
0.9311
154.33
0,5980
344.37
0,2667
899,80
0.2716
401.19
+0,070
.964.13
Base
0.8704
1275.23
0,5400
L575.36
0.1333
1003.57
0,2574
1586.95
0.000
0.00
GPT-O SS. 2013
TALE
0.8283
2664.41
0,5454
3878.87
0,2000
1354.67
0,2700
3262.47
+0,011
+1430.33
DRP
0,7880
902.50
0.6146
4137.00 .00
0.2245
4983.00
0,2715
1885.15
+0.024
+1616.64
SelfBudgeter
0.8610
1850.00
0.534n
0.1320
1256.00
0,2550
1298.00
.0.005
+312.47
ThinkLess
0,8740
1785.00
0,5410
2206:00
0,1600
1205.00
0,2580
1220.00
+0,008
+244.22
Ours
0,8764
377.17
0,5800
680.89
0.1667
1009.60
0.2794
691.71
+0.025
669.99
0.6075
2664.41
0,1480
1389988
0.0000
537.13
0.0699
1288.04
0.000
0,00
Magistral -Small-11. 1.1
TALE
0.7146
1516.86 86
0,3040
723.91
0.0333
967.43
0. 1544
748.18
+0.095
-480.77
DRP
0.6500
902.50
0,2100
1680.33
0.0450
1350,77
0,1120
1604.22
+0.048
-85.41
SeifBudgeter
0.6900
1850.00
0.2300
1520.00
0,0520
1256.00
0.1300
1298.00
+0,069
+11.13
ThinkLess
0.7200
1785.00
0,2500
1405 00
0.0600
1205.00
0,1450
1220.00
+0.087
66.112
Ours 0.7551
137.55
0.3040
98.20
0.0974
149.93
0.1551
109.23
+0.122
-1346.14
iteratively shortens the reasoning path by optimizing a token budget under budget and format reward signals. (4) ThinkLess (Fang et al., 2025); a reinforcement learning- method that optimizes reasoning by detecting critical thinking points and skipping low-value steps. It introduces a reward function that balances accuracy with token usage, enabling models to "think less" while maintaining performance. We use the same hyperparameters specified in their respective studies to ensure standardized evaluation conditions, enabling precise comparisons of each efficient reasoning method.
Results. As shown in table 1, FROST achieves the best overall performance across state-of-the-art efficient reasoning methods, delivering slight accuracy improvements while substantially reducing token usage in response generation. Specifically. FROST improves accuracy by an average of 70% and reduces token usage by 69.68% on the three base models, GPT-OSS-20B, Magistral- Small-1.1 and Phi-4-reasoning. Although TALE achieves the highest accuracy on certain tasks, this comes at the cost of significantly longer responses. This observation aligns with our assumption that excessively long or overly short responses can degrade model performance. By reducing token usage and focusing on high-attention sentences-i.e. critical reasoning traces-FROST lowers the probability of hallucination or misleading content and grounds responses essential reasoning. However, FROST may still occasionally prune low-attention but important reasoning steps, which explains why its accuracy is not always the best across all baselines.
6.1 SUPPLEMENTARY EXPERIMENTS
In this section, we conduct additional experiments examine the influence of our method's perfor- mance at different training stages and under different attention functions.
Efficiency of Different Activation Functions. To evaluate the contribution of Softmax, in FROST, we conduct experiments comparing FROST with different activation functions: vanilla Softmax, Sparsemax (Hu al., 2023; Martins & Astudillo, 2016), and Entmax15 (Wu et al., 2024; Correia et al., 2019). Here, Entmax15 is a special case of Tsallis o-entmax transformations, which interpolate between softmax and sparsemax. We evaluate these strategies on four datasets- MATH500, AIME24, and Minerva-
Phi-4-Reasoning. As shown in table 2, the results demon- strate that FROST achieves the best overall performance in both Pass @ accuracy and token usage. Specifically, the average accuracy increases by 15.65 %,, while the number of tokens decreases by 68.18% % compared to the base model. FROST also surpasses the overall performance of Sparsemax and Entmax 15, 5, which tend to sharpen both low- and high-attention sentences, potentially cutting
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Table 2: Performance of Different Activation Functions. We evaluate the impact of activation functions on method performance under the same training setup in FROST, using Phi-4-Reasoning across four mathematical datasets (GSM8K, MATH500, AIME24, and Minerva). Pass and token usage (#TK) are reported evaluation metrics, with variance omitted since it is consistently < 2%. Best results are shown bold, and second-best are underlined. In most settings, FROST achieves the best performance, with mn consistently ranking second.
