Conventional federated learning algorithms train a single global model by leveraging all participating clients' data. However, due to heterogeneity in client generative distributions and predictive models, these approaches may not appropriately approximate the predictive process, converge to an optimal state, or generalize to new clients. We study personalization and generalization in stateless cross-device federated learning setups assuming heterogeneity in client data distributions and predictive models. We first propose a hierarchical generative model and formalize it using Bayesian Inference. We then approximate this process using Variational Inference to train our model efficiently. We call this algorithm Federated Variational Inference (FedVI). We use PAC-Bayes analysis to provide generalization bounds for FedVI. We evaluate our model on FEMNIST and CIFAR-100 image classification and show that FedVI beats the state-of-the-art on both tasks.
Amortized variational inference is an often employed framework in simulation-based inference that produces a posterior approximation that can be rapidly computed given any new observation. Unfortunately, there are few guarantees about the quality of these approximate posteriors. We propose Conformalized Amortized Neural Variational Inference (CANVI), a procedure that is scalable, easily implemented, and provides guaranteed marginal coverage. Given a collection of candidate amortized posterior approximators, CANVI constructs conformalized predictors based on each candidate, compares the predictors using a metric known as predictive efficiency, and returns the most efficient predictor. CANVI ensures that the resulting predictor constructs regions that contain the truth with a user-specified level of probability. CANVI is agnostic to design decisions in formulating the candidate approximators and only requires access to samples from the forward model, permitting its use in likelihood-free settings. We prove lower bounds on the predictive efficiency of the regions produced by CANVI and explore how the quality of a posterior approximation relates to the predictive efficiency of prediction regions based on that approximation. Finally, we demonstrate the accurate calibration and high predictive efficiency of CANVI on a suite of simulation-based inference benchmark tasks and an important scientific task: analyzing galaxy emission spectra.
Stein Variational Gradient Descent (SVGD) is a nonparametric particle-based deterministic sampling algorithm. Despite its wide usage, understanding the theoretical properties of SVGD has remained a challenging problem. For sampling from a Gaussian target, the SVGD dynamics with a bilinear kernel will remain Gaussian as long as the initializer is Gaussian. Inspired by this fact, we undertake a detailed theoretical study of the Gaussian-SVGD, i.e., SVGD projected to the family of Gaussian distributions via the bilinear kernel, or equivalently Gaussian variational inference (GVI) with SVGD. We present a complete picture by considering both the mean-field PDE and discrete particle systems. When the target is strongly log-concave, the mean-field Gaussian-SVGD dynamics is proven to converge linearly to the Gaussian distribution closest to the target in KL divergence. In the finite-particle setting, there is both uniform in time convergence to the mean-field limit and linear convergence in time to the equilibrium if the target is Gaussian. In the general case, we propose a density-based and a particle-based implementation of the Gaussian-SVGD, and show that several recent algorithms for GVI, proposed from different perspectives, emerge as special cases of our unified framework. Interestingly, one of the new particle-based instance from this framework empirically outperforms existing approaches. Our results make concrete contributions towards obtaining a deeper understanding of both SVGD and GVI.
Generative model-based deep clustering frameworks excel in classifying complex data, but are limited in handling dynamic and complex features because they require prior knowledge of the number of clusters. In this paper, we propose a nonparametric deep clustering framework that employs an infinite mixture of Gaussians as a prior. Our framework utilizes a memoized online variational inference method that enables the "birth" and "merge" moves of clusters, allowing our framework to cluster data in a "dynamic-adaptive" manner, without requiring prior knowledge of the number of features. We name the framework as DIVA, a Dirichlet Process-based Incremental deep clustering framework via Variational Auto-Encoder. Our framework, which outperforms state-of-the-art baselines, exhibits superior performance in classifying complex data with dynamically changing features, particularly in the case of incremental features. We released our source code implementation at: https://github.com/Ghiara/diva
Federated Learning (FL) is a distributed learning scheme to train a shared model across clients. One common and fundamental challenge in FL is that the sets of data across clients could be non-identically distributed and have different sizes. Personalized Federated Learning (PFL) attempts to solve this challenge via locally adapted models. In this work, we present a novel framework for PFL based on hierarchical Bayesian modeling and variational inference. A global model is introduced as a latent variable to augment the joint distribution of clients' parameters and capture the common trends of different clients, optimization is derived based on the principle of maximizing the marginal likelihood and conducted using variational expectation maximization. Our algorithm gives rise to a closed-form estimation of a confidence value which comprises the uncertainty of clients' parameters and local model deviations from the global model. The confidence value is used to weigh clients' parameters in the aggregation stage and adjust the regularization effect of the global model. We evaluate our method through extensive empirical studies on multiple datasets. Experimental results show that our approach obtains competitive results under mild heterogeneous circumstances while significantly outperforming state-of-the-art PFL frameworks in highly heterogeneous settings. Our code is available at https://github.com/JunyiZhu-AI/confidence_aware_PFL.
