The m-out-of-n bootstrap--proposed by Bickel et al. [1992]--approximates the distribution of a statistic by repeatedly drawing msubsamples (m n) without replacement from an original sample of size n; it is now routinely used for robust inference with heavy-tailed data, bandwidth selection, and other large-sample applications. Despite this broad applicability across econometrics, biostatistics, and machine-learning workflows, rigorous parameter-free guarantees for the soundness of the m-out-of-n bootstrap when estimating sample quantiles have remained elusive. This paper establishes such guarantees by analysing the estimator of sample quantiles obtained from m-out-of-n resampling of a dataset of length n. We first prove a central limit theorem for a fully data-driven version of the estimator that holds under a mild moment condition and involves no unknown nuisance parameters. We then show that the moment assumption is essentially tight by constructing a counter-example in which the CLT fails. Strengthening the assumptions slightly, we derive an Edgeworth expansion that delivers exact convergence rates and, as a corollary, a Berry-Esséen bound on the bootstrap approximation error. Finally, we illustrate the scope of our results by obtaining parameter-free asymptotic distributions for practical statistics, including the quantiles for random walk MH, and rewards of ergodic MDP's, thereby demonstrating the usefulness of our theory in modern estimation and learning tasks.

In this paper, we address the challenges associated with merging low-rank adaptations of large neural networks. With the rise of parameter-efficient adaptation techniques, such as Low-Rank Adaptation (LoRA), model fine-tuning has become more accessible. While fine-tuning models with LoRA is highly efficient, existing merging methods often sacrifice this efficiency by merging fully-sized weight matrices. We propose the Core Space merging framework, which enables the merging of LoRA-adapted models within a common alignment basis, thereby preserving the efficiency of low-rank adaptation while substantially improving accuracy across tasks. We further provide a formal proof that projection into Core Space ensures no loss of information and provide a complexity analysis showing the efficiency gains. Extensive empirical results demonstrate that Core Space significantly improves existing merging techniques and achieves state-of-the-art results on both vision and language tasks while utilizing a fraction of the computational resources.

This paper introduces Completion Pruning Policy Optimization (CPPO) to accelerate the training of reasoning models based on Group Relative Policy Optimization (GRPO).

Large language model (LLM) routers improve the efficiency of multi-model systems by directing each query to the most appropriate model while leveraging the diverse strengths of heterogeneous LLMs. Most existing approaches frame routing as a classification problem based solely on the input query. While this reduces overhead by avoiding inference across all models, it overlooks valuable information that could be gleaned from potential outputs and fails to capture implicit intent or contextual nuances that often emerge only during response generation. These limitations can result in suboptimal routing decisions, particularly for complex or ambiguous queries that require deeper semantic understanding. To address this challenge, we propose Lookahead, a routing framework that "foresees" potential model outputs by predicting their latent representations and uses these predictions to guide model selection, thus enabling more informed routing without full inference. Within this framework, we implement two approaches based on causal and masked language models. Empirical evaluations across seven public benchmarks--spanning instruction following, mathematical reasoning, and code generation--show that Lookahead consistently outperforms existing routing baselines, achieving an average performance gain of 7.7% over the state-of-the-art.

Although Large Language Models (LLMs) have demonstrated remarkable progress, their proficiency in graph-related tasks remains notably limited, hindering the development of truly general-purpose models. Previous attempts, including pretraining graph foundation models or employing supervised fine-tuning, often face challenges such as the scarcity of large-scale, universally represented graph data. We introduce G1, a simple yet effective approach demonstrating that Reinforcement Learning (RL) on synthetic graph-theoretic tasks can significantly scale LLMs' graph reasoning abilities. To enable RL training, we curate Erdős, the largest graph reasoning dataset to date, comprising 50 diverse graph-theoretic tasks of varying difficulty levels, 100k training data and 5k test data, all drived from real-world graphs.

