Expectation propagation (EP) is a deterministic approximation algorithm that is often used to perform approximate Bayesian parameter learning. EP approximates the full intractable posterior distribution through a set of local approximations that are iteratively refined for each datapoint. EP can offer analytic and computational advantages over other approximations, such as V ariational Inference (VI), and is the method of choice for a number of models. The local nature of EP appears to make it an ideal candidate for performing Bayesian learning on large models in large-scale dataset settings. However, EP has a crucial limitation in this context: the number of approximating factors needs to increase with the number of data-points, N, which often entails a prohibitively large memory overhead. This paper presents an extension to EP, called stochastic expectation propagation (SEP), that maintains a global posterior approximation (like VI) but updates it in a local way (like EP). Experiments on a number of canonical learning problems using synthetic and real-world datasets indicate that SEP performs almost as well as full EP, but reduces the memory consumption by a factor of N . SEP is therefore ideally suited to performing approximate Bayesian learning in the large model, large dataset setting.

To address this issue, various approximate Bayesian inference methods have been proposed, such as Markov Chain Monte Carlo [MCMC, see e.g.

We introduce FedGVI, a probabilistic Federated Learning (FL) framework that is provably robust to both prior and likelihood misspecification. FedGVI addresses limitations in both frequentist and Bayesian FL by providing unbiased predictions under model misspecification, with calibrated uncertainty quantification. Our approach generalises previous FL approaches, specifically Partitioned Variational Inference (Ashman et al., 2022), by allowing robust and conjugate updates, decreasing computational complexity at the clients. We offer theoretical analysis in terms of fixed-point convergence, optimality of the cavity distribution, and provable robustness. Additionally, we empirically demonstrate the effectiveness of FedGVI in terms of improved robustness and predictive performance on multiple synthetic and real world classification data sets.

Massive MIMO (multiple-input multiple-output) detection is an important topic in wireless communication and various machine learning based methods have been developed recently for this task. Expectation propagation (EP) and its variants are widely used for MIMO detection and have achieved the best performance. However, EP-based solvers fail to capture the correlation between unknown variables, leading to loss of information, and in addition, they are computationally expensive. In this paper, we show that the real-valued system can be modeled as spectral signal convolution on graph, through which the correlation between unknown variables can be captured. Based on this analysis, we propose graph convolution-enhanced expectation propagation (GCEPNet), a graph convolution-enhanced EP detector. GCEPNet incorporates data-dependent attention scores into Chebyshev polynomial for powerful graph convolution with better generalization capacity. It enables a better estimation of the cavity distribution for EP and empirically achieves the state-of-the-art (SOTA) MIMO detection performance with much faster inference speed. To our knowledge, we are the first to shed light on the connection between the system model and graph convolution, and the first to design the data-dependent attention scores for graph convolution.

Expectation propagation (EP) is a deterministic approximation algorithm that is often used to perform approximate Bayesian parameter learning. EP approximates the full intractable posterior distribution through a set of local approximations that are iteratively refined for each datapoint. EP can offer analytic and computational advantages over other approximations, such as Variational Inference (VI), and is the method of choice for a number of models. The local nature of EP appears to make it an ideal candidate for performing Bayesian learning on large models in large-scale dataset settings. However, EP has a crucial limitation in this context: the number of approximating factors needs to increase with the number of datapoints, N, which often entails a prohibitively large memory overhead. This paper presents an extension to EP, called stochastic expectation propagation (SEP), that maintains a global posterior approximation (like VI) but updates it in a local way (like EP). Experiments on a number of canonical learning problems using synthetic and real-world datasets indicate that SEP performs almost as well as full EP, but reduces the memory consumption by a factor of N. SEP is therefore ideally suited to performing approximate Bayesian learning in the large model, large dataset setting.

We develop a message-passing algorithm for noisy matrix completion problems based on matrix factorization. The algorithm is derived by approximating message distributions of belief propagation with Gaussian distributions that share the same first and second moments. We also derive a memory-friendly version of the proposed algorithm by applying a perturbation treatment commonly used in the literature of approximate message passing. In addition, a damping technique, which is demonstrated to be crucial for optimal performance, is introduced without computational strain, and the relationship to the message-passing version of alternating least squares, a method reported to be optimal in certain settings, is discussed. Experiments on synthetic datasets show that while the proposed algorithm quantitatively exhibits almost the same performance under settings where the earlier algorithm is optimal, it is advantageous when the observed datasets are corrupted by non-Gaussian noise. Experiments on real-world datasets also emphasize the performance differences between the two algorithms.

Abstract--This paper proposes a new algorithm for Gaussian process classification based on posterior linearisation (PL). In PL, a Gaussian approximation to the posterior density is obtained iteratively using the best possible linearisation of the conditional mean of the labels and accounting for the linearisation error . Considering three widely-used likelihood functions, in general, PL provides lower classification errors in real data sets than expectation propagation and Laplace algorithms. Classification is an important problem with a high number of applications, for example, in handwriting and speech recognition, and medical diagnosis [1]. In (supervised) classification, a set of training data points with their corresponding classes are available to learn the underlying structure of the problem. Based on this information, the objective is to infer the classes of new data points. This classification problem can be posed using Gaussian processes (GPs) [2]-[8].

A common approach for Bayesian computation with big data is to partition the data into smaller pieces, perform local inference for each piece separately, and finally combine the results to obtain an approximation to the global posterior. Looking at this from the bottom up, one can perform separate analyses on individual sources of data and then combine these in a larger Bayesian model. In either case, the idea of distributed modeling and inference has both conceptual and computational appeal, but from the Bayesian perspective there is no general way of handling the prior distribution: if the prior is included in each separate inference, it will be multiply-counted when the inferences are combined; but if the prior is itself divided into pieces, it may not provide enough regularization for each separate computation, thus eliminating one of the key advantages of Bayesian methods. To resolve this dilemma, we propose expectation propagation (EP) as a general prototype for distributed Bayesian inference. The central idea is to factor the likelihood according to the data partitions, and to iteratively combine each factor with an approximate model of the prior and all other parts of the data, thus producing an overall approximation to the global posterior at convergence. In this paper, we give an introduction to EP and an overview of some recent developments of the method, with particular emphasis on its use in combining inferences from partitioned data. In addition to distributed modeling of large datasets, our unified treatment also includes hierarchical modeling of data with a naturally partitioned structure. The paper describes a general algorithmic framework, rather than a specific algorithm, and presents an example implementation for it.

Exponential family distributions are highly useful in machine learning since their calculation can be performed efficiently through natural parameters. The exponential family has recently been extended to the t-exponential family, which contains Student-t distributions as family members and thus allows us to handle noisy data well. However, since the t-exponential family is denied by the deformed exponential, we cannot derive an efficient learning algorithm for the t-exponential family such as expectation propagation (EP). In this paper, we borrow the mathematical tools of q-algebra from statistical physics and show that the pseudo additivity of distributions allows us to perform calculation of t-exponential family distributions through natural parameters. We then develop an expectation propagation (EP) algorithm for the t-exponential family, which provides a deterministic approximation to the posterior or predictive distribution with simple moment matching. We finally apply the proposed EP algorithm to the Bayes point machine and Student-t process classication, and demonstrate their performance numerically.