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Probabilistic Methods · Monte Carlo and Sampling Methods

Filip Ekström Kelvinius, Zheng Zhao, Fredrik Lindsten

A recent line of research has exploited pre-trained generative diffusion models as priors for solving Bayesian inverse problems. We contribute to this research direction by designing a sequential Monte Carlo method for linear-Gaussian inverse problems which builds on ``decoupled diffusion", where the generative process is designed such that larger updates to the sample are possible. The method is asymptotically exact and we demonstrate the effectiveness of our Decoupled Diffusion Sequential Monte Carlo (DDSMC) algorithm on both synthetic as well as protein and image data. Further, we demonstrate how the approach can be extended to discrete data.

Deep Learning · Large Language Models

Lennart Schneider, Martin Wistuba, Aaron Klein, Jacek Golebiowski, Giovanni Zappella, Felice Antonio Merra

Optimal prompt selection is crucial for maximizing large language model (LLM) performance on downstream tasks, especially in black-box settings where models are only accessible via APIs.Black-box prompt selection is challenging due to potentially large, combinatorial search spaces, absence of gradient information, and high evaluation cost of prompts on a validation set.We propose HbBoPs, a novel method that combines a structural-aware deep kernel Gaussian Process with Hyperband as a multi-fidelity scheduler to efficiently select prompts.HbBoPs uses embeddings of instructions and few-shot exemplars, treating them as modular components within prompts.This enhances the surrogate model's ability to predict which prompt to evaluate next in a sample-efficient manner.Hyperband improves query-efficiency by adaptively allocating resources across different fidelity levels, reducing the number of validation instances required for evaluating prompts.Extensive experiments across ten diverse benchmarks and three LLMs demonstrate that HbBoPs outperforms state-of-the-art methods in both performance and efficiency.

Probabilistic Methods · Gaussian Processes

Wanli Hong, Yuliang Shi, Jonathan Niles-Weed

Motivated by applications in trajectory inference and particle tracking, we introduce **Smooth Schrödinger Bridges**. Our proposal generalizes prior work by allowing the reference process in the multi-marginal Schrödinger Bridge problem to be a smooth Gaussian process, leading to more regular and interpretable trajectories in applications. Though naïvely smoothing the reference process leads to a computationally intractable problem, we identify a class of processes (including the Matérn processes) for which the resulting Smooth Schrödinger Bridge problem can be *lifted* to a simpler problem on phase space, which can be solved in polynomial time. We develop a practical approximation of this algorithm that outperforms existing methods on numerous simulated and real single-cell RNAseq datasets.

General Machine Learning · Causality

Quang-Duy Tran, Bao Duong, Phuoc Nguyen, Thin Nguyen

Telling apart the cause and effect between two random variables with purely observational data is a challenging problem that finds applications in various scientific disciplines. A key principle utilized in this task is the algorithmic Markov condition, which postulates that the joint distribution, when factorized according to the causal direction, yields a more succinct codelength compared to the anti-causal direction. Previous approaches approximate these codelengths by relying on simple functions or Gaussian processes (GPs) with easily evaluable complexity, compromising between model fitness and computational complexity. To overcome these limitations, we propose leveraging the variational Bayesian learning of neural networks as an interpretation of the codelengths. Consequently, we can enhance the model fitness while promoting the succinctness of the codelengths, while avoiding the significant computational complexity of the GP-based approaches. Extensive experiments on both synthetic and real-world benchmarks in cause-effect identification demonstrate the effectiveness of our proposed method, surpassing the overall performance of related complexity-based and structural causal model regression-based approaches.