GSM8K
MATH500
AIME24
Minerva
Method
Pass@P
#TT
Pass@1 1
#TK
Pass
#TK
Pass@1
#TK
Pass
#Tk
Base
0.9242
1017.70
0.5480
1721.95
0.0667
1017.70
0.2500
1898.86
0.4472
1414.05
Softmax
0.8317
1160.63
1379.52
0.1333
1909.07
0.2390
1934.72
0.4230
1595,99
Sparseuxx
0.8188
160,99
0.5120
451.59
0.1667
948.60
0.2647
580.84
0.4406
535.26
Entmax15
0.8984
163.75
0.5520
406.97
0.1667
876.63
0.2831
439.48
0.4751
471.71
Softmax; (PROST)
0,9311
154.33
0.5980
344.37
0.2667
899.80
0,2716
401.19
0.5169
449.92
off critical reasoning traces. In contrast, FROST is less prone to this issue. The only exception is the Minerva dataset, where Eutmax15 attains higher accuracy than FROST while maintaining a similar number of tokens. The underlying reason is difficult to explain at this stage, but it is pleasant surprise that, except for GSM8K, the overall performance of Sparsemax and Entmax15 does not decline significantly and in some cases even surpasses the base model. This offers a perspective contrary to that of Yang et (2025); Wang (2024).
Table 3: Outlier Removal Performance in FROST. We evaluate outlier removal performance on the AIME2024 dataset using the Reaasonngg model. As outlier metrics, we report the maximum infinity norm ||x|| and average kurtosis the activation tensors. To assess the proportion of critical traces, we also report the average sentence entropy before and after applying FROST. All results are reported with variance omitted, as consistently I 2%. Best results are shown in bold, and second-best results are underlined. In most settings, FROST achieves the best performance in outlier removal and yields higher average sentence entropy. These metrics demonstrate that our method effectively removes reasoning outliers, thereby improving both reasoning performance and efficiency.
Method
Maximum Infinity Norm |x|]
Average Kurtosis
Average Sentence Entropy 1
Pass@I 1
#Tk
Base
35.31
241.72
2.71
0.0667
1017.70
Sottmax
34,53
189.36
2.79
0.1333
1909.07
Sparsemax
34.06
152.18
2.93
0.1667
948.60
Entmax15
30.39
2.92
0.1667
876.63
FROST
29.67
21.54
3.07
0.2667
899.80
Outlier Removal Performance in FROST. To evaluate the performance of FROST in removing attention outliers, we employ two outlier-specific metrics: the maximum infinity norm ||x|| of the activation tensors x across all Transformer layers, and the average kurtosis of x, which together quantify the presence of outliers. In addition, to demonstrate that removing attention outliers increases the probability assigned to critical sentences, we introduce an entropy-based evaluation metric. Following Wang et al. (2025), token entropy serves tokens tend to exhibit higher entropy than non-critical ones. When a sentence contains more critical tokens, it is expected to exert stronger influence on final answer generation. Accordingly, use average sentence entropy to assess whether the reasoning traces in FROST become more critical after training. In our experiments, we analyze these metrics on the AIME2024 dataset using the Phi-4-Reasoning model and compare them with the base model. As shown in table 3, FROST effectively reduces outliers, evidenced by lower maximum infinity norm ||x||<c and average kurtosis values. Furthermore, the increase in average sentence entropy indicates that FROST strengthens the model's focus on critical reasoning traces, thereby improving reasoning efficiency. Specifically, we reduce the maximum infinity norm ||x|| by 15.97% % and the average kurtosis by 91.09%. % In addition, the average sentence entropy increases by 13.28 % compared to the base model. Additionally, the results show that reasoning outlier macriicsmaxximum infinity norm |xx|.>. and average kurtosis are closely related to model performance and average sentence entropy. Higher outlier values correspond to lower sentence entropy and less efficient reasoning traces. This further supports that the reasoning-oute outlier removal contributes to more efficient reasoning. The only exception is that the average sentence entropy of Sparsemax is similar to Entmax 155, 15, while
indicator of criticality: critical

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Published as a conference paper at ICLR 2026
the reasoning outlier values of Entmax15 much smaller than those of Sparsemax. A plausible explamation is that both Eutmax15 and Sparsemax act as sharpening activations that jointly suppress low- and high-valued attention scores. This bidirectional truncation inadvertently remove parts of crucial reasoning traces, lowering average sentence entropy and reducing Pass@) 1 performance. Meanwhile, attention outlier metrics such the maximum infinity norm and kurtosis primarily reflect internal activation dynamics rather than output quality, explaining their relative stability despite external performance declines. Since both activations reshape attention distributions similarly, their outputs also appear alike-with wwith comparable @ass@1 and entropy values values-thougl though Entmax15 smoother contraction yields slightly less degradation in outlier metrics. Overall, this indicates that excessive sharpening can eliminate valuable reasoning signals even while suppressing attention outliers, highlighting Softmax;ss advantage through selective tail contraction.
