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FROST: Filtering Reasoning Outliers with Attention for Efficient Reasoning

Haozheng Luo, Zhuolin Jiang, Md Zahid Hasan, Yan Chen, Soumalya Sarkar

arXiv:2601.19001v2cs.CLcs.AIcs.LG

TL;DR

Large reasoning models often generate irrelevant steps that increase token use and reduce reasoning efficiency. FROST uses attention-based outlier removal to prune uncritical paths, improving accuracy by 26.70% while reducing token usage by 69.68%.

  • Problem

    Large reasoning models generate redundant or irrelevant reasoning traces, creating unnecessary tokens and reducing reasoning efficiency.

  • Method

    FROST uses attention weights and Softmax1 during lightweight supervised fine-tuning to identify and suppress sentence-level reasoning outliers.

  • Results

    26.70% higher accuracy and 69.68% lower token usage are achieved across multiple benchmarks and base models.

  • Takeaways & Limitations

    FROST mitigates reasoning outliers while producing more efficient and accurate reasoning traces for mathematical problem solving.

  • Takeaways & Limitations

    FROST is currently restricted to mathematical reasoning tasks and relies solely on supervised fine-tuning.

Abstract

from arXiv · show

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 by 15.97% and the average kurtosis by 91.09% compared to the base model. Code is available at https://github.com/robinzixuan/FROST

1 INTRODUCTION

FROST identifies reasoning outliers as uncritical steps and uses attention-aware pruning, Softmax1, and supervised fine-tuning to produce shorter, reliable reasoning without sacrificing accuracy. Across benchmarks, it improves accuracy and efficiency while reducing attention outlier metrics and computation costs.

  • Motivation and approach: FROST introduces reasoning outliers and prunes uncritical reasoning steps characterized by low attention.It directs models toward critical reasoning steps, producing shorter and more reliable trajectories.
  • Method: FROST replaces Softmax with Softmax1 to suppress low attention weights while preserving high ones.Theoretical analysis shows this mechanism enhances the reasoning capacity of large reasoning models.
  • Method: FROST combines Softmax1 with supervised finetuning to enable efficient reasoning without sacrificing accuracy.The training strategy integrates Softmax1 with supervised fine-tuning on reasoning tasks.
  • Empirical results: 26.70% accuracy gain and 69.68% reasoning path length reduction are achieved across multiple benchmarks compared with base models.The empirical evaluation demonstrates FROST’s effectiveness across multiple benchmarks.
  • Empirical results: 15.97% reduction in maximum infinity norm and 91.09% reduction in average kurtosis are reported for attention outlier values.FROST also cuts inference time by at least 28.6% and reduces training time by 42.2% relative to other SFT baselines.

2 RELATED WORK

The paper situates FROST within research on reasoning models and efficient reasoning. Prior work improves reasoning through inference-time scaling, learning-to-reason, and methods that constrain or optimize reasoning paths.

  • Reasoning Models: Reasoning models such as DeepSeek-R1, OpenAI o1, and Gemini 2.0 Pro demonstrate strong capabilities, especially on mathematical and logical tasks.
  • Reasoning Models: Inference-time scaling methods include Chain-of-Thought, ReAct, Self-Ask, and agentic reasoning for enhancing model reasoning capabilities.
  • Reasoning Models: Learning-to-reason methods build reasoning ability through alignment, including reinforcement learning, supervised fine-tuning, and energy-based model reasoners.
  • Efficient Reasoning Methods: Efficient reasoning approaches address overthinking through prompt-based methods, supervised fine-tuning, and reinforcement learning.
  • Efficient Reasoning Methods: TALE limits each instance’s token budget to shorten reasoning paths while maintaining task accuracy.

3 REASONING OUTLIER

The section analyzes how attention is distributed across reasoning traces and shows that only a few traces strongly influence final-answer generation. It characterizes low-influence traces as reasoning outliers and motivates their removal with Softmax1 to improve reasoning efficiency while preserving or enhancing capacity.