We propose a new approach for sampling and Bayesian computation that uses the score of the target distribution to construct a transport from a given reference distribution to the target. Our approach is an infinite-dimensional Newton method, involving a linear PDE, for finding a zero of a ``score-residual'' operator. We prove sufficient conditions for convergence to a valid transport map. Our Newton iterates can be computed by exploiting fast solvers for elliptic PDEs, resulting in new algorithms for Bayesian inference and other sampling tasks. We identify elementary settings where score-operator Newton transport achieves fast convergence while avoiding mode collapse.
We present an iterative framework to improve the amortized approximations of posterior distributions in the context of Bayesian inverse problems, which is inspired by loop-unrolled gradient descent methods and is theoretically grounded in maximally informative summary statistics. Amortized variational inference is restricted by the expressive power of the chosen variational distribution and the availability of training data in the form of joint data and parameter samples, which often lead to approximation errors such as the amortization gap. To address this issue, we propose an iterative framework that refines the current amortized posterior approximation at each step. Our approach involves alternating between two steps: (1) constructing a training dataset consisting of pairs of summarized data residuals and parameters, where the summarized data residual is generated using a gradient-based summary statistic, and (2) training a conditional generative model -- a normalizing flow in our examples -- on this dataset to obtain a probabilistic update of the unknown parameter. This procedure leads to iterative refinement of the amortized posterior approximations without the need for extra training data. We validate our method in a controlled setting by applying it to a stylized problem, and observe improved posterior approximations with each iteration. Additionally, we showcase the capability of our method in tackling realistically sized problems by applying it to transcranial ultrasound, a high-dimensional, nonlinear inverse problem governed by wave physics, and observe enhanced posterior quality through better image reconstruction with the posterior mean.
Statistical shape modeling (SSM) enables population-based quantitative analysis of anatomical shapes, informing clinical diagnosis. Deep learning approaches predict correspondence-based SSM directly from unsegmented 3D images but require calibrated uncertainty quantification, motivating Bayesian formulations. Variational information bottleneck DeepSSM (VIB-DeepSSM) is an effective, principled framework for predicting probabilistic shapes of anatomy from images with aleatoric uncertainty quantification. However, VIB is only half-Bayesian and lacks epistemic uncertainty inference. We derive a fully Bayesian VIB formulation and demonstrate the efficacy of two scalable implementation approaches: concrete dropout and batch ensemble. Additionally, we introduce a novel combination of the two that further enhances uncertainty calibration via multimodal marginalization. Experiments on synthetic shapes and left atrium data demonstrate that the fully Bayesian VIB network predicts SSM from images with improved uncertainty reasoning without sacrificing accuracy.
Atmospheric Turbulence (AT) correction is a challenging restoration task as it consists of two distortions: geometric distortion and spatially variant blur. Diffusion models have shown impressive accomplishments in photo-realistic image synthesis and beyond. In this paper, we propose a novel deep conditional diffusion model under a variational inference framework to solve the AT correction problem. We use this framework to improve performance by learning latent prior information from the input and degradation processes. We use the learned information to further condition the diffusion model. Experiments are conducted in a comprehensive synthetic AT dataset. We show that the proposed framework achieves good quantitative and qualitative results.
We propose a novel hierarchical Bayesian approach to Federated Learning (FL), where our model reasonably describes the generative process of clients' local data via hierarchical Bayesian modeling: constituting random variables of local models for clients that are governed by a higher-level global variate. Interestingly, the variational inference in our Bayesian model leads to an optimisation problem whose block-coordinate descent solution becomes a distributed algorithm that is separable over clients and allows them not to reveal their own private data at all, thus fully compatible with FL. We also highlight that our block-coordinate algorithm has particular forms that subsume the well-known FL algorithms including Fed-Avg and Fed-Prox as special cases. Beyond introducing novel modeling and derivations, we also offer convergence analysis showing that our block-coordinate FL algorithm converges to an (local) optimum of the objective at the rate of $O(1/\sqrt{t})$, the same rate as regular (centralised) SGD, as well as the generalisation error analysis where we prove that the test error of our model on unseen data is guaranteed to vanish as we increase the training data size, thus asymptotically optimal.
