arXiv subjects

Igor Gitman

Publications and source records attributed to Igor Gitman.

At least 19 recordsLinked to original sources

arXiv2026

An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics

We study how model post-training and test-time inference design affect natural-language proof generation for hard olympiad mathematics. Starting from Nemotron 3 Ultra, we train two specialist checkpoints using supervised fine-tuning and reinforcement learning, and evaluate checkpoint choice, verification, and refinement. Based on these findings, we present an open-model test-time-compute pipeline. The system operates entirely in natural language, with no formal prover, external tools, or internet access. Three Nemotron 3 Ultra checkpoints - the general-availability model and two post-trained specialists - power an iterative search that generates, verifies, and refines candidate proofs; a separate high-compute stage then selects each final submission. The system scored 30 out of 42 points at IMO 2026, reaching the gold-medal threshold. We release the two post-trained checkpoints as well as the training data, the training and inference code, the submitted solutions, and Nemotron-IMO-Bench, a new benchmark of 200 novel olympiad-level problems.

cs.AI↗

arXiv2026

iGRPO: Self-Feedback-Driven LLM Reasoning

Large Language Models (LLMs) have shown promise in solving complex mathematical problems, yet they still fall short of producing accurate and consistent solutions. Reinforcement Learning (RL) is a framework for aligning these models with task-specific rewards, improving overall quality and reliability. Group Relative Policy Optimization (GRPO) is an efficient, value-function-free alternative to Proximal Policy Optimization (PPO) that leverages group-relative reward normalization. We introduce Iterative Group Relative Policy Optimization (iGRPO), a two-stage extension of GRPO that adds dynamic self-conditioning through model-generated drafts. In Stage 1, iGRPO samples multiple exploratory drafts and selects the highest-reward draft using the same scalar reward signal used for optimization. In Stage 2, it appends this best draft to the original prompt and applies a GRPO-style update on draft-conditioned refinements, training the policy to improve beyond its strongest prior attempt. Under matched rollout budgets, iGRPO consistently outperforms GRPO across base models (e.g., Nemotron-H-8B-Base-8K and DeepSeek-R1 Distilled), validating its effectiveness on diverse reasoning benchmarks. Moreover, applying iGRPO to OpenReasoning-Nemotron-7B trained on AceReason-Math achieves new state-of-the-art results of 85.62\% and 79.64\% on AIME24 and AIME25, respectively. Ablations further show that the refinement wrapper generalizes beyond GRPO variants, benefits from a generative judge, and alters learning dynamics by delaying entropy collapse. These results underscore the potential of iterative, self-feedback-based RL for advancing verifiable mathematical reasoning.

cs.AI↗

arXiv2026

Learning Generative Selection for Best-of-N

Scaling test-time compute via parallel sampling can substantially improve LLM reasoning, but is often limited by Best-of-N selection quality. Generative selection methods, such as GenSelect, address this bottleneck, yet strong selection performance remains largely limited to large models. We show that small reasoning models can acquire strong GenSelect capabilities through targeted reinforcement learning. To this end, we synthesize selection tasks from large-scale math and code instruction datasets by filtering to instances with both correct and incorrect candidate solutions, and train 1.7B-parameter models with DAPO to reward correct selections. Across math (AIME24, AIME25, HMMT25) and code (LiveCodeBench) reasoning benchmarks, our models consistently outperform prompting and majority-voting baselines, often approaching or exceeding much larger models. Moreover, these gains generalize to selecting outputs from stronger models despite training only on outputs from weaker models. Overall, our results establish reinforcement learning as a scalable way to unlock strong generative selection in small models, enabling efficient test-time scaling.

cs.LG↗

arXiv2025

Nemotron-Math: Efficient Long-Context Distillation of Mathematical Reasoning from Multi-Mode Supervision

