# 🧮 Fathom-R1-14B: $499 Training Recipe for Unlocking Math Reasoning at o4-mini level using R1-distilled-14B model under 16K context

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--- ## Overview Reasoning models often require high post-training budgets and extremely long reasoning chains(think 32k/64k) for maximising performance. Can we improve these models even if both these parameters are capped? To this end, we first introduce: **Fathom-R1-14B**, a 14-billion-parameter reasoning language model derived from Deepseek-R1-Distilled-Qwen-14B, post-trained at an affordable cost of only $499, and achieving SOTA mathematical reasoning performance within a 16K context window. On the latest olympiad level exams: AIME-25 and HMMT-25, our model not only **surpasses o3-mini-low, o1-mini and LightR1-14B(16k)** at pass@1 scores (averaged over 64 runs) but also delivers **performance rivaling closed-source o4-mini (low)** w.r.t cons@64 — all while staying within a **16K context window**. It achieves 52.71% Pass@1 accuracy on AIME2025 and 35.26% Pass@1 accuracy on HMMT25 (+7.2% and +5.2% improvement over the base model respectively). When provided with additional test-time compute in the form of cons@64, it achieves an impressive 76.7% accuracy on AIME2025 and 56.7% accuracy on HMMT25 (+13.4% and +6.7% improvement over the base model respectively). We perform supervised fine-tuning (SFT) on carefully curated datasets using a specific training approach, followed by model merging, achieving this performance at a total cost of just $499! We also introduce **Fathom-R1-14B-RS**, another model achieving performance comparable to our first, at a total post-training cost of just $967. It leverages post-training techniques—including reinforcement learning and supervised fine-tuning—in a multi-stage, cost-effective manner, followed by model merging. We are **open-sourcing our models, post-training recipes and datasets** which we believe will help the community to progress further in the reasoning domain. --- ## 🧪 Motivation Thinking longer during inference time has shown to unlock superior reasoning abilities and expert level performance on challenging queries and tasks. Since the open-source release of DeepSeek R1 series models, multiple open-source efforts [[s1](https://github.com/simplescaling/s1), [LIMO](https://github.com/GAIR-NLP/LIMO), [Light-R1](https://github.com/Qihoo360/Light-R1)] have focused on reproducing the results (easpecially at <=32B scale) either via distillation or RL based fine-tuning on top of non-reasoning models. Though in most cases, these efforts at best, could come close to the performance R1 series models but are unable to surpass them. In parallel, certain recent methods [[DeepScaleR](https://github.com/agentica-project/rllm), [DeepCoder](https://www.together.ai/blog/deepcoder), [Light-R1](https://github.com/Qihoo360/Light-R1)] started with the existing reasoning models and have managed to extend the performance of these models. However, the training runs for these methods are often costly and they rely on longer sequence lengths for higher accuracy. Given the latest findings [[Proof or Bluff ?](https://arxiv.org/abs/2503.21934), [Reasoning models don't always say what they think](https://assets.anthropic.com/m/71876fabef0f0ed4/original/reasoning_models_paper.pdf)] that raises the question on the correctness of the intermediate steps of the long COT in reasoning models, its important from interpretability, reliability and safety pov to ensure the reasoning chains are not inefficiently long. Hence, in this study, we work towards unlocking performance improvement of the reasoning models without training at very high (24k/32k) sequence length and restricting it to 16k context. We believe, while extremely long reasoning chains are still necessary for really challenging tasks, its also important to maximize the performance at lower context first before we proceed towards extending reasoning chains. ## Training Dataset We begin by curating a high-quality mathematical corpus from the following open-source datasets: - **Open-R1** - default subset - **Numina – Olympiads & AOPS_forum** (word problems, float type answers) - After rigorous deduplication and decontamination, we consolidated approximately **~100K unique problems**, forming the initial corpus for all subsequent trainings. ## 🏗️ Post-Training Strategies ### Training Recipe for