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Theory · Learning Theory

Itamar Harel, Yonathan Wolanowsky, Gal Vardi, Nati Srebro, Daniel Soudry

We analyze the generalization gap (gap between the training and test errors) when training a potentially over-parametrized model using a Markovian stochastic training algorithm, initialized from some distribution $\theta_0 \sim p_0$. We focus on Langevin dynamics with a positive temperature $\beta^{-1}$, i.e. gradient descent on a training loss $L$ with infinitesimal step size, perturbed with $\beta^{-1}$-variances Gaussian noise, and lightly regularized or bounded. There, we bound the generalization gap, *at any time during training*, by $\sqrt{(\beta\mathbb{E} L (\theta_0) + \log(1/\delta))/N}$ with probability $1-\delta$ over the dataset, where $N$ is the sample size, and $\mathbb{E} L(\theta_0)=O(1)$ with standard initialization scaling. In contrast to previous guarantees, we have no dependence on either training time or reliance on mixing, nor a dependence on dimensionality, gradient norms, or any other properties of the loss or model. This guarantee follows from a general analysis of any Markov process-based training that has a Gibbs-style stationary distribution. The proof is surprisingly simple, once we observe that the marginal distribution divergence from initialization remains bounded, as implied by a generalized second law of thermodynamics.

Optimization · Bayesian & Black-box Optimization

Artun Saday, Yaşar Cahit Yıldırım, Cem Tekin

We address the problem of Gaussian Process (GP) optimization in the presence of unknown and potentially varying adversarial perturbations. Unlike traditional robust optimization approaches that focus on maximizing performance under worst-case scenarios, we consider a robust satisficing objective, where the goal is to consistently achieve a predefined performance threshold $\tau$, even under adversarial conditions. We propose two novel algorithms based on distinct formulations of robust satisficing, and show that they are instances of a general robust satisficing framework. Further, each algorithm offers different guarantees depending on the nature of the adversary. Specifically, we derive two regret bounds: one that is sublinear over time, assuming certain conditions on the adversary and the satisficing threshold $\tau$, and another that scales with the perturbation magnitude but requires no assumptions on the adversary. Through extensive experiments, we demonstrate that our approach outperforms the established robust optimization methods in achieving the satisficing objective, particularly when the ambiguity set of the robust optimization framework is inaccurately specified.

Optimization · Bayesian & Black-box Optimization

Haowei Wang, Jingyi Wang, Zhongxiang Dai, Nai-Yuan Chiang, Szu Hui Ng, Cosmin Petra

Constrained Bayesian optimization (CBO) methods have seen significant success in black-box optimization with constraints. One of the most commonly used CBO methods is the constrained expected improvement (CEI) algorithm. CEI is a natural extension of expected improvement (EI) when constraints are incorporated. However, the theoretical convergence rate of CEI has not been established. In this work, we study the convergence rate of CEI by analyzing its simple regret upper bound. First, we show that when the objective function $f$ and constraint function $c$ are assumed to each lie in a reproducing kernel Hilbert space (RKHS), CEI achieves the convergence rates of $\mathcal{O} \left(t^{-\frac{1}{2}}\log^{\frac{d+1}{2}}(t) \right) \ \text{and }\ \mathcal{O}\left(t^{\frac{-\nu}{2\nu+d}} \log^{\frac{\nu}{2\nu+d}}(t)\right)$ for the commonly used squared exponential and Matérn kernels, respectively. Second, we show that when $f$ and $c$ are assumed to be sampled from Gaussian processes (GPs), CEI achieves the same convergence rates with a high probability. Numerical experiments are performed to validate the theoretical analysis.