6.2 GENERALIZABILITY OF MODEL
In this section, we evaluate the generalization ability of FROST on out-of-domain reasoning tasks to verify that its improvements do not harm, but rather preserve or enhance, the model's generation quality beyond the training domain. Using Phi-4-Reasoning as the base model, we test on three additional reasoning benchmarks- Leetrod (Xia et al., 2025b), LiveCodeBench (Jain et al., 2025), both coding and physical reasoning tasks. The results in
and UGPhysical (Xu et al., -025b)-cover table 4 show that FROST preserves- and even searroves--eeerlizzattti
to unseen reasoning tasks. This is expected because FROST filters out uncritical reasoning traces in a manner that generalizes beyond the specific tasks used during fine-tuning. Since FROST only replaces the attention activation with Softmax, and uses lightweight LoRA updates, the parameter shift is minimal, ensuring that the model's broader reasoning ability remains intact.
Table 4: FROST Generalization on Other Reasoning Tasks. We evaluate the generalization of FROST using Phi-4- Reasoning across three out-of-domain reasoning tasks (Code and Physics). Pass@I 1 accuracy and token usage (#Tk) are reported, with variance consistently S 2%. Best results are in bold, and second-best are underlined. FROST consistently achieves top performance, demonstrating strong generalization across reasoning domains.
Leetwode
LiveCodeBench
UGPhysics
Method
Pass I
#Tk
Pass @
#Tk
Pass@1
#Tk
Pass @
#Tk
Base
0.3222
2755.13
0.3248
3154.80
0.3172
2603.00
0.3214
2837.64
Softmax
0,3778
2106.85
0.3538
2909.07
0.3011
2622.52
0.3442
2546.15
FROST
0.3889
1163.06
0.3777
1967.56
0.3473
805.77
0.3713
1312.13
DISCUSSION AND CONCLUSION
We propose an attention-aware efficient reasoning method, FROST. Our approach introduces the concept of reasoning outliers, which contribute to high latency reasoning performance, and provides an outlier removal mechanism that enables LRMs to leverage lightweight supervised fine- tuning (SFT) for generating efficient and accurate reasoning traces in mathematical problem solving. Theoretically, we show that the existing outlier removal technique Softmax, is effective for reasoning outliers and can operate the sentence level. Empirically, FROST improves response accuracy by 26.70% and reduces token usage by 69.68 %. In addition, it decreases the maximum infinity norm || |x||a by 15.97% % and average kurtosis by 91.09% %, confirming its effectiveness in mitigating reasoning outliers.
Although FROST achieves strong performance in efficient reasoning, several limitations remain. First, our method is currently restricted mathematical reasoning tasks, while many reasoning models also target domains such as coding. Second, FROST relies solely on supervised fine-tuning and does not incorporate GRPO, which could further enhance efficiency. In future work, we plan to extend FROST to additional reasoning tasks, including coding, and to develop GRPO-based approach that builds on our current findings to further improve efficient reasoning performance.
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ACKNOWLEDGMENTS
HL would like to thank Dennis Wu for insightful discussions on theoretical analysis. This research was supported in part by the computational resources and staff contributions of the RTRC High Performance Computing Cluster. Additional support was provided by the Quest High Performance Computing Facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies.
ETHICAL STATEMENT
This work investigates reasoning outliers in large reasoning models (LRMs) and proposes an outlier- removal technique to mitigate attention outliers. In line with the ICLR Code of Ethics", we acknowledge that our method may inadvertently amplify biases present in training data, potentially leading to unfair outcomes for underrepresented groups. Prior studies have also noted that supervised fine-tuning (SFT) can induce shallow alignment and affect red-teaming protection, but this issue is outside the scope of our work. Our focus is on improving reasoning efficiency, and we believe this research does not raise serious ethical concerns.
REPRODUCIBILITY
To ensure reproducibility, we release anonymous open-source repository (link) ) containing the full implementation of FROST and selected baselines, with plans for full open-sourcing upon acceptance. All experiments are conducted with three random seeds, yielding stable results with standard deviations below 2%. We adopt a unified training setup using the Adam optimizer with learning rate 1 10 batch size 8, and batch size 256 for deployment. For low-rank adaptation, we set the LoRA A rank to 8 and LoRA a to 16. Detailed hyperparameters are provided in appendix E.2, and thcoretical proofs are included in appendix C.
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An Yang, Beichen Zhang, Binyuan Hui, Bofei Gao, Bowen Yu, Chengpeng Li, Dayiheng Liu, Jian- hong Tu, Jingren Zhou, Junyang Lin, et al. Qwen2. 5-math technical report: Toward mathematical expert model via self- eelloonprovement. arXiv preprint arXiv:2409.12122, 2024.
Lijie Yang, Zhihao Zhang, Arti Jain, Shijie Cao, Baihong Yuan, Yiwei Chen, Zhihao Jia, and Ravi Netravali. Less is more: Training-fiee sparse attention with global locality for efficient reasoning. arXiv preprint arXiv:2508.07110 2025.