High-quality mathematical reasoning supervision requires diverse reasoning styles, long-form traces, and effective tool integration, capabilities that existing datasets provide only in limited form. Leveraging the multi-mode generation ability of gpt-oss-120b, we introduce Nemotron-Math, a large-scale mathematical reasoning dataset containing 7.5M solution traces across high, medium, and low reasoning modes, each available both with and without Python tool-integrated reasoning (TIR). The dataset integrates 85K curated AoPS problems with 262K community-sourced StackExchange-Math problems, combining structured competition tasks with diverse real-world mathematical queries. We conduct controlled evaluations to assess the dataset quality. Nemotron-Math consistently outperforms the original OpenMathReasoning on matched AoPS problems. Incorporating StackExchange-Math substantially improves robustness and generalization, especially on HLE-Math, while preserving accuracy on math competition benchmarks. To support efficient long-context training, we develop a sequential bucketed strategy that accelerates 128K context-length fine-tuning by 2--3$\times$ without significant accuracy loss. Overall, Nemotron-Math enables state-of-the-art performance, including 100\% maj@16 accuracy on AIME 2024 and 2025 with Python TIR.

cs.AI↗

arXiv2025

Scaling Generative Verifiers For Natural Language Mathematical Proof Verification And Selection

Large language models have achieved remarkable success on final-answer mathematical problems, largely due to the ease of applying reinforcement learning with verifiable rewards. However, the reasoning underlying these solutions is often flawed. Advancing to rigorous proof-based mathematics requires reliable proof verification capabilities. We begin by analyzing multiple evaluation setups and show that focusing on a single benchmark can lead to brittle or misleading conclusions. To address this, we evaluate both proof-based and final-answer reasoning to obtain a more reliable measure of model performance. We then scale two major generative verification methods (GenSelect and LLM-as-a-Judge) to millions of tokens and identify their combination as the most effective framework for solution verification and selection. We further show that the choice of prompt for LLM-as-a-Judge significantly affects the model's performance, but reinforcement learning can reduce this sensitivity. However, despite improving proof-level metrics, reinforcement learning does not enhance final-answer precision, indicating that current models often reward stylistic or procedural correctness rather than mathematical validity. Our results establish practical guidelines for designing and evaluating scalable proof-verification and selection systems.

cs.AI↗

arXiv2025

NVIDIA Nemotron Nano 2: An Accurate and Efficient Hybrid Mamba-Transformer Reasoning Model

We introduce Nemotron-Nano-9B-v2, a hybrid Mamba-Transformer language model designed to increase throughput for reasoning workloads while achieving state-of-the-art accuracy compared to similarly-sized models. Nemotron-Nano-9B-v2 builds on the Nemotron-H architecture, in which the majority of the self-attention layers in the common Transformer architecture are replaced with Mamba-2 layers, to achieve improved inference speed when generating the long thinking traces needed for reasoning. We create Nemotron-Nano-9B-v2 by first pre-training a 12-billion-parameter model (Nemotron-Nano-12B-v2-Base) on 20 trillion tokens using an FP8 training recipe. After aligning Nemotron-Nano-12B-v2-Base, we employ the Minitron strategy to compress and distill the model with the goal of enabling inference on up to 128k tokens on a single NVIDIA A10G GPU (22GiB of memory, bfloat16 precision). Compared to existing similarly-sized models (e.g., Qwen3-8B), we show that Nemotron-Nano-9B-v2 achieves on-par or better accuracy on reasoning benchmarks while achieving up to 6x higher inference throughput in reasoning settings like 8k input and 16k output tokens. We are releasing Nemotron-Nano-9B-v2, Nemotron-Nano12B-v2-Base, and Nemotron-Nano-9B-v2-Base checkpoints along with the majority of our pre- and post-training datasets on Hugging Face.

cs.CL↗

arXiv2025

GenSelect: A Generative Approach to Best-of-N

Generative reward models with parallel sampling have enabled effective test-time scaling for reasoning tasks. Current approaches employ pointwise scoring of individual solutions or pairwise comparisons. However, pointwise methods underutilize LLMs' comparative abilities, while pairwise methods scale inefficiently with larger sampling budgets. We introduce GenSelect, where the LLM uses long reasoning to select the best solution among N candidates. This leverages LLMs' comparative strengths while scaling efficiently across parallel sampling budgets. For math reasoning, we demonstrate that reasoning models, such as QwQ and DeepSeek-R1-0528, excel at GenSelect, outperforming existing scoring approaches with simple prompting.

