Bayesian neural networks (BNNs) demonstrate promising success in improving the robustness and uncertainty quantification of modern deep learning. However, they generally struggle with underfitting at scale and parameter efficiency. On the other hand, deep ensembles have emerged as alternatives for uncertainty quantification that, while outperforming BNNs on certain problems, also suffer from efficiency issues. It remains unclear how to combine the strengths of these two approaches and remediate their common issues. To tackle this challenge, we propose a rank-1 parameterization of BNNs, where each weight matrix involves only a distribution on a rank-1 subspace. We also revisit the use of mixture approximate posteriors to capture multiple modes, where unlike typical mixtures, this approach admits a significantly smaller memory increase (e.g., only a 0.4% increase for a ResNet-50 mixture of size 10). We perform a systematic empirical study on the choices of prior, variational posterior, and methods to improve training. For ResNet-50 on ImageNet, Wide ResNet 28-10 on CIFAR-10/100, and an RNN on MIMIC-III, rank-1 BNNs achieve state-of-the-art performance across log-likelihood, accuracy, and calibration on the test sets and out-of-distribution variants.
Bar-Ilan University（巴伊兰大学）；Technion（以色列理工学院）；University of Illinois（伊利诺伊大学）
Generalization bounds which assess the difference between the true risk and the empirical risk have been studied extensively. However, to obtain bounds, current techniques use strict assumptions such as a uniformly bounded or a Lipschitz loss function. To avoid these assumptions, in this paper, we propose a new generalization bound for Bayesian deep nets by exploiting the contractivity of the Log-Sobolev inequalities. Using these inequalities adds an additional loss-gradient norm term to the generalization bound, which is intuitively a surrogate of the model complexity. Empirically, we analyze…
Bosch Center for Artificial Intelligence（博世人工智能中心）
Neural Ordinary Differential Equations (N-ODEs) are a powerful building block for learning systems, which extend residual networks to a continuous-time dynamical system. We propose a Bayesian version of N-ODEs that enables well-calibrated quantification of prediction uncertainty, while maintaining the expressive power of their deterministic counterpart. We assign Bayesian Neural Nets (BNNs) to both the drift and the diffusion terms of a Stochastic Differential Equation (SDE) that models the flow of the activation map in time. We infer the posterior on the BNN weights using a straightforward ad…
University of Tübingen（蒂宾根大学）；MPI for Intelligent Systems, Tübingen（蒂宾根马克斯·普朗克智能系统研究所）
A Bayesian treatment can mitigate overconfidence in ReLU nets around the training data. But far away from them, ReLU Bayesian neural networks (BNNs) can still underestimate uncertainty and thus be asymptotically overconfident. This issue arises since the output variance of a BNN with finitely many features is quadratic in the distance from the data region. Meanwhile, Bayesian linear models with ReLU features converge, in the infinite-width limit, to a particular Gaussian process (GP) with a variance that grows cubically so that no asymptotic overconfidence can occur. While this may seem of mos…
University of Oxford（牛津大学）；OATML, Department of Computer Science（OATML计算机科学系）
We challenge the longstanding assumption that the mean-field approximation for variational inference in Bayesian neural networks is severely restrictive, and show this is not the case in deep networks. We prove several results indicating that deep mean-field variational weight posteriors can induce similar distributions in function-space to those induced by shallower networks with complex weight posteriors. We validate our theoretical contributions empirically, both through examination of the weight posterior using Hamiltonian Monte Carlo in small models and by comparing diagonal- to structure…
Stanford University（斯坦福大学）；Facebook AI Research（Facebook人工智能研究院）；University of California, Los Angeles（加州大学洛杉矶分校）
A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from observational data. However, a point estimate may not accurately capture the uncertainty in inferring the underlying graph in practical scenarios, wherein the true DAG is non-identifiable and/or the observed dataset is limited. We propose Bayesian Causal Discovery Nets (BCD Nets), a variational inference framework for estimating a distribution over DAGs characterizi…
ETH Zurich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学院学习系统中心）；Technical University of Denmark（丹麦技术大学）；DeepMind（深度思维公司）
The generalized Gauss-Newton (GGN) approximation is often used to make practical Bayesian deep learning approaches scalable by replacing a second order derivative with a product of first order derivatives. In this paper we argue that the GGN approximation should be understood as a local linearization of the underlying Bayesian neural network (BNN), which turns the BNN into a generalized linear model (GLM). Because we use this linearized model for posterior inference, we should also predict using this modified model instead of the original one. We refer to this modified predictive as "GLM predi…
