FROSTICLR 2026 Read paper
Research paperICLR 2026

FROST: Filtering Reasoning Outliers with Attention for Efficient Reasoning

Shorter reasoning by suppressing low-attention steps.

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Authors

Haozheng Luo1,2,*Zhuolin Jiang2Md Zahid Hasan2,3,*Yan Chen1Soumalya Sarkar2

Work carried out at the RTX Technology Research Center during a Summer 2025 internship.

Affiliations

  • 1Northwestern University
  • 2RTX Technology Research Center
  • 3Iowa State University
01

Motivation

Large reasoning models often keep reasoning after the useful work is done. Many of those extra steps receive little attention when the model produces its answer.

Much of the extra length comes from repeated self-verification. Existing efficient-reasoning methods either prune at the token level, which can cut into a step that is naturally a whole sentence, or refine reasoning paths iteratively at the sentence level, which adds training cost and latency.

The paper looks at how much attention each sentence of a reasoning trace sends to the end of thinking. It calls sentences with low attention and a negligible contribution to the final answer reasoning outliers, and treats them like the attention outliers studied in earlier work.

Figure 1: the same complex-number question answered by the original GPT-oss-20B model and by FROST. The original response is long and contains repeated restarts and checks; the FROST response is much shorter and reaches the answer 540.
Figure 1. The same question answered by the original GPT-oss-20B model (left) and after FROST (right). Example from the paper. Source ↗

A few sentences carry most of the attention

For a sample GSM8K question, the paper splits a Phi-4-Reasoning trace into sentences and sums, for each sentence, the attention weight it contributes to the </think> token. In deeper layers, a few sentences stand out while most contribute almost nothing. These are often checks or repetitions of earlier steps.

Figure 3: bar charts of the summed attention from each of 24 sentences to the end-of-thinking token, for three layers of Phi-4-Reasoning. In the shallowest layer contributions are spread out; in the two deeper layers the final sentence dominates and most other sentences are close to zero.
Figure 3. Summed attention from each sentence of the trace to </think> in three layers of Phi-4-Reasoning. Original plots from the paper. Read the observation ↗
02

Methodology

View figure

Swap the attention normalizer for one that can push small weights to zero, then briefly fine-tune.

FROST replaces \(\operatorname{Softmax}\) with \(\operatorname{Softmax}_1\) in the attention layers of a pretrained reasoning model. It then runs supervised fine-tuning on math problems with worked solutions, so the model adapts to the new normalizer. LoRA keeps the update small.

Earlier uses of \(\operatorname{Softmax}_1\) for outlier removal needed training from scratch (OutEffHop) or multi-step continual learning (GERM). FROST starts from an existing checkpoint and needs only a short fine-tuning run.

FROST workflow. A large reasoning model's attention layer is changed from Softmax to Softmax1, which suppresses low-attention sentences. The model is then fine-tuned with SFT to produce an efficient reasoning model.
Figure 5. FROST replaces the vanilla \(\operatorname{Softmax}\) with an outlier-removal layer based on \(\operatorname{Softmax}_1\), then applies SFT to adapt the model. Workflow figure from the paper.
  1. 01 /

    Measure each step's contribution

    Split the trace into the question, the reasoning steps, and the answer. For each part, sum the attention its tokens send to the </think> token:

    \[ W_{\text{trace}} = \sum_{t_i \in T_{\text{trace}}} a_{iA}. \]
  2. 02 /

    Add a null option to attention

    \(\operatorname{Softmax}_1\) adds one to the denominator, so attention can put little weight on every candidate. Low weights move toward zero while large weights remain.

    \[ \operatorname{Softmax}_1(x_i) = \frac{\exp(x_i)}{\sum_{j}\exp(x_j) + 1}. \]
  3. 03 /

    Contract the tail at the sentence level

    With sentence scores \(s\) pooled from token scores and sentence attention \(\alpha=\sigma_1(s)\), the paper shows that the dominance ratio contracts by a factor \(\kappa\in(0,1)\) for heavy-tailed scores.

