Sampling multiple responses is a common way to improve LLM output quality, but it comes at the cost of additional computation. The key challenge is deciding when to stop generating new samples to balance accuracy gains against efficiency. To address this, we introduce BEACON (Bayesian Efficient Adaptive Criterion for Optimal N-stopping), a principled adaptive sampling framework grounded in Sequential Search with Bayesian Learning. BEACON sequentially generates responses from the policy LLM, updates posterior belief over reward distributions in real time without further training, and determines when to stop by weighing expected gains against computational cost. Sampling terminates once the marginal utility of further exploration no longer justifies the expense. We establish both theoretical optimality guarantees and practical tractability, and show empirically that BEACON reduces average sampling by up to 80% while maintaining response quality. We further demonstrate BEACON's utility for cost-efficient preference data generation and outline practical extensions, offering actionable insights for future researchers.

Reinforcement Learning with V erifiable Reward (RL VR) has significantly advanced the complex reasoning abilities of Large Language Models (LLMs). However, it struggles to break through the inherent capability boundaries of the base LLM, due to its essentially on-policy strategy coupled with LLM's immense action space and sparse reward. Critically, RL VR can lead to the capability boundary collapse, narrowing the LLM's problem-solving scope. To address this problem, we propose RL-PLUS, a novel hybrid-policy optimization approach for LLMs that synergizes internal exploitation with external data to achieve stronger reasoning capabilities and surpass the boundaries of base models. RL-PLUS integrates two core components, i.e., Multiple Importance Sampling to address distributional mismatch from external data, and Exploration-Based Advantage Function to guide the model towards high-value, unexplored reasoning paths. We provide both theoretical analysis and extensive experiments to demonstrate the superiority and gen-eralizability of our approach. Compared with existing RL VR methods, RL-PLUS achieves 1) state-of-the-art performance on six math reasoning benchmarks; 2) superior performance on six out-of-distribution reasoning tasks; 3) consistent and significant gains across diverse model families, with average relative improvements up to 69.2%. Moreover, the analysis of Pass@k curves indicates that RL-PLUS effectively resolves the capability boundary collapse problem.

Sim2Real aims at training policies in high-fidelity simulation environments and effectively transferring them to the real world. Despite the developments of accurate simulators and Sim2Real RL approaches, the policies trained purely in simulation often suffer significant performance drops when deployed in real environments. This drop is referred to as the Sim2Real performance gap. Current Sim2Real RL methods optimize the simulator accuracy and variability as proxies for real-world performance. However, these metrics do not necessarily correlate with the real-world performance of the policy as established theoretically and empirically in the literature. We propose a novel framework to address this issue by directly adapting the simulator parameters based on real-world performance. We frame this problem as a bi-level RL framework: the inner-level RL trains a policy purely in simulation, and the outer-level RL adapts the simulation model and in-sim reward parameters to maximize real-world performance of the in-sim policy. We derive and validate in simple examples the mathematical tools needed to develop bi-level RL algorithms that close the Sim2Real performance gap.

Unsupervised anomaly detection (UAD) plays a crucial role in neuroimaging for identifying deviations from healthy subject data and thus facilitating the diagnosis of neurological disorders. In this work, we focus on Bayesian flow networks (BFNs), a novel class of generative models, which have not yet been applied to medical imaging or anomaly detection. BFNs combine the strength of diffusion frameworks and Bayesian inference. We introduce AnoBFN, an extension of BFNs for UAD, designed to: i) perform conditional image generation under high levels of spatially correlated noise, and ii) preserve subject specificity by incorporating a recursive feedback from the input image throughout the generative process. We evaluate AnoBFN on the challenging task of Alzheimer's disease-related anomaly detection in FDG PET images. Our approach outperforms other state-of-the-art methods based on V AEs ( β -VAE), GANs (f-AnoGAN), and diffusion models (AnoDDPM), demonstrating its effectiveness at detecting anomalies while reducing false positive rates.

The prediction of learning curves for Natural Language Processing (NLP) models enables informed decision-making to meet specific performance objectives, while reducing computational overhead and lowering the costs associated with dataset acquisition and curation. In this work, we formulate the prediction task as a multitask learning problem, where each task's data is modelled as being organized within a two-layer hierarchy. To model the shared information and dependencies across tasks and hierarchical levels, we employ latent variable multi-output Gaussian Processes, enabling to account for task correlations and supporting zero-shot prediction of learning curves (LCs). We demonstrate that this approach facilitates the development of probabilistic scaling laws at lower costs. Applying an active learning strategy, LCs can be queried to reduce predictive uncertainty and provide predictions close to ground truth scaling laws. We validate our framework on three small-scale NLP datasets with up to $30$ LCs. These are obtained from nanoGPT models, from bilingual translation using mBART and Transformer models, and from multilingual translation using M2M100 models of varying sizes.