Shunyu Yao, Jeffrey Zhao, Dian Yu, Nan Du, Izhak Shafran, Karthik Narasimhan, and Yuan Cao. React: Synergizing reasoning and acting in language models. In International Conference Learning Representations (ICLR), 2023.
Jingyang Yi, Justin Wang, and Sida Li. Shorterbetter: Guiding reasoning models to find optimal infer- ence length for efficient reasoning. In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025.
Haoran Zhao, Yuchen Yan, Yongliang Shen, Haolei Xu, Wenqi Zhang, Kaitao Song, Jian Shao, Weiming Lu, Jun Xiao, and Yueting Zhuang. Let LRMs break free from overthinking via self- braking tuning. In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025a.
Yiran Zhao, Wenxuan Zhang, Yuxi Xie, Anirudh Goyal, Kenji Kawaguchi, and Michael Shich. Understanding and enhancing safety mechanisms of LLMs via spfety-specific neuron. In The Thirteenth International Conference on Learning Representations, 2025b.
Zhihan Zhou, Robert Riley, Satria Kautsar, Weimin Wu, Rob Egan, Steven Hofmeyr, Shira Goldhaber- Gordon, Mutian Yu, Harrison Ho, Fengchen Liu, et al. Genomcocean: An efficient genome foundation model trained on laage-scale metagenomic assemblies. bioRxiv, pp. 2025-01 2025.
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Supplementary Material
A Impact Statement
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B Additional Related Work
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C Proofs of Main Text
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C.1 lemma 5.1
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C.2 theorem 5.1
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C33 theorem 5.2
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D An Example of LRM Reasoning Traces
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E Experiment System and Implement Settings
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E.1
Computational Resources
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E.2
Hyperparameters
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F Additional Experiments
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F.I
Training and Test Time Comparison Attention Distributions of Activation Functions
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F2
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G Influence the attention dynamics of Softmax, during training and inference
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H Influence of Softmax, Across Layers
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I Extended Attention Heatmaps Across Additional Layers and Heads
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J Human Expert Evaluation
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K Disclosure of LLM Usage
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A IMPACT STATEMENT
We believe this methodology offers an opportunity strengthen the core large reasoning models by improving efficiency and enabling models to produce more critical reasoning traces. However, it may also amplify biases present in the training data, potentially leading to unfair or discriminatory outcomes for underrepresented groups
B ADDITIONAL RELATED WORK
Efficient Alignment. In recent years, foundation models (Zhou et al., 2025; He et al., 2025; 2024; Wang al., 2024; Touvron al., 2023) have shown strong capabilities in solving multitask problems. To further improve their performance on specific tasks, alignment techniques are essential for refining model behavior. However, traditional approaches like RLHF (Ouyang et al., 2022) and DPO (Rafailov et al., 2023) are computationally expensive. This highlights the urgent need for parameter-efficient fine-tuning methods that offer effective and economical alignment for foundation models. Several traditional methods demonstrate strong capabilities in aligning foundation models, including LoRA (Hu al., 2021) and QLoRA (Dettmers al., 2023). Building on this. Luo et al. (2025b) propose a LoRA variant that replaces the standard softmax layer with OutEfiHHp layers (Hu et al., 2024) improve the efficiency low-rank adaptation. However, all of these methods are heavily based on LoRA, and when adaptation is required for modules outside the attention architecture, the computational cost increases significantly. Zhao al. (2025b); Luo et al. (2025c) propose novel alignment methods that focus small subsets of neurons within foundation models. For example, Zhao et al. (2025b) identify key neurons with high influence on LLMs' jailbreak defense using latent representations, and fine-tune only these neurons using red-teaming datasets. Our method builds fast low-rank adaptation techniques (Luo et al., 2025b), further improving adaptation efficiency, and integrates them into SFT training optimize reasoning paths and produce efficient reasoning models.
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C PROOFS OF MAIN TEXT
C.I LEMMA 5.1
Proof of lemma 5.1. Monotonicity means that if we increase any input coordinate to ø. its output does not decrease. Let u = { hes, and w = )re,, If for each coordinate of there is not-smaller coordinate in w replaced by the smaller value, then by repeatedly applying coordinatewise monotonicity we obtain s; ((u) 2 d(w)= = sy. Order preservation (PI) then yields a; 2 aj.
C.2 THEOREM 5.1
Proof of theorem 5.1. By Lemma 5.1, sentence scores reflect dominance induced by token compatibilities under Φ. Applying Assumption SP3) directly to yields (2). Assumption 5. (P22 allows re-centering <- cl without changing a: thus (2) is invariant to any global shift and depends only on relative separations.
C.3 THEOREM 5.2
Proof of theorem 5.2. For (3), apply operator-norm submultiplicatiitty: [Wo(a,))) < |Wollop= as eellvell S B, By. To obtain (4), propagate the perturbation through differentiable layers with Jacobians Jz:
L | S [eeloo))) || WWolo & (II
B B. B
Finally, since Softmax; is 1-Lipschitz in foc + 1, the change in probabilities bounded by the logit change, yielding (5). Replacing II By with B (by definition of B) gives the stated B,B.B rate. If B,, B,, B are O(1), the rate simplifies to O(e).