Optimization · Zero-order and Black-box Optimization

Toby Boyne, Jose Pablo Folch, Robert Lee, Behrang Shafei, Ruth Misener

We perform Bayesian optimization using a Gaussian process perspective on Bayesian Additive Regression Trees (BART). Our BART Kernel (BARK) uses tree agreement to define a posterior over piecewise-constant functions, and we explore the space of tree kernels using a Markov chain Monte Carlo approach. Where BART only samples functions, the resulting BARK model obtains samples of Gaussian processes defining distributions over functions, which allow us to build acquisition functions for Bayesian optimization. Our tree-based approach enables global optimization over the surrogate, even for mixed-feature spaces. Moreover, where many previous tree-based kernels provide uncertainty quantification over function values, our sampling scheme captures uncertainty over the tree structure itself. Our experiments show the strong performance of BARK on both synthetic and applied benchmarks, due to the combination of our fully Bayesian surrogate and the optimization procedure.

Probabilistic Methods · Bayesian Models and Methods

Emilia Magnani, Marvin Pförtner, Tobias Weber, Philipp Hennig

Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs. However, to make them useful in high-stakes simulation scenarios, their inherent predictive error must be quantified reliably. We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator's predictions.We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief. Our framework provides a practical yet theoretically sound way to apply existing Bayesian deep learning methods such as the linearized Laplace approximation to neural operators. Just as the underlying neural operator, our approach is resolution-agnostic by design.The method adds minimal prediction overhead, can be applied post-hoc without retraining the network, and scales to large models and datasets.We evaluate these aspects in a case study on Fourier neural operators.

Optimization · Zero-order and Black-box Optimization

Leonard Papenmeier, Matthias Poloczek, Luigi Nardi

Recent work reported that simple Bayesian optimization (BO) methods perform well for high-dimensional real-world tasks, seemingly contradicting prior work and tribal knowledge. This paper investigates why. We identify underlying challenges that arise in high-dimensional BO and explain why recent methods succeed. Our empirical analysis shows that vanishing gradients caused by Gaussian process (GP) initialization schemes play a major role in the failures of high-dimensional Bayesian optimization (HDBO) and that methods that promote local search behaviors are better suited for the task. We find that maximum likelihood estimation (MLE) of GP length scales suffices for state-of-the-art performance. Based on this, we propose a simple variant of MLE called MSR that leverages these findings to achieve state-of-the-art performance on a comprehensive set of real-world applications. We present targeted experiments to illustrate and confirm our findings.

Theory · Everything Else

Alexander Moreno, Justin Xiao, Jonathan Mei

Structured Kernel Interpolation (SKI) scales Gaussian Processes (GPs) by approximating the kernel matrix via inducing point interpolation, achieving linear computational complexity. However, it lacks rigorous theoretical error analysis. This paper bridges this gap by proving error bounds for the SKI Gram matrix and examining their effect on hyperparameter estimation and posterior inference. We further provide a practical guide to selecting the number of inducing points under convolutional cubic interpolation: they should grow as \(n^{d/3}\) for error control. Crucially, we identify two dimensionality regimes for the SKI Gram matrix spectral norm error vs. complexity trade-off. For \(d<3\), \textit{any} error tolerance can achieve linear time for sufficiently large sample size. For \(d\geq 3\), the error must \textit{increase} with sample size for our guarantees to hold. Our analysis provides key insights into SKI's scalability-accuracy trade-offs, establishing precise conditions for achieving linear-time GP inference with controlled error.

Probabilistic Methods · Gaussian Processes

Xinxing Shi, Xiaoyu Jiang, Mauricio Álvarez

Gaussian Process (GP) Variational Autoencoders (VAEs) extend standard VAEs by replacing the fully factorised Gaussian prior with a GP prior, thereby capturing richer correlations among latent variables. However, performing exact GP inference in large-scale GPVAEs is computationally prohibitive, often forcing existing approaches to rely on restrictive kernel assumptions or large sets of inducing points. In this work, we propose a neighbour-driven approximation strategy that exploits local adjacencies in the latent space to achieve scalable GPVAE inference. By confining computations to the nearest neighbours of each data point, our method preserves essential latent dependencies, allowing more flexible kernel choices and mitigating the need for numerous inducing points. Through extensive experiments on tasks including representation learning, data imputation, and conditional generation, we demonstrate that our approach outperforms other GPVAE variants in both predictive performance and computational efficiency.