This paper proposes the Humanoid-inspired Structural Causal Model (HSCM), a novel causal framework inspired by human intelligence, designed to overcome the limitations of conventional domain generalization models. Unlike approaches that rely on statistics to capture data-label dependencies and learn distortion-invariant representations, HSCM replicates the hierarchical processing and multi-level learning of human vision systems, focusing on modeling fine-grained causal mechanisms. By disentangling and reweighting key image attributes such as color, texture, and shape, HSCM enhances generalization across diverse domains, ensuring robust performance and interpretability. Leveraging the flexibility and adaptability of human intelligence, our approach enables more effective transfer and learning in dynamic, complex environments. Through both theoretical and empirical evaluations, we demonstrate that HSCM outperforms existing domain generalization models, providing a more principled method for capturing causal relationships and improving model robustness. The code is available at https://github.com/lambett/HSCM.

Does semantic communication require a semantic information theory parallel to Shannon's information theory, or can Shannon's work be generalized for semantic communication? This paper advocates for the latter and introduces a semantic generalization of Shannon's information theory (G theory for short). The core idea is to replace the distortion constraint with the semantic constraint, achieved by utilizing a set of truth functions as a semantic channel. These truth functions enable the expressions of semantic distortion, semantic information measures, and semantic information loss. Notably, the maximum semantic information criterion is equivalent to the maximum likelihood criterion and similar to the Regularized Least Squares criterion. This paper shows G theory's applications to daily and electronic semantic communication, machine learning, constraint control, Bayesian confirmation, portfolio theory, and information value. The improvements in machine learning methods involve multilabel learning and classification, maximum mutual information classification, mixture models, and solving latent variables. Furthermore, insights from statistical physics are discussed: Shannon information is similar to free energy; semantic information to free energy in local equilibrium systems; and information efficiency to the efficiency of free energy in performing work. The paper also proposes refining Friston's minimum free energy principle into the maximum information efficiency principle. Lastly, it compares G theory with other semantic information theories and discusses its limitation in representing the semantics of complex data.

Symmetries are known to improve the empirical performance of machine learning models, yet theoretical guarantees explaining these gains remain limited. Prior work has focused mainly on compact group symmetries and often assumes that the data distribution itself is invariant, an assumption rarely satisfied in real-world applications. In this work, we extend generalization guarantees to the broader setting of non-compact symmetries, such as translations and to non-invariant data distributions. Building on the PAC-Bayes framework, we adapt and tighten existing bounds, demonstrating the approach on McAllester's PAC-Bayes bound while showing that it applies to a wide range of PAC-Bayes bounds. We validate our theory with experiments on a rotated MNIST dataset with a non-uniform rotation group, where the derived guarantees not only hold but also improve upon prior results. These findings provide theoretical evidence that, for symmetric data, symmetric models are preferable beyond the narrow setting of compact groups and invariant distributions, opening the way to a more general understanding of symmetries in machine learning.

Modern generative models, such as neural ordinary differential equations (neural ODEs) [4], transformers [25], and diffusion models [22], have demonstrated remarkable ability to learn and generate samples from complex, high-dimensional probability distributions. These architectures have achieved broad success in scientific computing, image processing, and data science, offering scalable frameworks for data-driven modeling. However, training and sampling in such spaces remain expensive and highly sensitive to architectural and optimization choices. Despite these advances, the curse of dimensionality continues to present a fundamental challenge in many real-world applications. Fortunately, numerous problems in scientific computing exhibit intrinsic structures, such as sparsity, low-rank representations, or approximate invariances, that can be interpreted as prior information about the underlying data or operators. Leveraging such priors within generative models offers a promising avenue to improve both computational efficiency and generalization. A classical way to incorporate prior information, such as sparsity or piecewise regularity, is through Bayesian modeling, where the posterior combines a prior distribution encoding structural knowledge with a likelihood function derived from observations.

Latent-variable energy-based models (LV-EBMs) assign a single normalized energy to joint pairs of observed data and latent variables, offering expressive generative modeling while capturing hidden structure. We recast maximum-likelihood training as a saddle problem over distributions on the latent and joint manifolds and view the inner updates as coupled Wasserstein gradient flows. The resulting algorithm alternates overdamped Langevin updates for a joint negative pool and for conditional latent particles with stochastic parameter ascent, requiring no discriminator or auxiliary networks. We prove existence and convergence under standard smoothness and dissi-pativity assumptions, with decay rates in KL divergence and Wasserstein-2 distance. The saddle-point view further yields an ELBO strictly tighter than bounds obtained with restricted amortized posteriors. Our method is evaluated on numerical approximations of physical systems and performs competitively against comparable approaches.