Fathom-R1-14B-v0.6 SFT on difficult questions and their reasoning chains has been shown to be effective for improving reasoning ability. For this checkpoint, we build on top of this. This training stage focuses on improving the model’s performance on **mathematical problems covering a spectrum of hard diffuculty level** through a iterative curriculum learning strategy at max 16k sequence length. Curriculum learning (CL) is a well-established technique for training LLMs, where the model is progressively exposed to more difficult tasks. The idea is to gradually scaffold more complex reasoning, thereby enhancing generalization and reducing overfitting. However, in our case we perform this in an iterative manner, which essentially means we do multiple iterations of CL. For the dataset preparation, we begin by annotating each question’s difficulty using **OpenAI's o3mini** model. We retain only those questions rated above average (in relative sense) and further filter them to include only those having **solve rates between certain range** (0.2 < pass_rate < 0.7). This yields the **Iterative Curriculum Learning dataset** comprising of 5K examples. Total H100 GPU Hours: 48 Cost: $136 ### Training Recipe for Fathom-R1-14B-v0.4-RS The core strategy used behind creating this checkpoint is a two-stage pipeline: First, leverage GRPO to improve reasoning of Deepseek-R1-Distilled-Qwen-14B at a lower sequence length, 6k, on a carefully curated dataset to ensure rapid improvement with minimal training steps. Second, we perform SFT, at max 16k tokens sequence length, on a carefully curated dataset of questions ( hard to very hard difficulty spectrum) and the corresponding shortest possible reasoning solution for each question. - **First Stage (Leveraging RL for effecient test-time thinking):** We start with curating a seed dataset which ensures the policy receives minumium reward while still having room for growth. The dataset comparises of questions having solve rates (at lower sequence length) between a certain range. This forms our **RL Compression dataset** comprising of 7.7K questions. Staring from DeepSeek-R1-Distill-Qwen-14B as the base model, we train the model using the GRPO algorithm, with a 6k token sequence length limit. We see a consistent increase in performance as the model learns to generate concise responses from the decreasing clip ratio, response length and increasing reward. The obtained model has learnt to generate responses below 6k tokens and outperforms the base model at lower token limits.

- **Second Stage (Leveraging SFT to improve reasoning efficiently at higher sequence length):** We build upon the RL checkpoint and perform SFT under a **16K context window** to encourage more detailed reasoning that would be required for solving more complex problems. For this stage, we strategically curate a dataset consisting of hard problems — specifically, questions with lower solve rates (0 < pass_rate <=0.4). Then, we obtain the shortest possible reasoning chains for these questions forming the **SFT Shortest Chains dataset** comprising of 9.5K examples. Through supervised fine-tuning on this dataset, the model is able to stablize its reasoning at sequence length upto 16K. The resulting model is named **Fathom-R1-14B-v0.4**, optimized for concise yet accurate mathematical reasoning. Total H100 GPU Hours: 293 Cost: $831 ### Training Recipe for Fathom-R1-14B-v0.4 Given the performance improvement we noticed during the second fine-tuning stage of developing Fathom-R1-14B-v0.4-RS and in attempt to further reduce the cost, we experiment by eliminating RL and directly performing second stage SFT on Deepseek-R1-Distilled-Qwen-14B base model. Total H100 GPU Hours: 128 Cost: $363 ## Model Merging Given v0.6 and v0.4 models have been developed by following different training methodologies, we perform linear merging to combine the strengths to obtain final 2 checkpoints. - **Fathom-R1-14B**: Obtained via merging Fathom-R1-14B-V0.6 (Iterative Curriculum SFT) and Fathom-R1-14B-V0.4 (SFT-Shortest-Chains) - **Fathom-R1-14B-RS**: Obtained via merging Fathom-R1-14B-V0.6 (Iterative Curriculum SFT) and Fathom-R1-14B-V0.4 (RL-compression + SFT-Shortest-Chains) ## 💰 Post-Training Cost We developed **Fathom-R1-14B** models using a focused, resource-efficient strategy that balances performance with compute budget. Below is the GPU time utilized and the cost