Applications · Robotics

Alexandre Capone, Ryan Cosner, Aaron Ames, Sandra Hirche

Control tasks with safety requirements under high levels of model uncertainty are increasingly common. Machine learning techniques are frequently used to address such tasks, typically by leveraging model error bounds to specify robust constraint-based safety filters. However, if the learned model uncertainty is very high, the corresponding filters are potentially invalid, meaning no control input satisfies the constraints imposed by the safety filter. While most works address this issue by assuming some form of safe backup controller, ours tackles it by collecting additional data on the fly using a Gaussian process bandit-type algorithm. We combine a control barrier function with a learned model to specify a robust certificate that ensures safety if feasible. Whenever infeasibility occurs, we leverage the control barrier function to guide exploration, ensuring the collected data contributes toward the closed-loop system safety. By combining a safety filter with exploration in this manner, our method provably achieves safety in a general setting that does not require any prior model or backup controller, provided that the true system lies in a reproducing kernel Hilbert space. To the best of our knowledge, it is the first safe learning-based control method that achieves this.

Probabilistic Methods · Gaussian Processes

Yuhao Liu, Yu Chen, Rui Hu, Longbo Huang

The stochastic interpolant framework offers a powerful approach for constructing generative models based on ordinary differential equations (ODEs) or stochastic differential equations (SDEs) to transform arbitrary data distributions. However, prior analyses of this framework have primarily focused on the continuous-time setting, assuming perfect solution of the underlying equations. In this work, we present the first discrete-time analysis of the stochastic interpolant framework, where we introduce a innovative discrete-time sampler and derive a finite-time upper bound on its distribution estimation error. Our result provides a novel quantification on how different factors, including the distance between source and target distributions and estimation accuracy, affect the convergence rate and also offers a new principled way to design efficient schedule for convergence acceleration. Finally, numerical experiments are conducted on the discrete-time sampler to corroborate our theoretical findings.

Deep Learning · Generative Models and Autoencoders

Ching-Hua Lee, Chouchang Yang, Jaejin Cho, Yashas Malur Saidutta, Rakshith Sharma Srinivasa, Yilin Shen, Hongxia Jin

Denoising diffusion probabilistic models (DDPMs) can be utilized to recover a clean signal from its degraded observation(s) by conditioning the model on the degraded signal. The degraded signals are themselves contaminated versions of the clean signals; due to this correlation, they may encompass certain useful information about the target clean data distribution. However, existing adoption of the standard Gaussian as the prior distribution in turn discards such information when shaping the prior, resulting in sub-optimal performance. In this paper, we propose to improve conditional DDPMs for signal restoration by leveraging a more informative prior that is jointly learned with the diffusion model. The proposed framework, called RestoreGrad, seamlessly integrates DDPMs into the variational autoencoder (VAE) framework, taking advantage of the correlation between the degraded and clean signals to encode a better diffusion prior. On speech and image restoration tasks, we show that RestoreGrad demonstrates faster convergence (5-10 times fewer training steps) to achieve better quality of restored signals over existing DDPM baselines and improved robustness to using fewer sampling steps in inference time (2-2.5 times fewer), advocating the advantages of leveraging jointly learned prior for efficiency improvements in the diffusion process.

Social Aspects

Hanshen Xiao, Zhen Yang, Edward Suh

This paper studies a range of AI/ML trust concepts, including memorization, data poisoning, and copyright, which can be modeled as constraints on the influence of data on a (trained) model, characterized by the outcome difference from a processing function (training algorithm). In this realm, we show that provable trust guarantees can be efficiently provided through a new framework termed Data-Specific Indistinguishability (DSI) to select trust-preserving randomization tightly aligning with targeted outcome differences, as a relaxation of the classic Input-Independent Indistinguishability (III). We establish both the theoretical and algorithmic foundations of DSI with the optimal multivariate Gaussian mechanism. We further show its applications to develop trustworthy deep learning with black-box optimizers. The experimental results on memorization mitigation, backdoor defense, and copyright protection show both the efficiency and effectiveness of the DSI noise mechanism.