We introduce TyXe, a Bayesian neural network library built on top of Pytorch and Pyro. Our leading design principle is to cleanly separate architecture, prior, inference and likelihood specification, allowing for a flexible workflow where users can quickly iterate over combinations of these components. In contrast to existing packages TyXe does not implement any layer classes, and instead relies on architectures defined in generic Pytorch code. TyXe then provides modular choices for canonical priors, variational guides, inference techniques, and layer selections for a Bayesian treatment of the…
Recently developed techniques have made it possible to quickly learn accurate probability density functions from data in low-dimensional continuous space. In particular, mixtures of Gaussians can be fitted to data very quickly using an accelerated EM algorithm that employs multiresolution kd-trees (Moore, 1999). In this paper, we propose a kind of Bayesian networks in which low-dimensional mixtures of Gaussians over different subsets of the domain's variables are combined into a coherent joint probability model over the entire domain. The network is also capable of modeling complex dependencie…
This paper addresses the estimation of parameters of a Bayesian network from incomplete data. The task is usually tackled by running the Expectation-Maximization (EM) algorithm several times in order to obtain a high log-likelihood estimate. We argue that choosing the maximum log-likelihood estimate (as well as the maximum penalized log-likelihood and the maximum a posteriori estimate) has severe drawbacks, being affected both by overfitting and model uncertainty. Two ideas are discussed to overcome these issues: a maximum entropy approach and a Bayesian model averaging approach. Both ideas ca…
Hierarchical Bayesian networks and neural networks with stochastic hidden units are commonly perceived as two separate types of models. We show that either of these types of models can often be transformed into an instance of the other, by switching between centered and differentiable non-centered parameterizations of the latent variables. The choice of parameterization greatly influences the efficiency of gradient-based posterior inference; we show that they are often complementary to eachother, we clarify when each parameterization is preferred and show how inference can be made robust. In t…
Aalto University（阿尔托大学）；University of Manchester（曼彻斯特大学）
We introduce implicit Bayesian neural networks, a simple and scalable approach for uncertainty representation in deep learning. Standard Bayesian approach to deep learning requires the impractical inference of the posterior distribution over millions of parameters. Instead, we propose to induce a distribution that captures the uncertainty over neural networks by augmenting each layer's inputs with latent variables. We present appropriate input distributions and demonstrate state-of-the-art performance in terms of calibration, robustness and uncertainty characterisation over large-scale, multi-…
In recent years, leveraging parallel and distributed computational resources has become essential to solve problems of high computational cost. Bayesian optimization (BO) has shown attractive results in those expensive-to-evaluate problems such as hyperparameter optimization of machine learning algorithms. While many parallel BO methods have been developed to search efficiently utilizing these computational resources, these methods assumed synchronous settings or were not scalable. In this paper, we propose a simple and scalable BO method for asynchronous parallel settings. Experiments are car…
Bayesian optimization is a technique for optimizing black-box target functions. At the core of Bayesian optimization is a surrogate model that predicts the output of the target function at previously unseen inputs to facilitate the selection of promising input values. Gaussian processes (GPs) are commonly used as surrogate models but are known to scale poorly with the number of observations. We adapt the Vecchia approximation, a popular GP approximation from spatial statistics, to enable scalable high-dimensional Bayesian optimization. We develop several improvements and extensions, including …
University of Toronto（多伦多大学）；Stanford Unviersity（斯坦福大学）
We perform scalable approximate inference in continuous-depth Bayesian neural networks. In this model class, uncertainty about separate weights in each layer gives hidden units that follow a stochastic differential equation. We demonstrate gradient-based stochastic variational inference in this infinite-parameter setting, producing arbitrarily-flexible approximate posteriors. We also derive a novel gradient estimator that approaches zero variance as the approximate posterior over weights approaches the true posterior. This approach brings continuous-depth Bayesian neural nets to a competitive …
The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of Causal Bayesian Networks has also been investigated in this context. After reviewing these newer relationships, we proceed to demonstrate that many of the ideas embodied in the concept of a ``causal model'' can be more generally expressed directly in terms of a partial order and a family of polynomial maps. The more conventional graphical constructions, when avai…