Reinforcement Learning · Deep RL

Shutong Ding, Ke Hu, Shan Zhong, Haoyang Luo, Weinan Zhang, Jingya Wang, Jun Wang, Ye Shi

Recent advances in reinforcement learning (RL) have demonstrated the powerful exploration capabilities and multimodality of generative diffusion-based policies. While substantial progress has been made in offline RL and off-policy RL settings, integrating diffusion policies into on-policy frameworks like PPO remains underexplored. This gap is particularly significant given the widespread use of large-scale parallel GPU-accelerated simulators, such as IsaacLab, which are optimized for on-policy RL algorithms and enable rapid training of complex robotic tasks. A key challenge lies in computing state-action log-likelihoods under diffusion policies, which is straightforward for Gaussian policies but intractable for flow-based models due to irreversible forward-reverse processes and discretization errors (e.g., Euler-Maruyama approximations). To bridge this gap, we propose GenPO, a generative policy optimization framework that leverages exact diffusion inversion to construct invertible action mappings. GenPO introduces a novel doubled dummy action mechanism that enables invertibility via alternating updates, resolving log-likelihood computation barriers. Furthermore, we also use the action log-likelihood for unbiased entropy and KL divergence estimation, enabling KL-adaptive learning rates and entropy regularization in on-policy updates. Extensive experiments on eight IsaacLab benchmarks, including legged locomotion (Ant, Humanoid, Anymal-D, Unitree H1, Go2), dexterous manipulation (Shadow Hand), aerial control (Quadcopter), and robotic arm tasks (Franka), demonstrate GenPO’s superiority over existing RL baselines. Notably, GenPO is the first method to successfully integrate diffusion policies into on-policy RL, unlocking their potential for large-scale parallelized training and real-world robotic deployment.

Probabilistic Methods · Gaussian Processes

Masahiro Kohjima

Shuffled regression is the problem of learning regression functions from shuffled data where the correspondence between the input features and target response is unknown. This paper proposes a probabilistic model for shuffled regression called Gaussian Process Shuffled Regression (GPSR). By introducing Gaussian processes as a prior of regression functions in function space via the kernel function, GPSR can express a wide variety of functions in a nonparametric manner while quantifying the uncertainty of the prediction. By adopting the Bayesian evidence maximization framework and a theoretical analysis of the connection between the marginal likelihood/predictive distribution of GPSR and that of standard Gaussian process regression (GPR), we derive an easy-to-implement inference algorithm for GPSR that iteratively applies GPR and updates the input-output correspondence. To reduce computation costs and obtain closed-form solutions for correspondence updates, we also develop a sparse approximate variant of GPSR using its weight space formulation, which can be seen as Bayesian shuffled linear regression with random Fourier features. Experiments on benchmark datasets confirm the effectiveness of our GPSR proposal.

Deep Learning · Generative Models and Autoencoders

Moongyu Jeon, Sangwoo Shin, Dongjae Jeon, Albert No

We present an information-theoretic framework for discrete diffusion models that yields principled estimators of log-likelihood using score-matching losses. Inspired by the I-MMSE identity for the Gaussian setup, we derive analogous results for the discrete setting. Specifically, we introduce the Information–Minimum Denoising Score Entropy (I-MDSE) relation, which links mutual information between data and its diffused version to the minimum denoising score entropy (DSE) loss. We extend this theory to masked diffusion and establish the Information–Minimum Denoising Cross-Entropy (I-MDCE) relation, connecting cross-entropy losses to mutual information in discrete masked processes. These results provide a time-integral decomposition of the log-likelihood of the data in terms of optimal score-based losses, showing that commonly used losses such as DSE and DCE are not merely variational bounds but tight and principled estimators of log-likelihood. The I-MDCE decomposition further enables practical extensions, including time-free formula, conditional likelihood estimation in prompt–response tasks, and coupled Monte Carlo estimation of likelihood ratios. Experiments on synthetic and real-world data confirm the accuracy, variance stability, and utility of our estimators. The code is publicly available at https://github.com/Dongjae0324/infodis.

