On the Geometry of Separation in Finite Gaussian Mixtures

We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.

Nemotron 3 Ultra: Open, Efficient Mixture-of-Experts Hybrid Mamba-Transformer Model for Agentic Reasoning

We introduce Nemotron 3 Ultra, a 550 billion total and 55 billion active parameter Mixture-of-Experts Hybrid Mamba-Attention language model. We pre-trained Nemotron 3 Ultra on 20 trillion text tokens, then extended the context length to 1M tokens, and post-trained using Supervised Fine Tuning (SFT), Reinforcement Learning (RL), and Multi-teacher On-Policy Distillation (MOPD). Nemotron 3 Ultra is our most capable model yet, employing multiple key technologies - LatentMoE, Multi Token Prediction (MTP), NVFP4 pre-training, multi-environment RLVR, MOPD, and reasoning budget control. Nemotron 3 Ultra achieves up to ~6x higher inference throughput as compared to state-of-the-art publicly available LLMs while attaining on-par accuracy. The state-of-the-art accuracy, high inference throughput, and 1M token context length make Nemotron 3 Ultra ideal for long-running autonomous agentic tasks. We open-source the base, post-trained, and quantized checkpoints, along with the training data and recipe on HuggingFace.

FIBER: A Differentially Private Optimizer with Filter-Aware Innovation Bias Correction

Differentially private (DP) training protects individual examples by adding noise to gradients, but the injected noise interacts nontrivially with adaptive optimizers. Recent DP methods temporally filter privatized gradients to reduce variance; however, filtering also changes the DP noise statistics seen by AdamW's second-moment accumulator. As a result, bias corrections derived for unfiltered DP noise, such as subtracting sigma_w squared, can become miscalibrated when filtering is present. We propose FiBeR, a DP optimizer designed for temporally filtered privatized gradients. FiBeR (i) performs denoising in innovation space by filtering the residual stream and integrating it to form the filtered gradient estimate, (ii) decouples the two-point observation geometry from the innovation gain to enable independent tuning, and (iii) introduces a filter-aware second-moment calibration that subtracts the attenuated DP noise contribution A(omega) sigma_w squared, where A(omega) is derived in closed form for the innovation filter and can be computed for general stable linear filters. Across vision and language benchmarks, FiBeR consistently demonstrates substantial improvements in the performance of DP optimizers, surpassing state-of-the-art results under equivalent privacy constraints on multiple tasks.

On Bayesian Softmax-Gated Mixture-of-Experts Models

Mixture-of-experts models provide a flexible framework for learning complex probabilistic input-output relationships by combining multiple expert models through an input-dependent gating mechanism. These models have become increasingly prominent in modern machine learning, yet their theoretical properties in the Bayesian framework remain largely unexplored. In this paper, we study Bayesian mixture-of-experts models, focusing on the ubiquitous softmax-based gating mechanism. Specifically, we investigate the asymptotic behavior of the posterior distribution for three fundamental statistical tasks: density estimation, parameter estimation, and model selection. First, we establish posterior contraction rates for density estimation, both in the regimes with a fixed, known number of experts and with a random learnable number of experts. We then analyze parameter estimation and derive convergence guarantees based on tailored Voronoi-type losses, which account for the complex identifiability structure of mixture-of-experts models. Finally, we propose and analyze two complementary strategies for selecting the number of experts. Taken together, these results provide one of the first systematic theoretical analyses of Bayesian mixture-of-experts models with softmax gating, and yield several theory-grounded insights for practical model design.

Nemotron 3 Super: Open, Efficient Mixture-of-Experts Hybrid Mamba-Transformer Model for Agentic Reasoning

We describe the pre-training, post-training, and quantization of Nemotron 3 Super, a 120 billion (active 12 billion) parameter hybrid Mamba-Attention Mixture-of-Experts model. Nemotron 3 Super is the first model in the Nemotron 3 family to 1) be pre-trained in NVFP4, 2) leverage LatentMoE, a new Mixture-of-Experts architecture that optimizes for both accuracy per FLOP and accuracy per parameter, and 3) include MTP layers for inference acceleration through native speculative decoding. We pre-trained Nemotron 3 Super on 25 trillion tokens followed by post-training using supervised fine tuning (SFT) and reinforcement learning (RL). The final model supports up to 1M context length and achieves comparable accuracy on common benchmarks, while also achieving up to 2.2x and 7.5x higher inference throughput compared to GPT-OSS-120B and Qwen3.5-122B, respectively. Nemotron 3 Super datasets, along with the base, post-trained, and quantized checkpoints, are open-sourced on HuggingFace.

Adaptive Stopping for Multi-Turn LLM Reasoning

Large Language Models (LLMs) increasingly rely on multi-turn reasoning and interaction, such as adaptive retrieval-augmented generation (RAG) and ReAct-style agents, to answer difficult questions. These methods improve accuracy by iteratively retrieving information, reasoning, or acting, but introduce a key challenge: \textbf{When should the model stop?} Existing approaches rely on heuristic stopping rules or fixed turn budgets and provide no formal guarantees that the final prediction still contains the correct answer. This limitation is particularly problematic in high-stakes domains such as finance and healthcare, where unnecessary turns increase cost and latency, while stopping too early risks incorrect decisions. Conformal prediction (CP) provides formal coverage guarantees, but existing LLM-CP methods only apply to a single model output and cannot handle multi-turn pipelines with adaptive stopping. To address this gap, we propose Multi-Turn Language Models with Conformal Prediction (MiCP), the first CP framework for multi-turn reasoning. MiCP allocates different error budgets across turns, enabling the model to stop early while maintaining an overall coverage guarantee. We demonstrate MiCP on adaptive RAG and ReAct, where it achieves the target coverage on both single-hop and multi-hop question answering benchmarks while reducing the number of turns, inference cost, and prediction set size. We further introduce a new metric that jointly evaluates coverage validity and answering efficiency.