  • Attention Distribution: In shallow layers, attention is relatively uniform, whereas deeper layers and later heads focus on specific reasoning-step and final-answer tokens.This pattern is illustrated with Phi-4-Reasoning on a sample GSM8K question.
  • Trace Contributions: A few reasoning traces strongly influence the final answer, while most contribute weakly and some contribute almost nothing.The contribution is measured through summed attention weights to the final answer token </think>.
  • Reasoning Outliers: Many negligible-contribution traces correspond to verification, self-checking, or repetition, forcing longer-than-necessary outputs and reducing reasoning efficiency.Extended reasoning may arise because developers encourage longer reasoning to maximize accuracy, while models can generate redundant or irrelevant information.
  • Outlier Removal: FROST uses the Softmax1 function to detect and eliminate reasoning outliers during the reasoning process.The method is motivated by similarities between reasoning outliers and attention outliers, with efficiency proved theoretically in section 5.
  • Theoretical Analysis: The theoretical analysis indicates that removing reasoning outliers with Softmax1 preserves and can even enhance the reasoning capacity of language reasoning models.The comparison uses Phi-4-Reasoning’s last-layer attention distribution in head 15 under vanilla attention and FROST attention.

4 FROST

FROST improves reasoning efficiency through supervised fine-tuning with reasoning-outlier removal, replacing vanilla Softmax with Softmax1 to focus on critical traces and suppress low-attention sentences.

  • Method: FROST applies supervised fine-tuning with reasoning outlier removal to enhance language reasoning model efficiency.Training uses math problems with detailed reasoning steps and answers.
  • Method: Softmax1 replaces vanilla Softmax during training, focusing the model on critical reasoning traces while suppressing outliers.The workflow then adapts model parameters to the new activation function through SFT.
  • Method: FROST removes outliers with only a few fine-tuning steps, unlike prior Softmax1 methods requiring training from scratch or multi-step continual learning.The comparison concerns Hu et al. (2024) and Luo et al. (2025b).
  • Method: The workflow significantly reduces the number of low-attention sentences.An outlier-removal layer based on Softmax1 precedes supervised fine-tuning.

5 THEORETICAL ANALYSIS

The theoretical analysis shows that Softmax1 can remove reasoning outliers at the sentence level while suppressing low-attention sentences during inference. Under stated structural assumptions, sentence-level outlier dominance contracts, and low-attention contributions become O(ε).

  • Sentence-level attention: Monotone pooling preserves sentence-score dominance, which Softmax1 transfers to sentence-level attention probabilities.If si ≥ sj, order preservation gives αi ≥ αj.
  • Softmax1 assumptions: Softmax1 is assumed to preserve order, remain shift-invariant, contract tails by κ ∈ (0, 1), and be smooth and positive.These properties comprise Assumption 5.1 for the mapping σ1 : Rm →∆m−1.
  • Outlier suppression: ∥α∥∞ median(α) ≤κ · ∥s∥∞ median(s) for heavy-tailed sentence scores, contracting relative outlier dominance at the sentence level.Here α = σ1(s), and κ ∈ (0, 1).
  • Deployment-time suppression: O(ε) bounds the effect of low-attention sentences when αi ≤ε, so they are effectively skipped at inference.The result assumes approximately constant operator-norm factors Bo, Bv, and B in practice.

6 EXPERIMENTAL STUDIES

FROST is evaluated across mathematical reasoning models and out-of-domain benchmarks using accuracy and token usage, with results showing substantial efficiency and accuracy gains. Additional studies examine activation functions, attention-outlier removal, and generalization beyond mathematics.

  • Experimental Setup: Experiments use Phi-4-Reasoning, Magistral-Small-1.1, and GPT-oss backbones, with FROST-finetuned checkpoints trained on mathematical reasoning data.Each evaluation is conducted three times with different random seeds, reporting averages and standard deviations.
  • Experimental Setup: Four out-of-domain mathematical benchmarks—GSM8K, MATH500, AIME24, and Minerva—measure reasoning efficiency and generalization.OpenR1 is used as the training corpus, and all datasets target mathematical question answering.
  • Experimental Setup: Pass@1 measures accuracy, while reasoning-response token count measures token efficiency; comparisons include TALE, DRP, SelfBudgeter, and ThinkLess.The baselines represent prompt-based, supervised fine-tuning, and reinforcement-learning approaches to efficient reasoning.
  • Main Results: 26.70% average accuracy improvement and 69.68% token-usage reduction are achieved across GPT-OSS-20B, Magistral-Small-1.1, and Phi-4-reasoning.FROST delivers the best overall performance among the evaluated state-of-the-art efficient reasoning methods, although TALE leads on some tasks.
  • Additional Studies: FROST is evaluated with vanilla Softmax, Sparsemax, and Entmax15, while further studies assess attention-outlier removal and generalization on coding and physical-reasoning tasks.The additional evaluations use four mathematical datasets for activation functions and LeetCode, LiveCodeBench, and UGPhysical for out-of-domain reasoning.