Motivated by the maneuvering target tracking with sensors such as radar and sonar, this paper considers the joint and recursive estimation of the dynamic state and the time-varying process noise covariance in nonlinear state space models. Due to the nonlinearity of the models and the non-conjugate prior, the state estimation problem is generally intractable as it involves integrals of general nonlinear functions and unknown process noise covariance, resulting in the posterior probability distribution functions lacking closed-form solutions. This paper presents a recursive solution for joint nonlinear state estimation and model parameters identification based on the approximate Bayesian inference principle. The stochastic search variational inference is adopted to offer a flexible, accurate, and effective approximation of the posterior distributions. We make two contributions compared to existing variational inference-based noise adaptive filtering methods. First, we introduce an auxiliary latent variable to decouple the latent variables of dynamic state and process noise covariance, thereby improving the flexibility of the posterior inference. Second, we split the variational lower bound optimization into conjugate and non-conjugate parts, whereas the conjugate terms are directly optimized that admit a closed-form solution and the non-conjugate terms are optimized by natural gradients, achieving the trade-off between inference speed and accuracy. The performance of the proposed method is verified on radar target tracking applications by both simulated and real-world data.
Owing to the cross-pollination between causal discovery and deep learning, non-statistical data (e.g., images, text, etc.) encounters significant conflicts in terms of properties and methods with traditional causal data. To unify these data types of varying forms, we redefine causal data from two novel perspectives and then propose three data paradigms. Among them, the indefinite data (like dialogues or video sources) induce low sample utilization and incapability of the distribution assumption, both leading to the fact that learning causal representation from indefinite data is, as of yet, largely unexplored. We design the causal strength variational model to settle down these two problems. Specifically, we leverage the causal strength instead of independent noise as the latent variable to construct evidence lower bound. By this design ethos, The causal strengths of different structures are regarded as a distribution and can be expressed as a 2D matrix. Moreover, considering the latent confounders, we disentangle the causal graph G into two relation subgraphs O and C. O contains pure relations between observed variables, while C represents the relations from latent variables to observed variables. We implement the above designs as a dynamic variational inference model, tailored to learn causal representation from indefinite data under latent confounding. Finally, we conduct comprehensive experiments on synthetic and real-world data to demonstrate the effectiveness of our method.
Fueled by the expressive power of deep neural networks, normalizing flows have achieved spectacular success in generative modeling, or learning to draw new samples from a distribution given a finite dataset of training samples. Normalizing flows have also been applied successfully to variational inference, wherein one attempts to learn a sampler based on an expression for the log-likelihood or energy function of the distribution, rather than on data. In variational inference, the unimodality of the reference Gaussian distribution used within the normalizing flow can cause difficulties in learning multimodal distributions. We introduce an extension of normalizing flows in which the Gaussian reference is replaced with a reference distribution that is constructed via a tensor network, specifically a matrix product state or tensor train. We show that by combining flows with tensor networks on difficult variational inference tasks, we can improve on the results obtained by using either tool without the other.
Bayesian neural networks (BNNs) have recently regained a significant amount of attention in the deep learning community due to the development of scalable approximate Bayesian inference techniques. There are several advantages of using a Bayesian approach: Parameter and prediction uncertainties become easily available, facilitating rigorous statistical analysis. Furthermore, prior knowledge can be incorporated. However, so far, there have been no scalable techniques capable of combining both structural and parameter uncertainty. In this paper, we apply the concept of model uncertainty as a framework for structural learning in BNNs and hence make inference in the joint space of structures/models and parameters. Moreover, we suggest an adaptation of a scalable variational inference approach with reparametrization of marginal inclusion probabilities to incorporate the model space constraints. Experimental results on a range of benchmark datasets show that we obtain comparable accuracy results with the competing models, but based on methods that are much more sparse than ordinary BNNs.
A digital twin (DT) leverages a virtual representation of the physical world, along with communication (e.g., 6G), computing (e.g., edge computing), and artificial intelligence (AI) technologies to enable many connected intelligence services. In order to handle the large amounts of network data based on digital twins (DTs), wireless systems can exploit the paradigm of semantic communication (SC) for facilitating informed decision-making under strict communication constraints by utilizing AI techniques such as causal reasoning. In this paper, a novel framework called causal semantic communication (CSC) is proposed for DT-based wireless systems. The CSC system is posed as an imitation learning (IL) problem, where the transmitter, with access to optimal network control policies using a DT, teaches the receiver using SC over a bandwidth limited wireless channel how to improve its knowledge to perform optimal control actions. The causal structure in the source data is extracted using novel approaches from the framework of deep end-to-end causal inference, thereby enabling the creation of a semantic representation that is causally invariant, which in turn helps generalize the learned knowledge of the system to unseen scenarios. The CSC decoder at the receiver is designed to extract and estimate semantic information while ensuring high semantic reliability. The receiver control policies, semantic decoder, and causal inference are formulated as a bi-level optimization problem within a variational inference framework. This problem is solved using a novel concept called network state models, inspired from world models in generative AI, that faithfully represents the environment dynamics leading to data generation. Simulation results demonstrate that the proposed CSC system outperforms state-of-the-art SC systems by achieving better semantic reliability and reduced semantic representation.