Rethinking Multinomial Logistic Mixture of Experts with Sigmoid Gating Function

The sigmoid gate in mixture-of-experts (MoE) models has been empirically shown to outperform the softmax gate across several tasks, ranging from approximating feed-forward networks to language modeling. Additionally, recent efforts have demonstrated that the sigmoid gate is provably more sample-efficient than its softmax counterpart under regression settings. Nevertheless, there are three notable concerns that have not been addressed in the literature, namely (i) the benefits of the sigmoid gate have not been established under classification settings; (ii) existing sigmoid-gated MoE models may not converge to their ground-truth; and (iii) the effects of a temperature parameter in the sigmoid gate remain theoretically underexplored. To tackle these open problems, we perform a comprehensive analysis of multinomial logistic MoE equipped with a modified sigmoid gate to ensure model convergence. Our results indicate that the sigmoid gate exhibits a lower sample complexity than the softmax gate for both parameter and expert estimation. Furthermore, we find that incorporating a temperature into the sigmoid gate leads to a sample complexity of exponential order due to an intrinsic interaction between the temperature and gating parameters. To overcome this issue, we propose replacing the vanilla inner product score in the gating function with a Euclidean score that effectively removes that interaction, thereby substantially improving the sample complexity to a polynomial order.

A Statistical Theory of Gated Attention through the Lens of Hierarchical Mixture of Experts

Self-attention has greatly contributed to the success of the widely used Transformer architecture by enabling learning from data with long-range dependencies. In an effort to improve performance, a gated attention model that leverages a gating mechanism within the multi-head self-attention has recently been proposed as a promising alternative. Gated attention has been empirically demonstrated to increase the expressiveness of low-rank mapping in standard attention and even to eliminate the attention sink phenomenon. Despite its efficacy, a clear theoretical understanding of gated attention's benefits remains lacking in the literature. To close this gap, we rigorously show that each entry in a gated attention matrix or a multi-head self-attention matrix can be written as a hierarchical mixture of experts. By recasting learning as an expert estimation problem, we demonstrate that gated attention is more sample-efficient than multi-head self-attention. In particular, while the former needs only a polynomial number of data points to estimate an expert, the latter requires exponentially many data points to achieve the same estimation error. Furthermore, our analysis also provides a theoretical justification for why gated attention yields higher performance when a gate is placed at the output of the scaled dot product attention or the value map rather than at other positions in the multi-head self-attention architecture.

Improving Minimax Estimation Rates for Contaminated Mixture of Multinomial Logistic Experts via Expert Heterogeneity

Contaminated mixture of experts (MoE) is motivated by transfer learning methods where a pre-trained model, acting as a frozen expert, is integrated with an adapter model, functioning as a trainable expert, in order to learn a new task. Despite recent efforts to analyze the convergence behavior of parameter estimation in this model, there are still two unresolved problems in the literature. First, the contaminated MoE model has been studied solely in regression settings, while its theoretical foundation in classification settings remains absent. Second, previous works on MoE models for classification capture pointwise convergence rates for parameter estimation without any guaranty of minimax optimality. In this work, we close these gaps by performing, for the first time, the convergence analysis of a contaminated mixture of multinomial logistic experts with homogeneous and heterogeneous structures, respectively. In each regime, we characterize uniform convergence rates for estimating parameters under challenging settings where ground-truth parameters vary with the sample size. Furthermore, we also establish corresponding minimax lower bounds to ensure that these rates are minimax optimal. Notably, our theories offer an important insight into the design of contaminated MoE, that is, expert heterogeneity yields faster parameter estimation rates and, therefore, is more sample-efficient than expert homogeneity.

Cite-While-You-Generate: Training-Free Evidence Attribution for Multimodal Clinical Summarization

Trustworthy clinical summarization requires not only fluent generation but also transparency about where each statement comes from. We propose a training-free framework for generation-time source attribution that leverages decoder attentions to directly cite supporting text spans or images, overcoming the limitations of post-hoc or retraining-based methods. We introduce two strategies for multimodal attribution: a raw image mode, which directly uses image patch attentions, and a caption-as-span mode, which substitutes images with generated captions to enable purely text-based alignment. Evaluations on two representative domains: clinician-patient dialogues (CliConSummation) and radiology reports (MIMIC-CXR), show that our approach consistently outperforms embedding-based and self-attribution baselines, improving both text-level and multimodal attribution accuracy (e.g., +15% F1 over embedding baselines). Caption-based attribution achieves competitive performance with raw-image attention while being more lightweight and practical. These findings highlight attention-guided attribution as a promising step toward interpretable and deployable clinical summarization systems.