Probabilistic Methods · Bayesian Models and Methods

Mohamad Al Ahdab, john leth, Zheng-Hua Tan

We study the Continuous-Discrete Kalman Filter (CD-KF) for State-Space Models (SSMs) where continuous-time dynamics are observed via multiple sensors with discrete, irregularly timed measurements. Our focus extends to scenarios in which the measurement process is coupled with the states of an auxiliary SSM. For instance, higher measurement rates may increase energy consumption or heat generation, while a sensor’s accuracy can depend on its own spatial trajectory or that of the measured target. Each sensor thus carries distinct costs and constraints associated with its measurement rate and additional constraints and costs on the auxiliary state. We model measurement occurrences as independent Poisson processes with sensor-specific rates and derive an upper bound on the mean posterior covariance matrix of the CD-KF along the mean auxiliary state. The bound is continuously differentiable with respect to the measurement rates, which enables efficient gradient-based optimization. Exploiting this bound, we propose a finite-horizon optimal control framework to optimize measurement rates and auxiliary-state dynamics jointly. We further introduce a deterministic method for scheduling measurement times from the optimized rates. Empirical results in state-space filtering and dynamic temporal Gaussian process regression demonstrate that our approach achieves improved trade-offs between resource usage and estimation accuracy.

Probabilistic Methods · Monte Carlo and Sampling Methods

Francis Bach, Saeed Saremi

Gaussian smoothing combined with a probabilistic framework for denoising via the empirical Bayes formalism, i.e., the Tweedie-Miyasawa formula (TMF), are the two key ingredients in the success of score-based generative models in Euclidean spaces. Smoothing holds the key for easing the problem of learning and sampling in high dimensions, denoising is needed for recovering the original signal, and TMF ties these together via the score function of noisy data. In this work, we extend this paradigm to the problem of learning and sampling the distribution of binary data on the Boolean hypercube by adopting Bernoulli noise, instead of Gaussian noise, as a smoothing device. We first derive a TMF-like expression for the optimal denoiser for the Hamming loss, where a score function naturally appears. Sampling noisy binary data is then achieved using a Langevin-like sampler which we theoretically analyze for different noise levels. At high Bernoulli noise levels sampling becomes easy, akin to log-concave sampling in Euclidean spaces. In addition, we extend the sequential multi-measurement sampling of Saremi et al. (2024) to the binary setting where we can bring the "effective noise" down by sampling multiple noisy measurements at a fixed noise level, without the need for continuous-time stochastic processes. We validate our formalism and theoretical findings by experiments on synthetic data and binarized images.

Applications · Computer Vision

Weihan Li, Linyun Zhou, YangJian, Shengxuming Zhang, Xiangtong Du, Xiuming Zhang, Jing Zhang, Chaoqing Xu, Mingli Song, Zunlei Feng

Pathology image segmentation plays a pivotal role in artificial digital pathology diagnosis and treatment. Existing approaches to pathology image segmentation are hindered by labor-intensive annotation processes and limited accuracy in tail-class identification, primarily due to the long-tail distribution inherent in gigapixel pathology images. In this work, we introduce the Laplace Diffusion Model, referred to as L-Diffusion, an innovative framework tailored for efficient pathology image segmentation. L-Diffusion utilizes multiple Laplace distributions, as opposed to Gaussian distributions, to model distinct components—a methodology supported by theoretical analysis that significantly enhances the decomposition of features within the feature space. A sequence of feature maps is initially generated through a series of diffusion steps. Following this, contrastive learning is employed to refine the pixel-wise vectors derived from the feature map sequence. By utilizing these highly discriminative pixel-wise vectors, the segmentation module achieves a harmonious balance of precision and robustness with remarkable efficiency. Extensive experimental evaluations demonstrate that L-Diffusion attains improvements of up to 7.16\%, 26.74\%, 16.52\%, and 3.55\% on tissue segmentation datasets, and 20.09\%, 10.67\%, 14.42\%, and 10.41\% on cell segmentation datasets, as quantified by DICE, MPA, mIoU, and FwIoU metrics. The source are available at https://github.com/Lweihan/LDiffusion.