7 DISCUSSION AND CONCLUSION

FROST is an attention-aware method that removes reasoning outliers to enable efficient and accurate reasoning traces. Its current scope is limited to mathematical reasoning and supervised fine-tuning, motivating extensions to coding and GRPO-based approaches.

  • FROST introduces reasoning outliers and an outlier-removal mechanism for efficient reasoning.The approach targets high latency in reasoning performance.
  • FROST enables lightweight supervised finetuning to generate efficient and accurate reasoning traces in mathematical problem solving.
  • FROST is restricted to mathematical reasoning tasks and relies solely on supervised fine-tuning without GRPO.
  • Future work will extend FROST to coding and develop a GRPO-based approach.

ETHICAL STATEMENT

The paper acknowledges potential bias amplification and unfair outcomes from removing attention outliers, while reporting reproducibility practices through open-source release and stable multi-seed experiments.

  • Limitations and ethics: FROST may inadvertently amplify training-data biases, potentially producing unfair outcomes for underrepresented groups.The authors frame this risk in accordance with the ICLR Code of Ethics.
  • Limitations and ethics: The authors note that prior work links supervised fine-tuning with shallow alignment and altered red-teaming protection, but leave this issue outside the stated scope.
  • Reproducibility: 3 random seeds produce stable results with standard deviations below 2%.The experiments use a unified AdamW setup with learning rate 1 × 10−5, training batch size 8, and deployment batch size 256.
  • Reproducibility: An anonymous open-source repository releases the full FROST implementation and selected baselines, with full open-sourcing planned upon acceptance.

Supplementary Material … C.1 LEMMA 5.1

The supplementary material presents a monotonicity-based proof of Lemma 5.1, alongside an impact statement on efficiency and potential bias and related-work motivation concerning computationally expensive alignment methods.

  • A IMPACT STATEMENT: The impact statement argues that the methodology could improve efficiency and enable more critical reasoning traces.It frames these effects as an opportunity to strengthen the core of large reasoning models.
  • A IMPACT STATEMENT: The impact statement warns that the methodology may amplify training-data biases and produce unfair or discriminatory outcomes for underrepresented groups.This is presented as a potential limitation of the approach.
  • B ADDITIONAL RELATED WORK: Efficient alignment is motivated by strong foundation-model multitask capabilities and the computational expense of traditional RLHF and DPO approaches.The passage identifies an urgent need for parameter-efficient methods, but the supplied text is truncated after that phrase.
  • C.1 LEMMA 5.1: The proof establishes that coordinatewise monotonicity implies s_i = ϕ(u) ≥ ϕ(w) = s_j when each coordinate in u has a corresponding not-smaller coordinate in w.It then applies order preservation (P1) to conclude α_i ≥ α_j.
  • C.1 LEMMA 5.1: Monotonicity is defined so increasing any input coordinate to ϕ cannot decrease the output.The proof uses repeated coordinatewise monotonicity applications.
  • C.1 LEMMA 5.1: Order preservation (P1) converts the score relation s_i ≥ s_j into the coefficient relation α_i ≥ α_j.This is the final inference in the proof of Lemma 5.1.

C.2 THEOREM 5.1 … F ADDITIONAL EXPERIMENTS

The paper proves invariance and perturbation bounds, illustrates critical and uncritical reasoning traces, specifies experimental settings, and reports additional experiments showing FROST surpasses efficient reasoning methods.