University at Buffalo（布法罗大学）；Duke University（杜克大学）
There has been recent interest in developing scalable Bayesian sampling methods such as stochastic gradient MCMC (SG-MCMC) and Stein variational gradient descent (SVGD) for big-data analysis. A standard SG-MCMC algorithm simulates samples from a discrete-time Markov chain to approximate a target distribution, thus samples could be highly correlated, an undesired property for SG-MCMC. In contrary, SVGD directly optimizes a set of particles to approximate a target distribution, and thus is able to obtain good approximations with relatively much fewer samples. In this paper, we propose a principl…
Center for Theoretical Neuroscience（理论神经科学中心）；Columbia University（哥伦比亚大学）
Bayesian optimization has emerged as a strong candidate tool for global optimization of functions with expensive evaluation costs. However, due to the dynamic nature of research in Bayesian approaches, and the evolution of computing technology, using Bayesian optimization in a parallel computing environment remains a challenge for the non-expert. In this report, I review the state-of-the-art in parallel and scalable Bayesian optimization methods. In addition, I propose practical ways to avoid a few of the pitfalls of Bayesian optimization, such as oversampling of edge parameters and over-explo…
Bayesian optimization (BO) is a model-based approach for gradient-free black-box function optimization. Typically, BO is powered by a Gaussian process (GP), whose algorithmic complexity is cubic in the number of evaluations. Hence, GP-based BO cannot leverage large amounts of past or related function evaluations, for example, to warm start the BO procedure. We develop a multiple adaptive Bayesian linear regression model as a scalable alternative whose complexity is linear in the number of observations. The multiple Bayesian linear regression models are coupled through a shared feedforward neur…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Technology Innovation Institute（技术创新研究院）
In deep active learning, it is especially important to choose multiple examples to markup at each step to work efficiently, especially on large datasets. At the same time, existing solutions to this problem in the Bayesian setup, such as BatchBALD, have significant limitations in selecting a large number of examples, associated with the exponential complexity of computing mutual information for joint random variables. We, therefore, present the Large BatchBALD algorithm, which gives a well-grounded approximation to the BatchBALD method that aims to achieve comparable quality while being more c…
University of Oxford（牛津大学）；University of British Columbia（不列颠哥伦比亚大学）；The Alan Turing Institute（艾伦·图灵研究所）
Bayesian inference via standard Markov Chain Monte Carlo (MCMC) methods is too computationally intensive to handle large datasets, since the cost per step usually scales like $Θ(n)$ in the number of data points $n$. We propose the Scalable Metropolis-Hastings (SMH) kernel that exploits Gaussian concentration of the posterior to require processing on average only $O(1)$ or even $O(1/\sqrt{n})$ data points per step. This scheme is based on a combination of factorized acceptance probabilities, procedures for fast simulation of Bernoulli processes, and control variate ideas. Contrary to many MCMC …
Bayesian optimization has recently emerged as a popular method for the sample-efficient optimization of expensive black-box functions. However, the application to high-dimensional problems with several thousand observations remains challenging, and on difficult problems Bayesian optimization is often not competitive with other paradigms. In this paper we take the view that this is due to the implicit homogeneity of the global probabilistic models and an overemphasized exploration that results from global acquisition. This motivates the design of a local probabilistic approach for global optimi…
We implement gradient-based variational inference routines for Wishart and inverse Wishart processes, which we apply as Bayesian models for the dynamic, heteroskedastic covariance matrix of a multivariate time series. The Wishart and inverse Wishart processes are constructed from i.i.d. Gaussian processes, existing variational inference algorithms for which form the basis of our approach. These methods are easy to implement as a black-box and scale favorably with the length of the time series, however, they fail in the case of the Wishart process, an issue we resolve with a simple modification…
We present a novel, scalable and Bayesian approach to modelling the occurrence of pairs of symbols (i,j) drawn from a large vocabulary. Observed pairs are assumed to be generated by a simple popularity based selection process followed by censoring using a preference function. By basing inference on the well-founded principle of variational bounding, and using new site-independent bounds, we show how a scalable inference procedure can be obtained for large data sets. State of the art results are presented on real-world movie viewing data.
