AIMO-2 Winning Solution: Building State-of-the-Art Mathematical Reasoning Models with OpenMathReasoning dataset

This paper presents our winning submission to the AI Mathematical Olympiad - Progress Prize 2 (AIMO-2) competition. Our recipe for building state-of-the-art mathematical reasoning models relies on three key pillars. First, we create a large-scale dataset comprising 540K unique high-quality math problems, including olympiad-level problems, and their 3.2M long-reasoning solutions. Second, we develop a novel method to integrate code execution with long reasoning models through iterative training, generation, and quality filtering, resulting in 1.7M high-quality Tool-Integrated Reasoning solutions. Third, we create a pipeline to train models to select the most promising solution from many candidates. We show that such generative solution selection (GenSelect) can significantly improve upon majority voting baseline. Combining these ideas, we train a series of models that achieve state-of-the-art results on mathematical reasoning benchmarks. To facilitate further research, we release our code, models, and the complete OpenMathReasoning dataset under a commercially permissive license.

Nemotron-H: A Family of Accurate and Efficient Hybrid Mamba-Transformer Models

As inference-time scaling becomes critical for enhanced reasoning capabilities, it is increasingly becoming important to build models that are efficient to infer. We introduce Nemotron-H, a family of 8B and 56B/47B hybrid Mamba-Transformer models designed to reduce inference cost for a given accuracy level. To achieve this goal, we replace the majority of self-attention layers in the common Transformer model architecture with Mamba layers that perform constant computation and require constant memory per generated token. We show that Nemotron-H models offer either better or on-par accuracy compared to other similarly-sized state-of-the-art open-sourced Transformer models (e.g., Qwen-2.5-7B/72B and Llama-3.1-8B/70B), while being up to 3$\times$ faster at inference. To further increase inference speed and reduce the memory required at inference time, we created Nemotron-H-47B-Base from the 56B model using a new compression via pruning and distillation technique called MiniPuzzle. Nemotron-H-47B-Base achieves similar accuracy to the 56B model, but is 20% faster to infer. In addition, we introduce an FP8-based training recipe and show that it can achieve on par results with BF16-based training. This recipe is used to train the 56B model. All Nemotron-H models will be released, with support in Hugging Face, NeMo, and Megatron-LM.

OpenMathInstruct-2: Accelerating AI for Math with Massive Open-Source Instruction Data

Mathematical reasoning continues to be a critical challenge in large language model (LLM) development with significant interest. However, most of the cutting-edge progress in mathematical reasoning with LLMs has become \emph{closed-source} due to lack of access to training data. This lack of data access limits researchers from understanding the impact of different choices for synthesizing and utilizing the data. With the goal of creating a high-quality finetuning (SFT) dataset for math reasoning, we conduct careful ablation experiments on data synthesis using the recently released \texttt{Llama3.1} family of models. Our experiments show that: (a) solution format matters, with excessively verbose solutions proving detrimental to SFT performance, (b) data generated by a strong teacher outperforms \emph{on-policy} data generated by a weak student model, (c) SFT is robust to low-quality solutions, allowing for imprecise data filtering, and (d) question diversity is crucial for achieving data scaling gains. Based on these insights, we create the OpenMathInstruct-2 dataset, which consists of 14M question-solution pairs ($\approx$ 600K unique questions), making it nearly eight times larger than the previous largest open-source math reasoning dataset. Finetuning the \texttt{Llama-3.1-8B-Base} using OpenMathInstruct-2 outperforms \texttt{Llama3.1-8B-Instruct} on MATH by an absolute 15.9\% (51.9\% $\rightarrow$ 67.8\%). Finally, to accelerate the open-source efforts, we release the code, the finetuned models, and the OpenMathInstruct-2 dataset under a commercially permissive license.

Nemotron-4 340B Technical Report

We release the Nemotron-4 340B model family, including Nemotron-4-340B-Base, Nemotron-4-340B-Instruct, and Nemotron-4-340B-Reward. Our models are open access under the NVIDIA Open Model License Agreement, a permissive model license that allows distribution, modification, and use of the models and its outputs. These models perform competitively to open access models on a wide range of evaluation benchmarks, and were sized to fit on a single DGX H100 with 8 GPUs when deployed in FP8 precision. We believe that the community can benefit from these models in various research studies and commercial applications, especially for generating synthetic data to train smaller language models. Notably, over 98% of data used in our model alignment process is synthetically generated, showcasing the effectiveness of these models in generating synthetic data. To further support open research and facilitate model development, we are also open-sourcing the synthetic data generation pipeline used in our model alignment process.