    \[ \frac{\|\alpha\|_\infty}{\operatorname{median}(\alpha)} \le \kappa \cdot \frac{\|s\|_\infty}{\operatorname{median}(s)}. \]
Notation

\(a_{iA}\) is the attention weight from token \(t_i\) to the </think> token, and \(T_{\text{trace}}\) is the set of tokens in one part of the trace. The sentence score \(s_i\) applies a monotone pooling operator, such as sum, mean, or max, to the token scores of sentence \(S_i\). \(\sigma_1\) is the \(\operatorname{Softmax}_1\) map over sentences. The contraction result is Theorem 5.1 and relies on Assumption 5.1. See Section 5 of the paper.

What changes after FROST

The paper compares one attention head in the last layer of Phi-4-Reasoning under vanilla attention and under FROST. It reports that attention to critical sentences is kept or strengthened, while the influence of reasoning outliers is reduced.

Figure 4: summed attention from each sentence to the end-of-thinking token for head 15 in layer 40, under vanilla attention (left) and FROST (right). The last sentence has the largest weight in both; the maximum rises from about 0.024 to about 0.035 under FROST.
Figure 4. Head 15, layer 40 of Phi-4-Reasoning: vanilla attention (left) and FROST (right). Note the different vertical scales. Original plots from the paper. Source ↗
What the analysis explains

Theorem 5.2 bounds how much a sentence with attention at most \(\varepsilon\) can move the output distribution. The bound is \(\varepsilon\) times the product of the layers' operator-norm bounds, so it is \(O(\varepsilon)\) when those norms are roughly constant. Under the stated assumptions, sentences that receive little attention have little effect at inference.

The results assume properties of the \(\operatorname{Softmax}_1\) map, such as order preservation and tail contraction, and bounded values and operator norms. The experiments test the trained models directly. Read Section 5 and its assumptions.

03

Selected results

Mathematical reasoning

Phi-4-Reasoning MATH500 · tokens per response

1721.95344.37

80.00% fewer tokens

Base model → FROST · Pass@1 0.5480 → 0.5980

Phi-4-Reasoning AIME24 · activation tensors

241.7221.54

91.09% lower average kurtosis

Base model → FROST

Comparison with efficient reasoning methods

FROST is compared with a prompt-based method (TALE), an SFT-based method (DRP), and two reinforcement-learning methods (SelfBudgeter and ThinkLess). FROST produces the shortest responses in most settings. It does not always reach the highest accuracy: TALE and DRP lead on some benchmarks, usually with much longer responses. The paper notes that FROST can occasionally prune a low-attention step that matters.

Pass@1 and tokens per response (#Tk) · Table 1
MethodGSM8KMATH500AIME24Minerva
Pass@1 ↑#Tk ↓Pass@1 ↑#Tk ↓Pass@1 ↑#Tk ↓Pass@1 ↑#Tk ↓
Phi-4-Reasoning
Base0.92421017.700.54801721.950.06671017.700.25001898.86
TALE0.95001716.600.58001874.430.29002069.970.26272093.17
DRP0.8340721.000.62002122.000.33336135.000.27011289.50
SelfBudgeter0.91891507.140.53471195.180.13421372.830.23572618.23
ThinkLess0.92791421.900.54141101.210.16081405.400.25751708.70
FROST0.9311154.330.5980344.370.2667899.800.2716401.19
GPT-oss-20B
Base0.87041275.230.54001575.360.13331003.570.25741586.95
TALE0.82832664.410.54543878.870.20001354.670.27003262.47
DRP0.7880902.500.61464137.000.22454983.000.27151885.15
SelfBudgeter0.86101850.000.53402285.000.13201256.000.25501298.00
ThinkLess0.87401785.000.54102206.000.16001205.000.25801220.00
FROST0.8764377.170.5800680.890.16671009.600.2794691.71
Magistral-Small-1.1
Base0.60752664.410.14801389.890.0000537.130.06991288.04
TALE0.71461516.860.3040723.910.0333967.430.1544748.18
DRP0.6500902.500.21001680.330.04501350.770.11201604.22
SelfBudgeter0.69001850.000.23001520.000.05201256.000.13001298.00
ThinkLess0.72001785.000.25001405.000.06001205.000.14501220.00
FROST0.7551137.550.304098.200.0974149.930.1551109.23

Each evaluation runs three times with different seeds; the paper omits variance because it is at most 2%. The paper also reports each method's average change relative to the base model. Averaged over the three backbones, the paper reports a 26.70% accuracy improvement and a 69.68% reduction in token usage relative to the base models.