D AN EXAMPLE OF LRM REASONING TRACES
In this section, we analyze the Phi-s-Reasnning response to the first question of AIME24, which is also illustrated in fig. 4. As shown in the color box in appendix D, traces S1 and S2 are classi- fied as uncritical. Although (actual walking time) plus minutes equals total time. ", its overall reasoning re- mains non-critical. Trace S3 represents critical reasoning step, where the model identifies the two key equations in the problem. Subsequently, from S4 to S19, the model enters a velffieriiicotiion phase, producing reasoning traces beginning with wait that reflect self-checking and correction. Starting from $20, the model resumes critical reasoning after the signal "We '1 produce final answer in a box. ", and by S24, it generates the final answer, concluding its reasoning process.
includes partially critical content such as "So the walking time
E EXPERIMENT SYSTEM AND IMPLEMENT SETTINGS
E. COMPUTATIONAL RESOURCES
We perform all experiments using two NVIDIA H100 GPUs with 80GB of memory and a 12-core INTEL(R) XEON(R) PLATINUM 8592 CPU operating 1.90GHz. Our code is developed in PyTorch and utilizes the Hugging Face Transformer Library for experimental execution. For running the LLMs, to 0.6 to balance consistency and performance.
the default system prompt provided by the official source and set the temperature
E.2 HYPERPARAMETERS
We present the hyperparameters used the fine-tuning stage for each model. We use Adam W (Loshchilov & Hutter, 2019) as the optimizer. Most other hyperparameters are kept consistent across all models and datasets, including a batch size of 256 during deployment and during training. In training, also gradient accumulation with 4 steps and set the weight decay 0.01 for
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Example of LRM Reasoning Traces
[Question] Every morning Aya walks km and then spends minutes in a coffee shop. At speed km/h, the total time is 4 hours (including the coffee stop). At speed 8 + 2 km/h, the total time is hours 24 minutes. If she walks at + 2 km/h, find the total time in minutes (including the coffee stop).
[Uneritical Trace S1] <think>We are given: "Every morning Aya goes for a 9-kilometer- coffee shop afterwards. When she walks constant speed of
long walk and stops 8 kilometers per hour, the walk takes her 4 hours, including / minutes spent in the coffee shop. When she walks 2 kilometers per hour, the walk takes her 2 hours and 24 minutes, including $1$ minutes spent the coffee shop. Suppose Aya walks at 8 + 1/2 kilometers per hour. Find the number of minutes the walk takes her, including the StS minutes spent in the coffee shop."
[Uneritical Trace S2] Let's denote: "Walking distance = km. There's coffee shop stop time minutes, but note that the given total times includes the coffee shop stop time. So the walking time (actual walking time) plus minutes equals total time. Let 8 be her walking speed (km/h) in first scenario. Then walking time (in hours) is 9/s. And the total time is hours."
[Critical Trace S3J But includes t minutes stop, so actual walking time is 4 hours minus t minutes (converted to hours, t/60 hours), so equation: 9/s = - t/60. (Equation 1) and 9/(s + 2) 2.4 r/60 (Equation 2)
[Uneritical Trace S4] But wait, let's check: speed = + 2. then walking time (in hours) is 9/(s + 2). Total time is hours 24 minutes, which is + 24/60 2.4 hours.
[Uncritical Trace S5] But wait, let's check: The problem said: "Suppose Aya walks at s+1/2 kilometers per hour. Find the number of minutes the walk takes her, including the minutes spent in the coffee shop."
[Uneritical Trace S6] Wait, hours 24 minutes = 144 minutes. [Uneritical Trace S7] Let me recale: 3 hours 24 minutes 2*60 + 24 = 120 24 144 minutes.
[Uneritical Trace S8] Let double-check the reasoning process.
[Uneritical Trace S9-S17]
[Uneritical Trace S18] I'll now produce step-by-step explanation.
[Uneritical Trace S19] We'll produce final answer in a box.
[Critical Trace S20] The two equations are given 9/s - 1/60 (Equation 1) and
9/(s + 2) = 2.4 - 1/60 (Equation 2).
[Critical Trace $21] Alternatively, we can solve for / in terms of from equation 1: / (4- = 240 5540s, and plug in equation 2: 9/(s+2) = 2.4-2242222 (2240-540/s/666 =
2.4 (4- 9/5) 2.4 4+ 9/s = (9/s) 1.6.
[Critical Trace S22] So equation becomes: 9/(s 2) = (9/s) - 1.6. Multiply both sides by s(s 2): 9s (9(s + 2)) 1..s(s + +2.