Deep Learning · Attention Mechanisms

Mana Sakai, Ryo Karakida, Masaaki Imaizumi

In modern theoretical analyses of neural networks, the infinite-width limit is often invoked to justify Gaussian approximations of neuron preactivations (e.g., via neural network Gaussian processes or Tensor Programs). However, these Gaussian-based asymptotic theories have so far been unable to capture the behavior of attention layers, except under special regimes such as infinitely many heads or tailored scaling schemes. In this paper, leveraging the Tensor Programs framework, we rigorously identify the infinite-width limit distribution of variables within a single attention layer under realistic architectural dimensionality and standard $1/\sqrt{n}$-scaling with $n$ dimensionality. We derive the exact form of this limit law without resorting to infinite-head approximations or tailored scalings, demonstrating that it departs fundamentally from Gaussianity. This limiting distribution exhibits non-Gaussianity from a hierarchical structure, being Gaussian conditional on the random similarity scores. Numerical experiments validate our theoretical predictions, confirming the effectiveness of our theory at finite width and accurate description of finite-head attentions. Beyond characterizing a standalone attention layer, our findings lay the groundwork for developing a unified theory of deep Transformer architectures in the infinite-width regime.

Applications · Time Series

Jiawen Wei, jiang lan, Pengbo Wei, Ziwen Ye, Teng Song, Chen Chen, Guangrui Ma

Time series data is ubiquitous, with forecasting applications spanning from finance to healthcare. Beyond popular deterministic methods, generative models are gaining attention due to advancements in areas like image synthesis and video generation, as well as their inherent ability to provide probabilistic predictions. However, existing generative approaches mostly involve recurrent generative operations or repeated denoising steps, making the prediction laborious, particularly for long-term forecasting. Most of them only conduct experiments for relatively short-term forecasting, with limited comparison to deterministic methods in long-term forecasting, leaving their practical advantages unclear. This paper presents TARFVAE, a novel generative framework that combines the Transformer-based autoregressive flow (TARFLOW) and variational autoencoder (VAE) for efficient one-step generative time series forecasting. Inspired by the rethinking that complex architectures for extracting time series representations might not be necessary, we add a flow module, TARFLOW, to VAE to promote spontaneous learning of latent variables that benefit predictions. TARFLOW enhances VAE's posterior estimation by breaking the Gaussian assumption, thereby enabling a more informative latent space. TARFVAE uses only the forward process of TARFLOW, avoiding autoregressive inverse operations and thus ensuring fast generation. During generation, it samples from the prior latent space and directly generates full-horizon forecasts via the VAE decoder. With simple MLP modules, TARFVAE achieves superior performance over state-of-the-art deterministic and generative models across different forecast horizons on benchmark datasets while maintaining efficient prediction speed, demonstrating its effectiveness as an efficient and powerful solution for generative time series forecasting. Our code is available at https://github.com/Gavine77/TARFVAE.

Probabilistic Methods · Gaussian Processes

Siavash Ameli, Chris van der Heide, Liam Hodgkinson, Michael Mahoney

Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in scale, the corresponding covariance and kernel matrices become increasingly large, often reaching magnitudes that make their direct formation impractical or impossible. Existing techniques typically rely on matrix-vector products, which can provide efficient approximations, if the matrix spectrum behaves well. However, in settings like distributed learning, or when the matrix is defined only indirectly, access to the full data set can be restricted to only very small sub-matrices of the original matrix. In these cases, the matrix of nominal interest is not even available as an implicit operator, meaning that even matrix-vector products may not be available. In such settings, the matrix is "impalpable", in the sense that we have access to only masked snapshots of it. We draw on principles from free probability theory to introduce a novel method of "free decompression" to estimate the spectrum of such matrices. Our method can be used to extrapolate from the empirical spectral densities of small submatrices to infer the eigenspectrum of extremely large (impalpable) matrices (that we cannot form or even evaluate with full matrix-vector products). We demonstrate the effectiveness of this approach through a series of examples, comparing its performance against known limiting distributions from random matrix theory in synthetic settings, as well as applying it to submatrices of real-world datasets, matching them with their full empirical eigenspectra.