Full results & protocol

Fewer outliers, higher sentence entropy

The paper measures the maximum infinity norm and average kurtosis of activation tensors across all layers, and uses average sentence entropy as a proxy for how critical the kept steps are. On AIME24 with Phi-4-Reasoning, FROST lowers the infinity norm by 15.97% and kurtosis by 91.09%, and raises sentence entropy by 13.28%, compared with the base model.

Outlier metrics · Table 3 · AIME24 · Phi-4-Reasoning
MethodMax. infinity norm ↓Avg. kurtosis ↓Avg. sentence entropy ↑Pass@1 ↑#Tk ↓
Base35.31241.722.710.06671017.70
Softmax34.53189.362.790.13331909.07
Sparsemax34.06152.182.930.1667948.60
Entmax30.3943.722.920.1667876.63
FROST29.6721.543.070.2667899.80

Softmax, Sparsemax, and Entmax use the same fine-tuning setup as FROST with a different attention normalizer. See Table 3 ↗

The choice of normalizer

With the training setup held fixed on Phi-4-Reasoning, fine-tuning with plain Softmax lowers average accuracy and lengthens responses. Sparsemax and Entmax shorten responses, but they sharpen both low and high attention scores and can cut into steps that matter. \(\operatorname{Softmax}_1\) has the best averages; Entmax is higher on Minerva.

Averages over GSM8K, MATH500, AIME24, and Minerva · Table 2 · Phi-4-Reasoning
NormalizerMean Pass@1 ↑Mean #Tk ↓
Base model0.44721414.05
Softmax0.42301595.99
Sparsemax0.4406535.26
Entmax0.4751471.71
\(\operatorname{Softmax}_1\) (FROST)0.5169449.92

Averages as reported in Table 2; the per-benchmark results are in the paper. See Table 2 and Section 6.1 ↗

04

Evaluation context

The models are fine-tuned on OpenR1 and evaluated on four out-of-domain math benchmarks: GSM8K, MATH500, AIME24, and Minerva. Accuracy is Pass@1, and efficiency is the number of tokens in the reasoning response. Baselines use the hyperparameters from their own papers.

Backbones
Phi-4-Reasoning · GPT-oss-20B · Magistral-Small-1.1
Fine-tuning
LoRA rank 8, alpha 16 · AdamW · learning rate \(10^{-5}\) · up to 5,000 steps · bfloat16
Decoding
Temperature 0.6 · top-p 0.9 · up to 4,096 generated tokens
Compute
2 NVIDIA H100 GPUs · 80GB each
Cost
Training 204 min, versus 353 for DRP and 1,186 for ThinkLess · AIME test time 3 min with GPT-oss-20B

On three additional tasks outside mathematics, LeetCode, LiveCodeBench, and UGPhysics, FROST fine-tuned on math data raises the mean Pass@1 of Phi-4-Reasoning from 0.3214 to 0.3713 and reduces mean tokens from 2837.64 to 1312.13. The paper notes two limitations: training targets mathematical reasoning only, and the method uses supervised fine-tuning without RL-based training such as GRPO.

Training details and additional results ↗
05

Citation

BibTeX
Download 
@inproceedings{luo2026frost,
  title     = {{FROST}: Filtering Reasoning Outliers with Attention for Efficient Reasoning},
  author    = {Haozheng Luo and Zhuolin Jiang and Md Zahid Hasan and
               Yan Chen and Soumalya Sarkar},
  booktitle = {The Fourteenth International Conference on Learning Representations},
  year      = {2026},
  url       = {https://openreview.net/forum?id=a9dngZLqGS}
}