Bayesian models are a powerful tool for studying complex data, allowing the analyst to encode rich hierarchical dependencies and leverage prior information. Most importantly, they facilitate a complete characterization of uncertainty through the posterior distribution. Practical posterior computation is commonly performed via MCMC, which can be computationally infeasible for high dimensional models with many observations. In this article we discuss the potential to improve posterior computation using ideas from machine learning. Concrete future directions are explored in vignettes on normalizing flows, Bayesian coresets, distributed Bayesian inference, and variational inference.
Priors are essential for reconstructing images from noisy and/or incomplete measurements. The choice of the prior determines both the quality and uncertainty of recovered images. We propose turning score-based diffusion models into principled image priors ("score-based priors") for analyzing a posterior of images given measurements. Previously, probabilistic priors were limited to handcrafted regularizers and simple distributions. In this work, we empirically validate the theoretically-proven probability function of a score-based diffusion model. We show how to sample from resulting posteriors by using this probability function for variational inference. Our results, including experiments on denoising, deblurring, and interferometric imaging, suggest that score-based priors enable principled inference with a sophisticated, data-driven image prior.
Brain-computer interfaces (BCIs), particularly the P300 BCI, facilitate direct communication between the brain and computers. The fundamental statistical problem in P300 BCIs lies in classifying target and non-target stimuli based on electroencephalogram (EEG) signals. However, the low signal-to-noise ratio (SNR) and complex spatial/temporal correlations of EEG signals present challenges in modeling and computation, especially for individuals with severe physical disabilities-BCI's primary users. To address these challenges, we introduce a novel Gaussian Latent channel model with Sparse time-varying effects (GLASS) under a fully Bayesian framework. GLASS is built upon a constrained multinomial logistic regression particularly designed for the imbalanced target and non-target stimuli. The novel latent channel decomposition efficiently alleviates strong spatial correlations between EEG channels, while the soft-thresholded Gaussian process (STGP) prior ensures sparse and smooth time-varying effects. We demonstrate GLASS substantially improves BCI's performance in participants with amyotrophic lateral sclerosis (ALS) and identifies important EEG channels (PO8, Oz, PO7, and Pz) in parietal and occipital regions that align with existing literature. For broader accessibility, we develop an efficient gradient-based variational inference (GBVI) algorithm for posterior computation and provide a user-friendly Python module available at https://github.com/BangyaoZhao/GLASS.
Numerical interactions leading to users sharing textual content published by others are naturally represented by a network where the individuals are associated with the nodes and the exchanged texts with the edges. To understand those heterogeneous and complex data structures, clustering nodes into homogeneous groups as well as rendering a comprehensible visualisation of the data is mandatory. To address both issues, we introduce Deep-LPTM, a model-based clustering strategy relying on a variational graph auto-encoder approach as well as a probabilistic model to characterise the topics of discussion. Deep-LPTM allows to build a joint representation of the nodes and of the edges in two embeddings spaces. The parameters are inferred using a variational inference algorithm. We also introduce IC2L, a model selection criterion specifically designed to choose models with relevant clustering and visualisation properties. An extensive benchmark study on synthetic data is provided. In particular, we find that Deep-LPTM better recovers the partitions of the nodes than the state-of-the art ETSBM and STBM. Eventually, the emails of the Enron company are analysed and visualisations of the results are presented, with meaningful highlights of the graph structure.
We propose the NeRF-LEBM, a likelihood-based top-down 3D-aware 2D image generative model that incorporates 3D representation via Neural Radiance Fields (NeRF) and 2D imaging process via differentiable volume rendering. The model represents an image as a rendering process from 3D object to 2D image and is conditioned on some latent variables that account for object characteristics and are assumed to follow informative trainable energy-based prior models. We propose two likelihood-based learning frameworks to train the NeRF-LEBM: (i) maximum likelihood estimation with Markov chain Monte Carlo-based inference and (ii) variational inference with the reparameterization trick. We study our models in the scenarios with both known and unknown camera poses. Experiments on several benchmark datasets demonstrate that the NeRF-LEBM can infer 3D object structures from 2D images, generate 2D images with novel views and objects, learn from incomplete 2D images, and learn from 2D images with known or unknown camera poses.