The global optimization of a high-dimensional black-box function under black-box constraints is a pervasive task in machine learning, control, and engineering. These problems are challenging since the feasible set is typically non-convex and hard to find, in addition to the curses of dimensionality and the heterogeneity of the underlying functions. In particular, these characteristics dramatically impact the performance of Bayesian optimization methods, that otherwise have become the de facto standard for sample-efficient optimization in unconstrained settings, leaving practitioners with evolu…
We present K-Means Batch Bayesian Optimization (KMBBO), a novel batch sampling algorithm for Bayesian Optimization (BO). KMBBO uses unsupervised learning to efficiently estimate peaks of the model acquisition function. We show in empirical experiments that our method outperforms the current state-of-the-art batch allocation algorithms on a variety of test problems including tuning of algorithm hyper-parameters and a challenging drug discovery problem. In order to accommodate the real-world problem of high dimensional data, we propose a modification to KMBBO by combining it with compressed sens…
Datasets are growing not just in size but in complexity, creating a demand for rich models and quantification of uncertainty. Bayesian methods are an excellent fit for this demand, but scaling Bayesian inference is a challenge. In response to this challenge, there has been considerable recent work based on varying assumptions about model structure, underlying computational resources, and the importance of asymptotic correctness. As a result, there is a zoo of ideas with few clear overarching principles. In this paper, we seek to identify unifying principles, patterns, and intuitions for scalin…
We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear …
We describe Bayesian Layers, a module designed for fast experimentation with neural network uncertainty. It extends neural network libraries with drop-in replacements for common layers. This enables composition via a unified abstraction over deterministic and stochastic functions and allows for scalability via the underlying system. These layers capture uncertainty over weights (Bayesian neural nets), pre-activation units (dropout), activations ("stochastic output layers"), or the function itself (Gaussian processes). They can also be reversible to propagate uncertainty from input to output. W…
University of Michigan（密歇根大学）；Duke University（杜克大学）
Analysis of heterogeneous patterns in complex spatio-temporal data finds usage across various domains in applied science and engineering, including training autonomous vehicles to navigate in complex traffic scenarios. Motivated by applications arising in the transportation domain, in this paper we develop a model for learning heterogeneous and dynamic patterns of velocity field data. We draw from basic nonparameric Bayesian modeling elements such as hierarchical Dirichlet process and infinite hidden Markov model, while the smoothness of each homogeneous velocity field element is captured with…
Sandia National Laboratories（桑迪亚国家实验室）；Xilinx Inc.（赛灵思公司）
Bayesian optimization (BO) is a flexible and powerful framework that is suitable for computationally expensive simulation-based applications and guarantees statistical convergence to the global optimum. While remaining as one of the most popular optimization methods, its capability is hindered by the size of data, the dimensionality of the considered problem, and the nature of sequential optimization. These scalability issues are intertwined with each other and must be tackled simultaneously. In this work, we propose the Scalable$^3$-BO framework, which employs sparse GP as the underlying surr…
While deep neural networks have become the go-to approach in computer vision, the vast majority of these models fail to properly capture the uncertainty inherent in their predictions. Estimating this predictive uncertainty can be crucial, for example in automotive applications. In Bayesian deep learning, predictive uncertainty is commonly decomposed into the distinct types of aleatoric and epistemic uncertainty. The former can be estimated by letting a neural network output the parameters of a certain probability distribution. Epistemic uncertainty estimation is a more challenging problem, and…
IBM Research（IBM研究（中心））；Dublin, Ireland（爱尔兰都柏林（注意：Dublin, Ireland是地址，非机构，按要求不应提取，此处原数据中可能存在格式错误，正确机构应为IBM Research，而Dublin, Ireland是地点信息，根据提取规则，不应提取地址，因此仅保留IBM Research））
We introduce a new Bayesian multi-class support vector machine by formulating a pseudo-likelihood for a multi-class hinge loss in the form of a location-scale mixture of Gaussians. We derive a variational-inference-based training objective for gradient-based learning. Additionally, we employ an inducing point approximation which scales inference to large data sets. Furthermore, we develop hybrid Bayesian neural networks that combine standard deep learning components with the proposed model to enable learning for unstructured data. We provide empirical evidence that our model outperforms the co…
Large multilayer neural networks trained with backpropagation have recently achieved state-of-the-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a large number of hyperparameters to the data, lack of calibrated probabilistic predictions, and a tendency to overfit the training data. In principle, the Bayesian approach to learning neural networks does not have these problems. However, existing Bayesian techniques lack scalability to large dataset and network sizes. In this work we present a novel scalabl…
Bayesian Optimisation (BO) refers to a suite of techniques for global optimisation of expensive black box functions, which use introspective Bayesian models of the function to efficiently search for the optimum. While BO has been applied successfully in many applications, modern optimisation tasks usher in new challenges where conventional methods fail spectacularly. In this work, we present Dragonfly, an open source Python library for scalable and robust BO. Dragonfly incorporates multiple recently developed methods that allow BO to be applied in challenging real world settings; these include…