Theory · Reinforcement Learning and Planning

Jasmine Bayrooti, Sattar Vakili, Amanda Prorok, Carl Henrik Ek

Thompson sampling (TS) is a powerful and widely used strategy for sequential decision-making, with applications ranging from Bayesian optimization to reinforcement learning (RL). Despite its success, the theoretical foundations of TS remain limited, particularly in settings with complex temporal structure such as RL. We address this gap by establishing no-regret guarantees for TS using models with Gaussian marginal distributions. Specifically, we consider TS in episodic RL with joint Gaussian process (GP) priors over rewards and transitions. We prove a regret bound of $\mathcal{\tilde{O}}(\sqrt{KH\Gamma(KH)})$ over $K$ episodes of horizon $H$, where $\Gamma(\cdot)$ captures the complexity of the GP model. Our analysis addresses several challenges, including the non-Gaussian nature of value functions and the recursive structure of Bellman updates, and extends classical tools such as the elliptical potential lemma to multi-output settings. This work advances the understanding of TS in RL and highlights how structural assumptions and model uncertainty shape its performance in finite-horizon Markov Decision Processes.

Computer Vision · 3D Rendering & Reconstruction

Chensheng Dai, Shengjun Zhang, Min Chen, Yueqi Duan

3D Gaussian Splatting (3DGS) has demonstrated impressive performance in 3D scene reconstruction. Beyond novel view synthesis, it shows great potential for multi-view surface reconstruction. Existing methods employ optimization-based reconstruction pipelines that achieve precise and complete surface extractions. However, these approaches typically require dense input views and high time consumption for per-scene optimization. To address these limitations, we propose SurfaceSplat, a feed-forward framework that generates efficient and generalizable pixel-aligned Gaussian surfel representations from sparse-view images. We observe that conventional feed-forward structures struggle to recover accurate geometric attributes of Gaussian surfels because the spatial frequency of pixel-aligned primitives exceeds Nyquist sampling rates. Therefore, we propose a cross-view feature aggregation module based on the Nyquist sampling theorem. Specifically, we first adapt the geometric forms of Gaussian surfels with spatial sampling rate-guided low-pass filters. We then project the filtered surfels across all input views to obtain cross-view feature correlations. By processing these correlations through a specially designed feature fusion network, we can finally regress Gaussian surfels with precise geometry. Extensive experiments on DTU reconstruction benchmarks demonstrate that our model achieves comparable results with state-of-the-art methods, and predict Gaussian surfels within 1 second, offering a 100× speedup without costly per-scene training.

Probabilistic Methods

Wanpeng Zhang, Yuhao Fang, Xihang Qiu, Jiarong Cheng, Jialong Hong, Bin Zhai, Qing Zhou, Yao Lu, Ye Zhang, Chun Li

Diffusion-based generative models have long depended on Gaussian priors, with little exploration of alternative distributions. We introduce a Proper Hölder-Kullback Dirichlet framework that uses time-varying multiplicative transformations to define both forward and reverse diffusion processes. Moving beyond conventional reweighted evidence lower bounds (ELBO) or Kullback–Leibler upper bounds (KLUB), we propose two novel divergence measures: the Proper Hölder Divergence (PHD) and the Proper Hölder–Kullback (PHK) divergence, the latter designed to restore symmetry missing in existing formulations. When optimizing our Dirichlet diffusion model with PHK, we achieve a Fréchet Inception Distance (FID) of 2.78 on unconditional CIFAR-10. Comprehensive experiments on natural-image datasets validate the generative strengths of model and confirm PHK’s effectiveness in model training. These contributions expand the diffusion-model family with principled non-Gaussian processes and effective optimization tools, offering new avenues for versatile, high-fidelity generative modeling.