cs.LG↗

arXiv2025

The Challenge of Teaching Reasoning to LLMs Without RL or Distillation

Reasoning-capable language models achieve state-of-the-art performance in diverse complex tasks by generating long, explicit Chain-of-Thought (CoT) traces. While recent works show that base models can acquire such reasoning traces via reinforcement learning or distillation from stronger models like DeepSeek-R1, previous works demonstrate that even short CoT prompting without fine-tuning is able to improve reasoning. We ask whether long CoT can be induced in a base model using only prompting or minimal tuning. Using just 20 long CoT examples from the reasoning model \texttt{QwQ-32B-Preview}, we lightly fine-tune the base model \texttt{Qwen2.5-32B}. The resulting model outperforms the much larger \texttt{Qwen2.5-Math-72B-Instruct}, showing that a handful of high-quality examples can unlock strong reasoning capabilities. We further explore using CoT data from non-reasoning models and human annotators, enhanced with prompt engineering, multi-pass editing, and structural guidance. However, neither matches the performance of reasoning model traces, suggesting that certain latent qualities of expert CoT are difficult to replicate. We analyze key properties of reasoning data, such as problem difficulty, diversity, and answer length, that influence reasoning distillation. While challenges remain, we are optimistic that carefully curated human-written CoT, even in small quantities, can activate reasoning behaviors in base models. We release our human-authored dataset across refinement stages and invite further investigation into what makes small-scale reasoning supervision so effective.

cs.AI↗

arXiv2025

Llama-Nemotron: Efficient Reasoning Models

We introduce the Llama-Nemotron series of models, an open family of heterogeneous reasoning models that deliver exceptional reasoning capabilities, inference efficiency, and an open license for enterprise use. The family comes in three sizes -- Nano (8B), Super (49B), and Ultra (253B) -- and performs competitively with state-of-the-art reasoning models such as DeepSeek-R1 while offering superior inference throughput and memory efficiency. In this report, we discuss the training procedure for these models, which entails using neural architecture search from Llama 3 models for accelerated inference, knowledge distillation, and continued pretraining, followed by a reasoning-focused post-training stage consisting of two main parts: supervised fine-tuning and large scale reinforcement learning. Llama-Nemotron models are the first open-source models to support a dynamic reasoning toggle, allowing users to switch between standard chat and reasoning modes during inference. To further support open research and facilitate model development, we provide the following resources: 1. We release the Llama-Nemotron reasoning models -- LN-Nano, LN-Super, and LN-Ultra -- under the commercially permissive NVIDIA Open Model License Agreement. 2. We release the complete post-training dataset: Llama-Nemotron-Post-Training-Dataset. 3. We also release our training codebases: NeMo, NeMo-Aligner, and Megatron-LM.

cs.CL↗

arXiv2025

NeMo-Inspector: A Visualization Tool for LLM Generation Analysis

Adapting Large Language Models (LLMs) to novel tasks and enhancing their overall capabilities often requires large, high-quality training datasets. Synthetic data, generated at scale, serves a valuable alternative when real-world data is scarce or difficult to obtain. However, ensuring the quality of synthetic datasets is challenging, as developers must manually inspect and refine numerous samples to identify errors and areas for improvement. This process is time-consuming and requires specialized tools. We introduce NeMo-Inspector, an open-source tool designed to simplify the analysis of synthetic datasets with integrated inference capabilities. We demonstrate its effectiveness through two real-world cases. Analysis and cleaning of the synthetically generated GSM-Plus dataset with NeMo-Inspector led to a significant decrease in low-quality samples from 46.99% to 19.51%. The tool also helped identify and correct generation errors in OpenMath models, improving accuracy by 1.92% on the MATH dataset and by 4.17% on the GSM8K dataset for a Meta-Llama-3-8B model fine-tuned on synthetic data generated from Nemotron-4-340B.

cs.LG↗

arXiv2025

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.

cs.AI↗

arXiv2025

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. We are releasing Nemotron-H base model checkpoints with support in Hugging Face and NeMo.

cs.CL↗

arXiv2024

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 equally-sized 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.

cs.CL↗

arXiv2024

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.

cs.CL↗

arXiv2024

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.

cs.CL↗

arXiv2023

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.

eess.AS↗

arXiv2023

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.

eess.AS↗

arXiv2019

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.

cs.LG↗