The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scale of massive modern datasets, and modifications to make them so are typically model-specific, require expert tuning, and can break theoretical guarantees on inferential quality. Building on the Bayesian coresets framework, this work instead takes advantage of data redundancy to s…
The graph structure of a Bayesian network (BN) can be learned from data using the well-known score-and-search approach. Previous work has shown that incorporating structured representations of the conditional probability distributions (CPDs) into the score-and-search approach can improve the accuracy of the learned graph. In this paper, we present a novel approach capable of learning the graph of a BN and simultaneously modelling linear and non-linear local probabilistic relationships between variables. We achieve this by a combination of feature selection to reduce the search space for local …
Explosive growth in data and availability of cheap computing resources have sparked increasing interest in Big learning, an emerging subfield that studies scalable machine learning algorithms, systems, and applications with Big Data. Bayesian methods represent one important class of statistic methods for machine learning, with substantial recent developments on adaptive, flexible and scalable Bayesian learning. This article provides a survey of the recent advances in Big learning with Bayesian methods, termed Big Bayesian Learning, including nonparametric Bayesian methods for adaptively inferr…
University of California, Berkeley（加州大学伯克利分校）；Intel Corporation（英特尔公司）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Celeste is a procedure for inferring astronomical catalogs that attains state-of-the-art scientific results. To date, Celeste has been scaled to at most hundreds of megabytes of astronomical images: Bayesian posterior inference is notoriously demanding computationally. In this paper, we report on a scalable, parallel version of Celeste, suitable for learning catalogs from modern large-scale astronomical datasets. Our algorithmic innovations include a fast numerical optimization routine for Bayesian posterior inference and a statistically efficient scheme for decomposing astronomical optimizati…
In this paper we propose the first non-parametric Bayesian model using Gaussian Processes to make inference on Poisson Point Processes without resorting to gridding the domain or to introducing latent thinning points. Unlike competing models that scale cubically and have a squared memory requirement in the number of data points, our model has a linear complexity and memory requirement. We propose an MCMC sampler and show that our model is faster, more accurate and generates less correlated samples than competing models on both synthetic and real-life data. Finally, we show that our model easil…
Ericsson Research（爱立信研究院）；KTH Royal Institute of Technology（KTH皇家理工学院）
Cell-free multi-user multiple input multiple output networks are a promising alternative to classical cellular architectures, since they have the potential to provide uniform service quality and high resource utilisation over the entire coverage area of the network. To realise this potential, previous works have developed radio resource management mechanisms using various optimisation engines. In this work, we consider the problem of overall ergodic spectral efficiency maximisation in the context of uplink-downlink data power control in cell-free networks. To solve this problem in large networ…
Carnegie Mellon University（卡内基梅隆大学）；Massachusetts Institute of Technology（麻省理工学院）
In this paper we present decomposable priors, a family of priors over structure and parameters of tree belief nets for which Bayesian learning with complete observations is tractable, in the sense that the posterior is also decomposable and can be completely determined analytically in polynomial time. This follows from two main results: First, we show that factored distributions over spanning trees in a graph can be integrated in closed form. Second, we examine priors over tree parameters and show that a set of assumptions similar to (Heckerman and al. 1995) constrain the tree parameter priors…
We propose a scalable Bayesian preference learning method for jointly predicting the preferences of individuals as well as the consensus of a crowd from pairwise labels. Peoples' opinions often differ greatly, making it difficult to predict their preferences from small amounts of personal data. Individual biases also make it harder to infer the consensus of a crowd when there are few labels per item. We address these challenges by combining matrix factorisation with Gaussian processes, using a Bayesian approach to account for uncertainty arising from noisy and sparse data. Our method exploits …
We propose Macau, a powerful and flexible Bayesian factorization method for heterogeneous data. Our model can factorize any set of entities and relations that can be represented by a relational model, including tensors and also multiple relations for each entity. Macau can also incorporate side information, specifically entity and relation features, which are crucial for predicting sparsely observed relations. Macau scales to millions of entity instances, hundred millions of observations, and sparse entity features with millions of dimensions. To achieve the scale up, we specially designed sam…
In this paper we introduce a novel online time series forecasting model we refer to as the pM-GP filter. We show that our model is equivalent to Gaussian process regression, with the advantage that both online forecasting and online learning of the hyper-parameters have a constant (rather than cubic) time complexity and a constant (rather than squared) memory requirement in the number of observations, without resorting to approximations. Moreover, the proposed model is expressive in that the family of covariance functions of the implied latent process, namely the spectral Matern kernels, have …
Norwegian Computing Center（挪威计算中心）；University of Oslo（奥斯陆大学）
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 Bayesian approach: Parameter and prediction uncertainty become easily available, facilitating rigid statistical analysis. Furthermore, prior knowledge can be incorporated. However so far there have been no scalable techniques capable of combining both model (structural) and parameter uncertainty. In this paper we introduce the concept of model uncertainty in BNN…