[Critical Trace S23] Now, when Aya walks at speed + 1/2 2.5 + 0.5 = 3 km/h, then walking time (in hours) 9/3 = 3 hours. But total time includes coffee shop stop time minutes, which is 24 minutes 24/60 0.4 hours. So total time is 3 + 0.4 3.4 hours, which is hours 24 minutes.
[Critical Trace S24] I'll produce final answer:
204
</think>
all training runs. A learning rate of 1e-5 is used for all models during fine-tuning. For low-rank adaptation, we use a LoRA rank of 8 and LoRA alpha set to 16. In FROST, we set the maximum training steps to 5,000. All supervised fine-tuning and GRPO training are conducted using mixed precision with 16 In deployment, we set the temperature to 0.6 for all models with top-p sampling 0.9. For evaluation, except TALE.
maximum generation length of 4096 across all models,
F ADDITIONAL EXPERIMENTS
In this section, we present additional experiments demonstrating that FROST surpasses current state-of-the-ard efficient reasoning methods.
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F.1 TRAINING AND TEST TIME COMPARISON
We conduct experiments to measure the training and inference time of each baseline and compare their computational costs with FROST. For evaluation, test time measured on the AIME tasks with the GPT-OSS-200 model, while training time is reported using the respective datasets specified in each baseline's 's original paper. All experiments are conducted on the same computational resources, described in appendix E.
Table 5: Comparison of Training and Test Time Costs Across Methods. We conduct experiments to measure the training and test time of each method. For test-time evaluation, use the AIME dataset with the GPT-OSS-20B model. Best results are shown in bold, and second-best are underlined.
Method
TALE
DRP
ThinkLess
FROST
Training Time (m)
353
1186
204
Test Time (m)
56
18.5
4.2
3
As shown in table 5, FROST achieves the fastest training time among all methods, while also minimizing computation cost and inference time during deployment. This demonstrates that our approach not only accelerates training but also reduces deployment overhead.
F.2 ATTENTION DISTRIBUTIONS OF ACTIVATION FUNCTIONS
conduct an additional experiment to analyze the attention distribution of GPT-OSS-20B on sample from the GSM8K dataset. As shown in fig. 6, FROST effectively removes large number of low-attention sentences while retaining significant ones. In contrast, the vanilla model produces many sentences with low attention weights, and Sparsemax and Entmax15 retain only one two sentences, often aggressively discarding important reasoning traces. This visualization provides an explanation consistent with the performance results reported in table 2.
Figure 6: Attention Distribution of Each Activation Function.
G INFLUENCE THE ATTENTION DYNAMICS OF Softmax, DURING TRAINING AND INFERENCE
We observe that incorporating Softmaxı significantly influences both training and inference attention dynamics across transformer layers. During supervised fine-tuning (SFT), Sofumax, enforces tail contraction by suppressing low-attention activations, which stabilizes gradients and reduces the variance of updates propagated through residual connections. This effect leads to faster convergence of LoRA adapters, as the low-rank parameter subspace more efficiently aligns with critical attention directions, improving overall adaptation coverage within fewer training steps. This observation is consistent with Luo et al. (2025b); Hu al. (2024). Across layers, Softmax, reshapes the attention handscape-shill layers become more selective in contextual grounding, while deeper layers exhibit higher entropy concentration around critical reasoning traces. During inference, this sharpening propagates forward, effectively filtering redundant reasoning sentences while maintaining coherence. Together, these behaviors demonstrate that Softmax, not only enhances efficient reasoning but also accelerates LoRA-SFF optimization by improving the representational focus of each attention head.
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H INFLUENCE OF Softmax, ACROSS LAYERS
We analyze the effect Softmaxı across transformer layers by visualizing the attention distributions of head 15 for both vanilla Softmax and Softmax1. As shown in figs. 7 and 8, Softmax, consistently suppresses attention outliers, leading to smoother and more stable activations across the network. In lower layers, Softmaxı contracts heavy tails and mitigates rare extreme peaks, enhancing local feature mixing with higher-entropy and reduced kurtosis distributions. In higher layers, it suppresses residual long-range spikes and sharpens focus semantically relevant tokens, yielding sparser yet more stable attention and clearer causal information flow.
Vasilla
FROST
Figure 7: Theoretical Analysis of Reasoning Outlier Removal in All Layers
I EXTENDED ATTENTION HEATMAPS ACROSS ADDITIONAL LAYERS AND HEADS
In this section, we present extended attention heatmaps covering additional layers and heads. Specifi- cally, we analyze Layers 0, 5, 15, 25, 30, 35, and 39 and Heads 0, 5, 10, 15, 20, 25, 30, 35, and 39 provide a more comprehensive view of attention evolution across the network. The corresponding observations are illustrated in fig. 9.