Applications · Health

Bin Pu, Liwen Wang, Xingbo Dong, Xingguo Lv, ZHE JIN

Accurate anatomical structure detection is a critical preliminary step for diagnosing diseases characterized by structural abnormalities. In clinical practice, medical experts frequently adjust the zoom level of medical images to obtain comprehensive views for diagnosis. This common interaction results in significant variations in the apparent scale of anatomical structures across different images or fields of view. However, the information embedded in these zoom-induced scale changes is often overlooked by existing detection algorithms. In addition, human organs possess a priori, fixed topological knowledge. To overcome this limitation, we propose ZR-DETR, a zoom-aware probabilistic framework tailored for medical object detection. ZR-DETR uniquely incorporates scale-sensitive zoom embeddings, anatomical relation constraints, and a Gaussian Process-based detection head. This architecture enables the framework to jointly model semantic context, enforce anatomical plausibility, and quantify detection uncertainty. Empirical validation across three diverse medical imaging benchmarks demonstrates that ZR-DETR consistently outperforms strong baselines in both single-domain and unsupervised domain adaptation scenarios.

Zahra Babaiee, Peyman M. Kiasari, Daniela Rus, Radu Grosu

A recent study has proposed the ``Master Key Filters Hypothesis" for convolutional neural network filters. This paper extends this hypothesis by radically constraining its scope to a single set of just 8 universal filters that depthwise separable convolutional networks inherently converge to. While conventional DS-CNNs employ thousands of distinct trained filters, our analysis reveals these filters are predominantly linear shifts (ax+b) of our discovered universal set. Through systematic unsupervised search, we extracted these fundamental patterns across different architectures and datasets. Remarkably, networks initialized with these 8 unique frozen filters achieve over 80\% ImageNet accuracy, and even outperform models with thousands of trainable parameters when applied to smaller datasets. The identified master key filters closely match Difference of Gaussians (DoGs), Gaussians, and their derivatives, structures that are not only fundamental to classical image processing but also strikingly similar to receptive fields in mammalian visual systems. Our findings provide compelling evidence that depthwise convolutional layers naturally gravitate toward this fundamental set of spatial operators regardless of task or architecture. This work offers new insights for understanding generalization and transfer learning through the universal language of these master key filters.

Deep Learning · Generative Models and Autoencoders

Yirong Shen, Lu GAN, Cong Ling

Generative models that maximize model likelihood have gained traction in many practical settings. Among them, perturbation-based approaches underpin many state-of-the-art likelihood estimation models, yet they often face slow convergence and limited theoretical understanding. In this paper, we derive a tighter likelihood bound for noise-driven models to improve both the accuracy and efficiency of maximum likelihood learning. Our key insight extends the classical Kullback–Leibler (KL) divergence–Fisher information relationship to arbitrary noise perturbations, going beyond the Gaussian assumption and enabling structured noise distributions. This formulation allows flexible use of randomized noise distributions that naturally account for sensor artifacts, quantization effects, and data distribution smoothing, while remaining compatible with standard diffusion training. Treating the diffusion process as a Gaussian channel, we further express the mismatched entropy between data and model, showing that the proposed objective upper-bounds the negative log-likelihood (NLL). In experiments, our models achieve competitive NLL on CIFAR-10 and state-of-the-art results on ImageNet across multiple resolutions, all without data augmentation, and the framework extends naturally to discrete data.

Applications · Chemistry and Drug Discovery

Shayan Shekarforoush, David Lindell, Marcus Brubaker, David Fleet

Cryo-EM is a transformational paradigm in molecular biology where computational methods are used to infer 3D molecular structure at atomic resolution from extremely noisy 2D electron microscope images. At the forefront of research is how to model the structure when the imaged particles exhibit non-rigid conformational flexibility and compositional variation where parts are sometimes missing. We introduce a novel 3D reconstruction framework with a hierarchical Gaussian mixture model, inspired in part by Gaussian Splatting for 4D scene reconstruction. In particular, the structure of the model is grounded in an initial process that infers a part-based segmentation of the particle, providing essential inductive bias in order to handle both conformational and compositional variability. The framework, called \methodName, is shown to reveal biologically meaningful structures on complex experimental datasets, and establishes a new state-of-the-art on CryoBench, a benchmark for cryo-EM heterogeneity methods.

