Denoising diffusion models have driven significant progress in the field of Bayesian inverse problems. Recent approaches use pre-trained diffusion models as priors to solve a wide range of such problems, only leveraging inference-time compute and thereby eliminating the need to retrain task-specific models on the same dataset. To approximate the posterior of a Bayesian inverse problem, a diffusion model samples from a sequence of intermediate posterior distributions, each with an intractable likelihood function. This work proposes a novel mixture approximation of these intermediate distributions. Since direct gradient-based sampling of these mixtures is infeasible due to intractable terms, we propose a practical method based on Gibbs sampling. We validate our approach through extensive experiments on image inverse problems, utilizing both pixel- and latent-space diffusion priors, as well as on source separation with an audio diffusion model. The code is available at https://www.github.com/badr-moufad/mgdm

This paper introduces Type 2 Tobit Bayesian Additive Regression Trees (TOBART-2). BART can produce accurate individual-specific treatment effect estimates. However, in practice estimates are often biased by sample selection. We extend the Type 2 Tobit sample selection model to account for nonlinearities and model uncertainty by including sums of trees in both the selection and outcome equations. A Dirichlet Process Mixture distribution for the error terms allows for departure from the assumption of bivariate normally distributed errors. Soft trees and a Dirichlet prior on splitting probabilities improve modeling of smooth and sparse data generating processes. We include a simulation study and an application to the RAND Health Insurance Experiment data set.

We propose a novel Bregman descent algorithm for minimizing a convex function that is expressed as the sum of a differentiable part (defined over an open set) and a possibly nonsmooth term. The approach, referred to as the Variable Bregman Majorization-Minimization (VBMM) algorithm, extends the Bregman Proximal Gradient method by allowing the Bregman function used in the divergence to adaptively vary at each iteration, provided it satisfies a majorizing condition on the objective function. This adaptive framework enables the algorithm to approximate the objective more precisely at each iteration, thereby allowing for accelerated convergence compared to the traditional Bregman Proximal Gradient descent. We establish the convergence of the VBMM algorithm to a minimizer under mild assumptions on the family of metrics used. Furthermore, we introduce a novel application of both the Bregman Proximal Gradient method and the VBMM algorithm to the estimation of the multidimensional parameters of a Dirichlet distribution through the maximization of its log-likelihood. Numerical experiments confirm that the VBMM algorithm outperforms existing approaches in terms of convergence speed.

I think the distributed setting and all the subtleties that come with it should have been explored better. I was left wondering about the communication costs of an architecture needed for this to work, and potential issues with that. One obvious question would be, how would the iterate updates at each node be impacted by random noise injected into the w_{js} being passed around in the comms channels and/or random drops / missed updates. The only difference between the convergence discussion in \S4.1 and other works in the literature that use similar machinery seems to be the formulations for the extra constants/iterate weights in the distributed setting. This reduces the novelty/significance of that section somewhat, in my opinion.

The paper adresses the important problem of Bayesian inference in a distributed setting, via a decentralized Langevin algorithm. Although the method is a natural extension of existing algorithms, its simplicity is an advantage, and the theoretical analysis is nontrivial. After considering the author's response, all reviewers agreed that the paper will make a nice contribution to Neurips.

We address the challenge of minimizing true risk in multi-node distributed learning. These systems are frequently exposed to both inter-node and intra-node label shifts, which present a critical obstacle to effectively optimizing model performance while ensuring that data remains confined to each node. To tackle this, we propose the Versatile Robust Label Shift (VRLS) method, which enhances the maximum likelihood estimation of the test-to-train label density ratio. VRLS incorporates Shannon entropy-based regularization and adjusts the density ratio during training to better handle label shifts at the test time. In multi-node learning environments, VRLS further extends its capabilities by learning and adapting density ratios across nodes, effectively mitigating label shifts and improving overall model performance. Experiments conducted on MNIST, Fashion MNIST, and CIFAR-10 demonstrate the effectiveness of VRLS, outperforming baselines by up to 20% in imbalanced settings. These results highlight the significant improvements VRLS offers in addressing label shifts. Our theoretical analysis further supports this by establishing high-probability bounds on estimation errors.

In many practical applications, regression models are employed to uncover relationships between predictors and a response variable, yet the common assumption of constant error variance is frequently violated. This issue is further compounded in high-dimensional settings where the number of predictors exceeds the sample size, necessitating regularization for effective estimation and variable selection. To address this problem, we propose the Heteroscedastic Double Bayesian Elastic Net (HDBEN), a novel framework that jointly models the mean and log-variance using hierarchical Bayesian priors incorporating both $\ell_1$ and $\ell_2$ penalties. Our approach simultaneously induces sparsity and grouping in the regression coefficients and variance parameters, capturing complex variance structures in the data. Theoretical results demonstrate that proposed HDBEN achieves posterior concentration, variable selection consistency, and asymptotic normality under mild conditions which justifying its behavior. Simulation studies further illustrate that HDBEN outperforms existing methods, particularly in scenarios characterized by heteroscedasticity and high dimensionality.

Building neural reward models from human preferences is a pivotal component in reinforcement learning from human feedback (RLHF) and large language model alignment research. Given the scarcity and high cost of human annotation, how to select the most informative pairs to annotate is an essential yet challenging open problem. In this work, we highlight the insight that an ideal comparison dataset for reward modeling should balance exploration of the representation space and make informative comparisons between pairs with moderate reward differences. Technically, challenges arise in quantifying the two objectives and efficiently prioritizing the comparisons to be annotated. To address this, we propose the Fisher information-based selection strategies, adapt theories from the classical experimental design literature, and apply them to the final linear layer of the deep neural network-based reward modeling tasks. Empirically, our method demonstrates remarkable performance, high computational efficiency, and stability compared to other selection methods from deep learning and classical statistical literature across multiple open-source LLMs and datasets. Further ablation studies reveal that incorporating cross-prompt comparisons in active reward modeling significantly enhances labeling efficiency, shedding light on the potential for improved annotation strategies in RLHF.

This paper focuses on linear regression models with non-conjugate sparsity-inducing regularizers such as lasso and group lasso. Although empirical Bayes approach enables us to estimate the regularization parameter, little is known on the properties of the estimators. In particular, there are many unexplained aspects regarding the specific conditions under which the mechanism of automatic relevance determination (ARD) occurs. In this paper, we derive the empirical Bayes estimators for the group lasso regularized linear regression models with a limited number of parameters. It is shown that the estimators diverge under a certain condition, giving rise to the ARD mechanism. We also prove that empirical Bayes methods can produce ARD mechanism in general regularized linear regression models and clarify the conditions under which models such as ridge, lasso, and group lasso can produce ARD mechanism.

This paper proposes a hierarchical Bayesian multitask learning model that is applicable to the general multi-task binary classification learning problem where the model assumes a shared sparsity structure across different tasks. We derive a computationally efficient inference algorithm based on variational inference to approximate the posterior distribution. We demonstrate the potential of the new approach on various synthetic datasets and for predicting human health status based on microbiome profile. Our analysis incorporates data pooled from multiple microbiome studies, along with a comprehensive comparison with other benchmark methods. Results in synthetic datasets show that the proposed approach has superior support recovery property when the underlying regression coefficients share a common sparsity structure across different tasks. Our experiments on microbiome classification demonstrate the utility of the method in extracting informative taxa while providing well-calibrated predictions with uncertainty quantification and achieving competitive performance in terms of prediction metrics. Notably, despite the heterogeneity of the pooled datasets (e.g., different experimental objectives, laboratory setups, sequencing equipment, patient demographics), our method delivers robust results.