HUMAN EXPERT EVALUATION
We invite three computer science students specializing in reasoning models to annotate reasoning traces generated by the original and FROST-trained models. We then compare the traces pruned by FROST and evaluate their criticality based on relevance and contribution to the final answer. Averaging across all evaluators. FROST achieves 92% accuracy in correctly removing non-critical reasoming traces. Only 8% of reasoning traces are incorrectly removed, which significantly degrades final-answer accuracy. These mistakenly pruned traces are typically long and contain repeated information that supports self-verification and error correction. However, they also provide critical content such as key equations the end of trace. This observation suggests potential explanation experiment shown
for why FROST achieves the second-best Pass @ I score in the table 1.
K DISCLOSURE OF LLM USAGE
In our paper and project, we use large language models (LLMs) to help revise the text for greater conciseness and precision.
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Figure 8: Attention Distribution of Softmax, Acruss All Layers
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Figure 9: Extended Attention Heatmaps Across Additional Layers and Heads
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Reviews:
[{"review_id": "WYe7dhYhvv", "reviewer": "Reviewer_vRQV", "summary": "LRMs typically generate a large number of noncritical steps, inevitably introducing inefficiencies and potential inaccuracies. The authors propose FROST, a method for accelerating and strengthening inference. They first introduce the concept of inference outliers: noncritical steps with low attention weights and low entropy. Technically, the authors use softmax1 to detect and eliminate inference outliers. Preliminary experiments effectively reveal the problem FROST aims to address, and comprehensive experiments demonstrate its effectiveness.", "questions": "1.The x and y axes of all the graphs are unclear, especially in graph 2, where the y-axis of the two upper subgraphs is even all zero due to missing parts. Could you please make the graphs clearer?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.The authors clearly articulate and demonstrate the problem of outliers in reasoning through premise experiments, while also providing a simple yet highly effective solution. This is easy to understand and well-intentioned.\n2.The experimental results on multiple datasets are significant and compared with the state-of-the-art algorithms, convincingly demonstrating the effectiveness of FROST.\n3.In addition to basic performance experiments, the paper also includes a large number of validation experiments, such as using some metrics to try to quantify the performance of outlier removal.", "weaknesses": "1.In Figure 4, the attention weight of S2 is also significantly compressed, while S3 is actually improved. I understand that S24 is the key reasoning path and has been improved. However, further analysis of S2 and S3 is necessary.\n2.The authors demonstrate in Table 3 that removing attention outliers increases the probability assigned to critical sentences. Token entropy is used as an indicator of criticality. While most experiments meet expectations, Entmax15 and Sparsemax exhibit unexpected performance. The authors offer no explanation. This is necessary to clarify whether removing inference outliers actually improves reasoning ability. Please try to provide possible explanations as to whether there is token entropy or some other fluctuation.", "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": "yTZujXCpKo", "reviewer": "Reviewer_ncQk", "summary": "This paper proposes to improve LLM’s mathematical reasoning capabilities by suppressing less important tokens. Specifically, it adopts the softmax1 function, which is less prone to long-tailed distribution. Theoretical proof is provided to validate the usefulness of the approach on sentence-level pruning. Experimental results on several datasets demonstrate the advantages of the method in efficient token usage and enhanced performance.", "questions": "(1) Please justify the technical contribution of the proposed studies over prior ones using softmax1 activation.\n\n(2) How does the observation in Section 3 motivate the use of softmax1? \n\n(3) How does incorporating softmax1 affect the attention behaviors across layers for both inference and during training?\n\n(4) How would the proposed method reduce the computational overhead in practice?\n\n(5) While I am aware of the focus on mathematical reasoning in this paper, the requirements of supervised fine-tuning makes me concerned about the potential issues on model generalization. For instance, does fine-tuning on mathematical dataset hurt the model performance on other reasoning tasks? How would training models with softmax1 from scratch work, compared to post-hoc fine-tuning?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "(1) It is an important topic to develop efficient LLMs that adaptively reduce computational overheads based on inputs.\n\n(2) The paper draws several observations on the attention characteristics of reasoning models, which facilitates understanding on their underlying decision-making process.\n\n(3) The proposed method shows generalizability across different models and datasets.", "weaknesses": "(1) Prior studies (e.g., Xiao et al, 2024, denoted as [ref1] in the remaining review) have already studied the use of softmax1 activation in developing efficient LLMs. Adopting the same methods on a specific reasoning scenario introduces rather limited technical contributions. In addition, the observations in the paper are also similar to previous ones, for instance, Figure 2 is similar to Figure 7 in [ref1], and [ref1] also pointed out the focus on specific tokens in deeper layers (initial tokens vs think token in this paper).  \n\n(2) The paper highlights different characteristics of reasoning in Section 3 (e.g., progressive refinement and reasoning traces), nevertheless, I found the proposed method disconnected from these observations. For instance, how does softmax1 affect the attention distribution in different layers? Does softmax1 enhance or reduce the progressive refinement? Figure 4 only shows the theoretical analysis of softmax1 on deep layers, does it hold true in practical experiments? What kinds of tokens does softmax1 activation prune out?