Mingxu Chai, Ziyu Shen, Chong Zhang, Yue Zhang, Xiao Wang, Shihan Dou, Jihua Kang, Jiazheng Zhang, Qi Zhang

Document parsing involves layout element detection and recognition, essential for extracting information. However, existing methods often employ multiple models for these tasks, leading to increased system complexity and maintenance overhead. While some models attempt to unify detection and recognition, they often fail to address the intrinsic differences in data representations, thereby limiting performance in document processing. Our research reveals that recognition relies on discrete tokens, whereas detection relies on continuous coordinates, leading to challenges in gradient updates and optimization. To bridge this gap, we propose the Gaussian-Kernel Cross-Entropy Loss (GK-CEL), enabling generative frameworks to handle both tasks simultaneously. Building upon GK-CEL, we propose DocFusion, a unified document parsing model with only 0.28B parameters. Additionally, we construct the DocLatex-1.6M dataset to provide high-quality training support. Experimental results show that DocFusion, equipped with GK-CEL, performs competitively across four core document parsing tasks, validating the effectiveness of our unified approach.

Rongguang Ye, Ming Tang

Existing pruning methods for large language models (LLMs) focus on achieving high compression rates while maintaining model performance. Although these methods have demonstrated satisfactory performance in handling a single user’s compression request, their processing time increases linearly with the number of requests, making them inefficient for real-world scenarios with multiple simultaneous requests. To address this limitation, we propose a Univeral Model for Customized Compression (UniCuCo) for LLMs, which introduces a StratNet that learns to map arbitrary requests to their optimal pruning strategy. The challenge in training StratNet lies in the high computational cost of evaluating pruning strategies and the non-differentiable nature of the pruning process, which hinders gradient backpropagation for StratNet updates. To overcome these challenges, we leverage a Gaussian process to approximate the evaluation process. Since the gradient of the Gaussian process is computable, we can use it to approximate the gradient of the non-differentiable pruning process, thereby enabling StratNet updates. Experimental results show that UniCuCo is 28 times faster than baselines in processing 64 requests, while maintaining comparable accuracy to baselines.

Deep Learning · Algorithms

Ganchao Wei, Li Ma

Flow matching (FM) is a family of training algorithms for fitting continuous normalizing flows (CNFs). Conditional flow matching (CFM) exploits the fact that the marginal vector field of a CNF can be learned by fitting least-squares regression to the conditional vector field specified given one or both ends of the flow path. In this paper, we extend the CFM algorithm by defining conditional probability paths along "streams'', instances of latent stochastic paths that connect data pairs of source and target, which are modeled with Gaussian process (GP) distributions. The unique distributional properties of GPs help preserve the ``simulation-free'' nature of CFM training. We show that this generalization of the CFM can effectively reduce the variance in the estimated marginal vector field at a moderate computational cost, thereby improving the quality of the generated samples under common metrics. Additionally, adopting the GP on the streams allows for flexibly linking multiple correlated training data points (e.g., time series). We empirically validate our claim through both simulations and applications to image and neural time series data.

Probabilistic Methods · Gaussian Processes

Henry Moss, Sebastian Ober, Tom Diethe

Bayesian optimisation in the latent space of a VAE is a powerful framework for optimisation tasks over complex structured domains, such as the space of valid molecules. However, existing approaches tightly couple the surrogate and generative models, which can lead to suboptimal performance when the latent space is not tailored to specific tasks, which in turn has led to the proposal of increasingly sophisticated algorithms. In this work, we explore a new direction, instead proposing a decoupled approach that trains a generative model and a GP surrogate separately, then combines them via a simple yet principled Bayesian update rule. This separation allows each component to focus on its strengths— structure generation from the VAE and predictive modelling by the GP. We show that our decoupled approach improves our ability to identify high-potential candidates in molecular optimisation problems under constrained evaluation budgets.