Marvin Li, Aayush Karan, Sitan Chen

Large language models can exhibit unexpected behavior in the blink of an eye. In a recent computer use demo, a language model switched from coding to Googling pictures of Yellowstone, and these sudden shifts in behavior have also been observed in reasoning patterns and jailbreaks. This phenomenon is not unique to autoregressive models: in diffusion models, key features of the final output are decided in narrow ``critical windows'' of the generation process. In this work we develop a simple, unifying theory to explain this phenomenon. Using the formalism of stochastic localization for generative models, we show that it emerges generically as the generation process localizes to a sub-population of the distribution it models. While critical windows have been studied at length in diffusion models, existing theory heavily relies on strong distributional assumptions and the particulars of Gaussian diffusion. In contrast to existing work our theory (1) applies to autoregressive and diffusion models; (2) makes very few distributional assumptions; (3) quantitatively improves previous bounds even when specialized to diffusions; and (4) requires basic mathematical tools. Finally, we validate our predictions empirically for LLMs and find that critical windows often coincide with failures in problem solving for various math and reasoning benchmarks.

Weihan Li, Yule Wang, Chengrui Li, Anqi Wu

Understanding and constructing brain communications that capture dynamic communications across multiple regions is fundamental to modern system neuroscience, yet current methods struggle to find time-varying region-level communications or scale to large neural datasets with long recording durations. We present a novel framework using Markovian Gaussian Processes to learn brain communications with time-varying temporal delays from multi-region neural recordings, named Adaptive Delay Model (ADM). Our method combines Gaussian Processes with State Space Models and employs parallel scan inference algorithms, enabling efficient scaling to large datasets while identifying concurrent communication patterns that evolve over time. This time-varying approach captures how brain region interactions shift dynamically during cognitive processes. Validated on synthetic and multi-region neural recordings datasets, our approach discovers both the directionality and temporal dynamics of neural communication. This work advances our understanding of distributed neural computation and provides a scalable tool for analyzing dynamic brain networks. Code is available at https://github.com/BRAINML-GT/Adaptive-Delay-Model.

Shogo Iwazaki, Shion Takeno

We study the Gaussian process (GP) bandit problem, whose goal is to minimize regret under an unknown reward function lying in some reproducing kernel Hilbert space (RKHS). The maximum posterior variance analysis is vital in analyzing near-optimal GP bandit algorithms such as maximum variance reduction (MVR) and phased elimination (PE).Therefore, we first show the new upper bound of the maximum posterior variance, which improves the dependence of the noise variance parameters of the GP. By leveraging this result, we refine the MVR and PE to obtain (i) a nearly optimal regret upper bound in the noiseless setting and (ii) regret upper bounds that are optimal with respect to the RKHS norm of the reward function. Furthermore, as another application of our proposed bound, we analyze the GP bandit under the time-varying noise variance setting, which is the kernelized extension of the linear bandit with heteroscedastic noise. For this problem, we show that MVR and PE-based algorithms achieve noise variance-dependent regret upper bounds, which matches our regret lower bound.

Reinforcement Learning · Deep RL

Vivienne Huiling Wang, Tinghuai Wang, Joni Pajarinen

Hierarchical reinforcement learning (HRL) learns to make decisions on multiple levels of temporal abstraction. A key challenge in HRL is that the low-level policy changes over time, making it difficult for the high-level policy to generate effective subgoals. To address this issue, the high-level policy must capture a complex subgoal distribution while also accounting for uncertainty in its estimates. We propose an approach that trains a conditional diffusion model regularized by a Gaussian Process (GP) prior to generate a complex variety of subgoals while leveraging principled GP uncertainty quantification. Building on this framework, we develop a strategy that selects subgoals from both the diffusion policy and GP's predictive mean. Our approach outperforms prior HRL methods in both sample efficiency and performance on challenging continuous control benchmarks.