Ensembles of generative large language models (LLMs) are a promising way to compensate for individual model limitations, integrating the strengths of different LLMs. Existing LLM ensemble methods, however, face limitations such as first-token delay and challenges in long-range semantic collaboration between models, Moreover, they typically assume equal voting weights for all models during ensemble, ignoring task-specific performance differences among models. In this work, we propose SpecEM, a training-free, plug-and-play LLM ensemble framework that dynamically adjusts each model's model contribution in real time based on task performance. Inspired by speculative decoding, SpecEM iteratively performs drafting and verification, allowing models to collaborate semantically at the segment level for integrated output. Furthermore, we introduce an online feedback mechanism with multiplicative weight updates, where each model's voting weight is adjusted on-the-fly according to how often it outperforms others during verification stage, ensuring that stronger models exert greater influence during ensembling. Experimental results on five LLM families (ranging from 7B to 72B parameters) and six benchmark datasets, spanning open-domain instruction following, reasoning, commonsense, demonstrate consistent performance improvements compared to state-of-the-art LLM ensemble methods.

Process Reward Models (PRMs) have shown promise in enhancing the mathematical reasoning capabilities of Large Language Models (LLMs) through Test-Time Scaling (TTS). However, their integration into multimodal reasoning remains largely unexplored. In this work, we take the first step toward unlocking the potential of PRMs in multimodal mathematical reasoning. We identify three key challenges: (i) the scarcity of high-quality reasoning data constrains the capabilities of foundation Multimodal Large Language Models (MLLMs), which imposes further limitations on the upper bounds of TTS and reinforcement learning (RL); (ii) a lack of automated methods for process labeling within multimodal contexts persists; (iii) the employment of process rewards in unimodal RL faces issues like reward hacking, which may extend to multimodal scenarios. To address these issues, we introduce URSA, a three-stage Unfolding multimodal pRocessSupervision Aided training framework. We first construct MMathCoT-1M, a high-quality large-scale multimodal Chain-of-Thought (CoT) reasoning dataset, to build a stronger math reasoning foundation MLLM, URSA-8B. Subsequently, we go through an automatic process to synthesize process supervision data, which emphasizes both logical correctness and perceptual consistency. We introduce DualMath-1.1M to facilitate the training of URSA-8B-RM.

Learning how the world works is central to building AI agents that can adapt to complex environments. Traditional world models based on deep learning demand vast amounts of training data, and do not flexibly update their knowledge from sparse observations. Recent advances in program synthesis using Large Language Models (LLMs) give an alternate approach which learns world models represented as source code, supporting strong generalization from little data. To date, application of program-structured world models remains limited to natural language and grid-world domains. We introduce a novel program synthesis method for effectively modeling complex, non-gridworld domains by representing a world model as an exponentially-weighted product of programmatic experts (PoE-World) synthesized by LLMs. We show that this approach can learn complex, stochastic world models from just a few observations. We evaluate the learned world models by embedding them in a model-based planning agent, demonstrating efficient performance and generalization to unseen levels on Atari's Pong and Montezuma's Revenge.

We introduce a novel class of score-based diffusion processes that operate directly in the representation space of Lie groups. Leveraging the framework of Generalized Score Matching, we derive a class of Langevin dynamics that decomposes as a direct sum of Lie algebra representations, enabling the modeling of any target distribution on any (non-Abelian) Lie group. Standard score-matching emerges as a special case of our framework when the Lie group is the translation group. We prove that our generalized generative processes arise as solutions to a new class of paired stochastic differential equations (SDEs), introduced here for the first time.

Recent long-form video-language understanding benchmarks have driven progress in video large multimodal models (Video-LMMs). However, the scarcity of wellannotated long videos has left the training of hour-long Video-LMMs underexplored. To close this gap, we present VideoMarathon, a large-scale hour-long video instruction-following dataset. This dataset includes around 9,700 hours of long videos sourced from diverse domains, ranging from 3 to 60 minutes per video. Specifically, it contains 3.3M high-quality QA pairs, spanning six fundamental topics: temporality, spatiality, object, action, scene, and event. Compared to existing video instruction datasets, VideoMarathon significantly extends training video durations up to 1 hour, and supports 22 diverse tasks requiring both short-and long-term video comprehension. Building on VideoMarathon, we propose Hour-LLaVA, a powerful and efficient Video-LMM for hour-scale video-language modeling. It enables hour-long video training and inference at 1-FPS sampling by leveraging a memory augmentation module, which adaptively integrates question-relevant and spatiotemporally informative semantics from the cached full video context. In our experiments, Hour-LLaVA achieves the best performance on multiple representative long video-language benchmarks, demonstrating the high quality of the VideoMarathon dataset and the superiority of the Hour-LLaVA model.