\n\n(3) Softmax1 only suppresses the contribution of tokens with lower weights, instead of pruning it from the computational graph, thus there may not be actual reduction of computational overhead.\n\n(4) The method requires supervised finetuning to replace softmax with softmax1 in pretrained models. This could be relatively impractical for large-scale experiments, and may also affect the generalizability of pretrained models (due to finetuning on specific datasets).", "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": "rfy1FCqswr", "reviewer": "Reviewer_ej9v", "summary": "This paper introduces FROST, a novel method designed to improve the efficiency and accuracy of Large Reasoning Models (LRMs). The authors identify a key problem in current LRMs: the generation of \"reasoning outliers,\" which are defined as uncritical, irrelevant, or redundant steps in a reasoning chain. These outliers increase computational cost (token usage) and can negatively impact the final answer's accuracy.\n\nThe core of FROST is a simple yet effective architectural modification: replacing the standard Softmax function in the self-attention mechanism with Softmax_1. This function has the property of aggressively suppressing small input values (driving low attention weights toward zero) while preserving large ones. By doing so, FROST prunes the influence of outlier sentences during generation. The authors combine this modification with a lightweight Supervised Fine-Tuning (SFT) process to adapt the model to the new attention dynamics.", "questions": "- The paper uses the maximum infinity norm and average kurtosis as metrics to quantify the presence of attention outliers. Have the authors considered or compared other statistical measures for identifying outlier activations? Why were these specific metrics chosen, and are they considered optimal for this task?\n- The results show a significant reduction in token count. Could you provide a qualitative example or analysis of what kind of reasoning steps are typically pruned? Are they mostly redundant self-corrections, or does the model sometimes remove steps that could be considered part of a valid, albeit longer, reasoning path?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper's primary strength lies in its empirical results. The method achieves a compelling combination of significantly reduced token usage while simultaneously improving accuracy across multiple benchmarks and models. This is a strong and desirable outcome for any efficiency-focused technique.\n- The approach is methodologically simple and elegant: swapping an activation function and performing a short SFT run. This makes the method practical and easily reproducible.\n- The supplementary experiments are valuable. The ablation study comparing Softmax1 with other activation functions (Table 2) effectively supports the authors' design choices, and the analysis of outlier metrics like kurtosis and infinity norm (Table 3) provides concrete evidence that the method is working as intended.", "weaknesses": "- My main concern is the limited novelty of the core mechanism. The paper frames the use of the Softmax1 function as a key contribution for removing \"reasoning outliers.\" However, this exact function and its properties for suppressing outlier/low-value attention scores were previously discussed in other contexts, notably in Evan Miller's 2021 blog post \"Attention is Off by One.\" While applying this function to reasoning chains is new, the underlying technique for attention modification is not, which should be more clearly acknowledged.\n- The paper introduces \"reasoning outliers\" as sentences with low attention and negligible contribution. While the concept is intuitive, the method for identifying them seems to be implicitly handled by Softmax1 rather than through an explicit detection step. This makes the framing feel a bit like a post-hoc justification for using a pre-existing technique.", "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": "cgrm5b839b", "reviewer": "Reviewer_oTRF", "summary": "This paper proposes a novel training method that introduces a softmax-based activation function to suppress the attention weights of outlier tokens while preserving important reasoning paths.\nExperimental results demonstrate that the proposed approach outperforms existing baselines and can be effectively applied to strong reasoning models.", "questions": "**[Q1]** Please verify whether the proposed observations and improvements generalize to other reasoning models (e.g., R1-distill models).\n\n**[Q2]** In Figure 2, why were layers 0, 30, and 39 and specifically the first and last heads selected for visualization? A quantitative analysis across all layers and heads would make the claim more convincing.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The proposed method is simple yet effective, significantly reducing token usage compared to prior approaches.\n- The theoretical analysis seems solid and support for the design and interpretation of the method.\n- Reproducibility is high — the authors provide full code and detailed implementation information.", "weaknesses": "**[W1] Restricted experimental scope.** Despite the interesting ideas, both the analyses and experiments are conducted on a limited number of models. In particular, the observations are primarily focused on the Phi-reasoning model, making the study somewhat specialized and reducing the general applicability of the findings.\n\n**[W2] Limited discussion of related work.** There are already recent studies [1, 2] analyzing internal attention patterns in reasoning models. A deeper discussion comparing this work with those, and clarifying its novelty relative to them, would strengthen the contribution. Notably, the analysis of the attention distribution around the end of thinking token (Figure 3) appears similar to [1].\n\n[1] Choi et al., Think Clearly: Improving Reasoning via Redundant Token Pruning, EMNLP 2025 (Findings) \n\n[2] Cai et al., R-KV: Redundancy-aware KV Cache Compression for Reasoning Models, NeurIPS 2025", "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)