OpenMathInstruct-1: A 1.8 Million Math Instruction Tuning Dataset

Recent work has shown the immense potential of synthetically generated datasets for training large language models (LLMs), especially for acquiring targeted skills. Current large-scale math instruction tuning datasets such as MetaMathQA (Yu et al., 2024) and MAmmoTH (Yue et al., 2024) are constructed using outputs from closed-source LLMs with commercially restrictive licenses. A key reason limiting the use of open-source LLMs in these data generation pipelines has been the wide gap between the mathematical skills of the best closed-source LLMs, such as GPT-4, and the best open-source LLMs. Building on the recent progress in open-source LLMs, our proposed prompting novelty, and some brute-force scaling, we construct OpenMathInstruct-1, a math instruction tuning dataset with 1.8M problem-solution pairs. The dataset is constructed by synthesizing code-interpreter solutions for GSM8K and MATH, two popular math reasoning benchmarks, using the recently released and permissively licensed Mixtral model. Our best model, OpenMath-CodeLlama-70B, trained on a subset of OpenMathInstruct-1, achieves a score of 84.6% on GSM8K and 50.7% on MATH, which is competitive with the best gpt-distilled models. We release our code, models, and the OpenMathInstruct-1 dataset under a commercially permissive license.

Confidence-based Ensembles of End-to-End Speech Recognition Models

The number of end-to-end speech recognition models grows every year. These models are often adapted to new domains or languages resulting in a proliferation of expert systems that achieve great results on target data, while generally showing inferior performance outside of their domain of expertise. We explore combination of such experts via confidence-based ensembles: ensembles of models where only the output of the most-confident model is used. We assume that models' target data is not available except for a small validation set. We demonstrate effectiveness of our approach with two applications. First, we show that a confidence-based ensemble of 5 monolingual models outperforms a system where model selection is performed via a dedicated language identification block. Second, we demonstrate that it is possible to combine base and adapted models to achieve strong results on both original and target data. We validate all our results on multiple datasets and model architectures.

Powerful and Extensible WFST Framework for RNN-Transducer Losses

This paper presents a framework based on Weighted Finite-State Transducers (WFST) to simplify the development of modifications for RNN-Transducer (RNN-T) loss. Existing implementations of RNN-T use CUDA-related code, which is hard to extend and debug. WFSTs are easy to construct and extend, and allow debugging through visualization. We introduce two WFST-powered RNN-T implementations: (1) "Compose-Transducer", based on a composition of the WFST graphs from acoustic and textual schema -- computationally competitive and easy to modify; (2) "Grid-Transducer", which constructs the lattice directly for further computations -- most compact, and computationally efficient. We illustrate the ease of extensibility through introduction of a new W-Transducer loss -- the adaptation of the Connectionist Temporal Classification with Wild Cards. W-Transducer (W-RNNT) consistently outperforms the standard RNN-T in a weakly-supervised data setup with missing parts of transcriptions at the beginning and end of utterances. All RNN-T losses are implemented with the k2 framework and are available in the NeMo toolkit.

Understanding the Role of Momentum in Stochastic Gradient Methods

The use of momentum in stochastic gradient methods has become a widespread practice in machine learning. Different variants of momentum, including heavy-ball momentum, Nesterov's accelerated gradient (NAG), and quasi-hyperbolic momentum (QHM), have demonstrated success on various tasks. Despite these empirical successes, there is a lack of clear understanding of how the momentum parameters affect convergence and various performance measures of different algorithms. In this paper, we use the general formulation of QHM to give a unified analysis of several popular algorithms, covering their asymptotic convergence conditions, stability regions, and properties of their stationary distributions. In addition, by combining the results on convergence rates and stationary distributions, we obtain sometimes counter-intuitive practical guidelines for setting the learning rate and momentum parameters.

OpenSeq2Seq: extensible toolkit for distributed and mixed precision training of sequence-to-sequence models

We present OpenSeq2Seq -- an open-source toolkit for training sequence-to-sequence models. The main goal of our toolkit is to allow researchers to most effectively explore different sequence-to-sequence architectures. The efficiency is achieved by fully supporting distributed and mixed-precision training. OpenSeq2Seq provides building blocks for training encoder-decoder models for neural machine translation and automatic speech recognition. We plan to extend it with other modalities in the future.

Novel Prediction Techniques Based on Clusterwise Linear Regression

In this paper we explore different regression models based on Clusterwise Linear Regression (CLR). CLR aims to find the partition of the data into $k$ clusters, such that linear regressions fitted to each of the clusters minimize overall mean squared error on the whole data. The main obstacle preventing to use found regression models for prediction on the unseen test points is the absence of a reasonable way to obtain CLR cluster labels when the values of target variable are unknown. In this paper we propose two novel approaches on how to solve this problem. The first approach, predictive CLR builds a separate classification model to predict test CLR labels. The second approach, constrained CLR utilizes a set of user-specified constraints that enforce certain points to go to the same clusters. Assuming the constraint values are known for the test points, they can be directly used to assign CLR labels. We evaluate these two approaches on three UCI ML datasets as well as on a large corpus of health insurance claims. We show that both of the proposed algorithms significantly improve over the known CLR-based regression methods. Moreover, predictive CLR consistently outperforms linear regression and random forest, and shows comparable performance to support vector regression on UCI ML datasets. The constrained CLR approach achieves the best performance on the health insurance dataset, while enjoying only $\approx 20$ times increased computational time over linear regression.