incurred | Model Weights | GPU Hours (H100) | Cost(USD) | |----------------------------|------------------|------| | Fathom-R1-14B-V0.4-RS | 293 | 831 | | Fathom-R1-14B-V0.4 | 128 | 363 | | Fathom-R1-14B-V0.6 | 48 | 136 | | Fathom-R1-14B-RS | 341 | 967 | | **Fathom-R1-14B** | **176** | **499** | So, the final Fathom-R1-14B took just 499$ to be trained overall! This low training cost highlights the efficiency of our method — enabling high-level mathematical reasoning comparable to **o4-mini** in **$499** , all within a **16k sequence length budget**. --- ## 📊 Evaluation We evaluate Fathom‑R1-14B using the same metrics and sampling configuration introduced in the DeepSeek‑R1 paper, namely **pass@1** and **cons@64**. However, our evaluation is conducted under a reduced output budget of 16,384 tokens, compared to DeepSeek‑R1’s 32,768 tokens, to better reflect practical deployment constraints. - **pass@1**: Pass@1 is computed as the average correctness over k sampled solution chains per problem (in our experiments we keep k=64). - **cons@64**: Assesses consistency by sampling 64 reasoning chains per question and computing the majority vote accuracy. **Evaluation Configuration**: - Temperature: 0.6 - top_p: 0.95 - Number of sampled chains: 64 - Context: 16,384 tokens This setup allows us to benchmark Fathom-R1-14B’s reasoning performance and stability under realistic memory and inference budgets, while maintaining compatibility with the DeepSeek‑R1 evaluation protocol. We utilize the evaluation framework provided by the [LIMO](https://github.com/GAIR-NLP/LIMO) repository to run inference and compute metrics. For detailed instructions and implementation details, please refer to [`eval/README.md`](https://github.com/FractalAIResearchLabs/Fathom-R1/blob/main/eval/readme.md). --- ## Results We evaluate and compare **Fathom‑R1-14B** with several baseline models across 3 challenging benchmarks: **AIME25**, **HMMT25**, and **GPQA**. For each, we report `pass@1` and `cons@64`, following the same evaluation configuration. | Model | AIME25 | | HMMT25 | | |------------------|----------------|---------------|----------------|---------------| | | pass@1 | cons@64 | pass@1 | cons@64 | | **Closed-Source Models** | | | | | | o1‑mini | 50.71 | 63.33 | 35.15 | 46.67 | | o3‑mini‑low | 42.60 | 53.33 | 26.61 | 33.33 | | o3‑mini‑medium | 72.24 | 83.33 | 49.21 | 60.00 | | o4-mini-low | 60.20 | 76.67 | 39.11 | 53.33 | | o1‑preview | 33.33 | 36.67 | 17.78 | 20.00 | | gpt‑4.5‑preview | 34.44 | 40.00 | 16.67 | 20.00 | | **Open-Source Models** | | | | | | DeepSeek-R1-Distill-Qwen-14B | 45.50 | 63.33 | 30.00 | 50.00 | | DeepSeek-R1-Distill-Qwen-32B | 49.64 | 73.33 | 33.02 | 53.33 | | DeepSeekR1‑670B | 61.25 | 83.33 | 42.19 | 56.67 | | LightR1‑14B | 51.15 | 76.67 | 33.75 | 50.00 | | Fathom‑R1-14B-V0.4-RS | 50.94 | 73.33 | 33.70 | 40.00 | | Fathom‑R1-14B-V0.4 | 50.94 | 70.00 | 34.53 | 56.67 | | Fathom‑R1-14B-V0.6 | 50.63 | 76.67 | 32.19 | 50.00 | | Fathom‑R1-14B-RS | 52.03 | 76.67 | 35.00 | 53.33 | | **Fathom‑R1-14B** | **52.71** | **76.67** | **35.26** | **56.67** | **Fathom‑R1-14B** demonstrates highly competitive performance across all datasets, improving over the original R1-distilled models while closely matching or surpassing other strong baselines in several settings. On both AIME 25 and HMMT 25, our model shows the highest pass@1 as well as cons@64 scores among all the open-source models (including the bigger R1-Distilled-32B model), with R1-670B being the only exception. In fact, we observe that Fathom-R1-14B is superior to the first two generations of OpenAI's mini-reasoning models, including **o1-mini** and **o3-mini-low-** and it's performance closely matches that of newly released **o4-mini-low** (self-consistency decoding). --- ## 🌍 Generalization Beyond Math: GPQA-Diamond Notably, we also observe out-of-domain improvement in **GPQA-Diamond**, even though there wasn't a single instance of non-math questions in our training data. This indicates that our training methodology mentioned above and training on math wuestions facilitates generalization across diverse domains, a finding similar to what LightR1-14B & LIMO had observed. #### ✅ GPQA Benchmark Comparison (16k) | **Model** | **pass@1** | **cons@64** | |-------------------|------------|-------------| | DeepSeek-R1-Distill-Qwen-14B | 54.19 | 64.14 | | LightR1‑14B | 56.94 | 65.15 | | Fathom‑R1-14B-RS | 59.13 | 66.16 | | **Fathom‑R1-14B** | **59.46** | **66.16** | --- ## ✂️ Ablation Study on Token Efficiency To assess reasoning token efficiency, we compare the **average response token count**, under 16k context length, across AIME25, and HMMT25. On AIME25, Fathom‑R1-14B-RS uses 10% fewer response tokens than LightR1-14B despite having higher pass@1. HMMT25 questions are relatively tougher compared to AIME'25 and tougher questions usually require more thinking tokens. On HMMT25, Fathom‑R1-14B-RS uses 4.5% fewer response tokens than LightR1-14B despite having higher pass@1. #### Average Response Length (Tokens) | Model | AIME25 | HMMT25 | |------------------|--------|--------| | LightR1-14B | 11330 | 12680 | | DeepSeek-R1-Distill-Qwen-14B | 10878 | 12263 | | Fathom‑R1-14B-V0.4 | 10570 | 11950 | | Fathom‑R1-14B | 10956 | 12125 | | **Fathom‑R1-14B-RS** | **10083** | **12100** | --- ## Data Decontimination Both benchmarks used (AIME 25 and HMMT 25) were released a few weeks after the release of the base model, ensuring no contamination occurred during the model's pre-training. The dataset corpora (Numina-Math 1.5 & OpenR1-Math) were released around the same time as these exams, with a cutoff date no later than 2024. Additionally, we conduct checks to verify there is no contamination in the training data. --- ## Release Assets - Training Recipe Blog: [🤗 $499 training recipe for creating Fathom-R1-14B](https://huggingface.co/FractalAIResearch/Fathom-R1-14B) - Final Merged Models: [🤗 Fathom-R1-14B](https://huggingface.co/FractalAIResearch/Fathom-R1-14B), [🤗 Fathom-R1-14B-RS](https://huggingface.co/FractalAIResearch/Fathom-R1-14B-RS) - Intermediate Models: [🤗 Fathom-R1-14B-V0.6](https://huggingface.co/FractalAIResearch/Fathom-R1-14B-V0.6), [🤗 Fathom-R1-14B-V0.4](https://huggingface.co/FractalAIResearch/Fathom-R1-14B-V0.4), [🤗 Fathom-R1-14B-V0.4-RS](https://huggingface.co/FractalAIResearch/Fathom-R1-14B-V0.4-RS) - Fathom-R1-14B Datasets: [🤗 V0.6-Iterative-Curriculum-Learning](https://huggingface.co/datasets/FractalAIResearch/Fathom-V0.6-Iterative-Curriculum-Learning), [🤗 V0.4-SFT-Shortest-Chains](https://huggingface.co/datasets/FractalAIResearch/Fathom-V0.4-SFT-Shortest-Chains), [🤗 V0.4-RL-Compression](https://huggingface.co/datasets/FractalAIResearch/Fathom-V0.4-RL-Compression) --- ## 📜 License This repository and all the release assets are available under the MIT License, underscoring our dedication to open and inclusive AI innovation. By freely sharing our work, we aim to democratize AI technology, empowering researchers, developers, and enthusiasts everywhere to use, adapt, and expand upon it without limitation. This open and permissive approach promotes global collaboration, accelerates innovation, and enriches the AI community as a whole. ## Acknowledgments We would like to acknowledge the following works for enabling our project: - [Deepseek-R1-Distill-Qwen-14B](https://huggingface.co/deepseek-ai/DeepSeek-R1-Distill-Qwen-14B) - [NuminaMath-1.5](https://huggingface.co/datasets/AI-MO/NuminaMath-1.5) - [OpenR1-Math](https://huggingface.co/datasets/open-r1/OpenR1-Math-220k) - [360-LLAMA-Factory](https://github.com/Qihoo360/360-LLaMA-Factory) - [verl](https://github.com/volcengine/verl) - [LIMO](https://github.com/GAIR-NLP/LIMO) - [FuseAI](https://github.com/fanqiwan/FuseAI) --- ## 📖 Citation ```bibtex @misc{fathom14b2025, title={Fathom-R1: $499 Training Recipe for Unlocking Math Reasoning at o4-mini level with just 14B parameters under 16K context}, author={Kunal Singh and Pradeep Moturi and Ankan Biswas and Siva Gollapalli and Sayandeep Bhowmick}, howpublished={\url{https://huggingface.co/FractalAIResearch/Fathom-R1-14B}}, note={Hugging Face}, year={2025} } ``` ## About Project Ramanujan We initiated Project Ramanujan approximately one year ago, aiming to unlock intelligence and enhance AI agents by pushing the boundaries of advanced reasoning. Our key accomplishments include: - ICLR'25 & NeurIPS'24-MATH-AI: [SBSC: Step-By-Step Coding for Improving Mathematical Olympiad Performance](https://arxiv.org/abs/2502.16666) - Winners of HackerCupAI@NeurIPS'24 & ICLR'25-VerifAI: [Stress Testing Based Self-Consistency For Olympiad Programming](https://openreview.net/forum?id=7SlCSjhBsq) - CVPR'25-MULA: [TRISHUL: Towards Region Identification and Screen Hierarchy Understanding for Large VLM based GUI Agents ](https://arxiv.org/abs/2502.08226)) - Silver Medal in AIMO'24