  • C.2 THEOREM 5.1: Theorem 5.1 derives sentence-score relation (2) from token-compatibility dominance and shows invariance under global score shifts.Re-centering s ← s − c1 leaves α unchanged, so (2) depends only on relative separations.
  • C.3 THEOREM 5.2: Theorem 5.2 bounds the output perturbation by Bo ε Bv and propagates it through L differentiable layers using Jacobians.The proof applies operator-norm submultiplicativity before propagating the perturbation through the network.
  • C.3 THEOREM 5.2: O(BoBvBLε) is the resulting probability-change rate, simplifying to O(ε) when Bo, Bv, and B are O(1).The simplification follows from replacing the product of layer bounds with BL and using the Softmax1 Lipschitz property.
  • D AN EXAMPLE OF LRM REASONING TRACES: In the Phi-4-Reasoning AIME24 example, traces S1 and S2 are classified as uncritical, although S2 contains partially critical content.The example identifies S2’s walking-time statement as partially critical while judging the overall trace non-critical.
  • D AN EXAMPLE OF LRM REASONING TRACES: Trace S3 identifies two key equations, S4–S19 perform self-verification, and S20–S24 resume critical reasoning and produce the final answer.The transition back to critical reasoning follows the signal “We’ll produce final answer in a box.”
  • E.1 COMPUTATIONAL RESOURCES: The experiments use two NVIDIA H100 80GB GPUs, a 12-core Intel Xeon CPU, PyTorch, Hugging Face Transformers, and temperature 0.6.The default official system prompt is used for the LLMs.
  • E.2 HYPERPARAMETERS: Batch sizes are 256 during deployment and 8 during training, with four-step gradient accumulation and weight decay 0.01.Fine-tuning uses AdamW, learning rate 1e−5, LoRA rank 8, LoRA alpha 16, and maximum training steps 5,000.
  • F ADDITIONAL EXPERIMENTS: FROST’s additional experiments demonstrate that it surpasses current state-of-the-art efficient reasoning methods.The passage states this as the purpose and conclusion of the additional experiments.

F.1 TRAINING AND TEST TIME COMPARISON … I EXTENDED ATTENTION HEATMAPS ACROSS ADDITIONAL LAYERS AND HEADS

Across five consecutive sections, FROST is reported to reduce computational overhead and low-attention reasoning content while Softmax1 stabilizes attention dynamics across training and layers. Extended heatmaps further examine attention evolution across selected layers and heads.

  • F.1 TRAINING AND TEST TIME COMPARISON: Training and test-time costs are measured across baselines and FROST, with test-time evaluation on AIME using GPT-OSS-20B.Training time uses the respective datasets specified in each baseline’s original paper.
  • F.1 TRAINING AND TEST TIME COMPARISON: FROST achieves the fastest training time among all methods while minimizing computation cost and inference time during deployment.The experiments compare methods under the same computational resources, using AIME with GPT-OSS-20B for test-time evaluation.
  • F.2 ATTENTION DISTRIBUTIONS OF ACTIVATION FUNCTIONS: FROST removes many low-attention sentences while retaining significant ones, unlike the vanilla model’s numerous low-attention sentences and Sparsemax or Entmax15’s aggressive retention of only one to two sentences.The comparison is visualized on a GSM8K sample with GPT-OSS-20B.
  • G INFLUENCE THE ATTENTION DYNAMICS OF Softmax1 DURING TRAINING: During supervised fine-tuning, Softmax1 suppresses low-attention activations, stabilizing gradients and reducing update variance through residual connections.The passage reports faster convergence of LoRA adapters as the low-rank parameter subspace aligns more efficiently with critical attention directions.
  • H INFLUENCE OF Softmax1 ACROSS LAYERS: Across transformer layers, Softmax1 consistently suppresses attention outliers and produces smoother, more stable activations than vanilla Softmax.In lower layers, it contracts heavy tails, mitigates rare extreme peaks, and yields higher-entropy, reduced-kurtosis distributions.
  • H INFLUENCE OF Softmax1 ACROSS LAYERS: In lower layers, Softmax1 contracts heavy tails and mitigates rare extreme peaks, enhancing local feature mixing with higher-entropy and reduced-kurtosis distributions.The analysis visualizes head 15 for vanilla Softmax and Softmax1 across transformer layers.
  • I EXTENDED ATTENTION HEATMAPS ACROSS ADDITIONAL LAYERS AND HEADS: Extended attention heatmaps analyze Layers 0, 5, 15, 25, 30, 35, and 39 together with Heads 0, 5, 10, 15, 20, 25, 30, 35, and 39.The observations are illustrated in figure 9 to provide a broader view of attention evolution across the network.

J HUMAN EXPERT EVALUATION

Three computer science students evaluated reasoning traces from the original and FROST-trained models, finding that FROST accurately removes most non-critical traces while occasionally deleting harmful ones.

  • Human expert evaluation: Three computer science students specializing in reasoning models annotated traces from the original and FROST-trained models.They compared traces pruned by FROST and assessed criticality by relevance and contribution to the final answer.
  • Human expert evaluation: 92% accuracy was achieved in correctly removing non-critical reasoning traces.This figure is averaged across all evaluators.
  • Human expert evaluation: 8% of reasoning traces were incorrectly removed, significantly degrading final-answer accuracy.
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