Stochastic variational inference offers an attractive option as a default method for differentiable probabilistic programming. However, the performance of the variational approach depends on the choice of an appropriate variational family. Here, we introduce automatic structured variational inference (ASVI), a fully automated method for constructing structured variational families, inspired by the closed-form update in conjugate Bayesian models. These convex-update families incorporate the forward pass of the input probabilistic program and can therefore capture complex statistical dependencies. Convex-update families have the same space and time complexity as the input probabilistic program and are therefore tractable for a very large family of models including both continuous and discrete variables. We validate our automatic variational method on a wide range of low- and high-dimensional inference problems. We find that ASVI provides a clear improvement in performance when compared with other popular approaches such as the mean-field approach and inverse autoregressive flows. We provide an open source implementation of ASVI in TensorFlow Probability.
Columbia University（哥伦比亚大学）；Adobe Research（Adobe研究院）；Adobe Systems Incorporated（奥多比系统公司）
Beta process is the standard nonparametric Bayesian prior for latent factor model. In this paper, we derive a structured mean-field variational inference algorithm for a beta process non-negative matrix factorization (NMF) model with Poisson likelihood. Unlike the linear Gaussian model, which is well-studied in the nonparametric Bayesian literature, NMF model with beta process prior does not enjoy the conjugacy. We leverage the recently developed stochastic structured mean-field variational inference to relax the conjugacy constraint and restore the dependencies among the latent variables in t…
Theory and computation for structured variational inference
Department of Statistics, Columbia University（哥伦比亚大学统计系）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）；Foundations of Data Science, Yale University（耶鲁大学数据科学基础）
Donders Centre for Cognition, Radboud University（拉德堡德大学唐德斯认知中心）；One Planet（一个星球（研究机构））
The automation of probabilistic reasoning is one of the primary aims of machine learning. Recently, the confluence of variational inference and deep learning has led to powerful and flexible automatic inference methods that can be trained by stochastic gradient descent. In particular, normalizing flows are highly parameterized deep models that can fit arbitrarily complex posterior densities. However, normalizing flows struggle in highly structured probabilistic programs as they need to relearn the forward-pass of the program. Automatic structured variational inference (ASVI) remedies this prob…
In this note we present a generative model of natural images consisting of a deep hierarchy of layers of latent random variables, each of which follows a new type of distribution that we call rectified Gaussian. These rectified Gaussian units allow spike-and-slab type sparsity, while retaining the differentiability necessary for efficient stochastic gradient variational inference. To learn the parameters of the new model, we approximate the posterior of the latent variables with a variational auto-encoder. Rather than making the usual mean-field assumption however, the encoder parameterizes a …
Variational inference is a scalable technique for approximate Bayesian inference. Deriving variational inference algorithms requires tedious model-specific calculations; this makes it difficult to automate. We propose an automatic variational inference algorithm, automatic differentiation variational inference (ADVI). The user only provides a Bayesian model and a dataset; nothing else. We make no conjugacy assumptions and support a broad class of models. The algorithm automatically determines an appropriate variational family and optimizes the variational objective. We implement ADVI in Stan (…
Google Research（谷歌研究院）；Georgia Institute of Technology（佐治亚理工学院）
Automatic Differentiation Variational Inference (ADVI) is a useful tool for efficiently learning probabilistic models in machine learning. Generally approximate posteriors learned by ADVI are forced to be unimodal in order to facilitate use of the reparameterization trick. In this paper, we show how stratified sampling may be used to enable mixture distributions as the approximate posterior, and derive a new lower bound on the evidence analogous to the importance weighted autoencoder (IWAE). We show that this "SIWAE" is a tighter bound than both IWAE and the traditional ELBO, both of which are…
University of Warwick（华威大学）；The Alan Turing Institute（艾伦·图灵研究所）；CSIRO’s Data61（澳大利亚联邦科学与工业研究组织Data61研究所）；UNSW（新南威尔士大学）；Centre for Translational Data Science（转化数据科学中心）；The University of Sydney（悉尼大学）
We propose a scalable framework for inference in an inhomogeneous Poisson process modeled by a continuous sigmoidal Cox process that assumes the corresponding intensity function is given by a Gaussian process (GP) prior transformed with a scaled logistic sigmoid function. We present a tractable representation of the likelihood through augmentation with a superposition of Poisson processes. This view enables a structured variational approximation capturing dependencies across variables in the model. Our framework avoids discretization of the domain, does not require accurate numerical integrati…
Approximate Bayesian Computation (ABC) is a framework for performing likelihood-free posterior inference for simulation models. Stochastic Variational inference (SVI) is an appealing alternative to the inefficient sampling approaches commonly used in ABC. However, SVI is highly sensitive to the variance of the gradient estimators, and this problem is exacerbated by approximating the likelihood. We draw upon recent advances in variance reduction for SV and likelihood-free inference using deterministic simulations to produce low variance gradient estimators of the variational lower-bound. By the…
Data Science Institute（数据科学研究所）；Columbia University（哥伦比亚大学）；Princeton University（普林斯顿大学）
Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient …
The stochastic variational inference (SVI) paradigm, which combines variational inference, natural gradients, and stochastic updates, was recently proposed for large-scale data analysis in conjugate Bayesian models and demonstrated to be effective in several problems. This paper studies a family of Bayesian latent variable models with two levels of hidden variables but without any conjugacy requirements, making several contributions in this context. The first is observing that SVI, with an improved structured variational approximation, is applicable under more general conditions than previousl…
Mean-field variational inference (MFVI) has been widely applied in large scale Bayesian inference. However MFVI, which assumes a product distribution on the latent variables, often leads to objective functions with many local optima, making optimization algorithms sensitive to initialization. In this paper, we study the advantage of structured variational inference for the two class Stochastic Blockmodel. The variational distribution is constructed to have pairwise dependency structure on the nodes of the network. We prove that, in a broad density regime and for general random initializations,…
We develop a general variational inference method that preserves dependency among the latent variables. Our method uses copulas to augment the families of distributions used in mean-field and structured approximations. Copulas model the dependency that is not captured by the original variational distribution, and thus the augmented variational family guarantees better approximations to the posterior. With stochastic optimization, inference on the augmented distribution is scalable. Furthermore, our strategy is generic: it can be applied to any inference procedure that currently uses the mean-f…
We develop flexible methods of deriving variational inference for models with complex latent variable structure. By splitting the variables in these models into "global" parameters and "local" latent variables, we define a class of variational approximations that exploit this partitioning and go beyond Gaussian variational approximation. This approximation is motivated by the fact that in many hierarchical models, there are global variance parameters which determine the scale of local latent variables in their posterior conditional on the global parameters. We also consider parsimonious parame…
It is difficult to use subsampling with variational inference in hierarchical models since the number of local latent variables scales with the dataset. Thus, inference in hierarchical models remains a challenge at large scale. It is helpful to use a variational family with structure matching the posterior, but optimization is still slow due to the huge number of local distributions. Instead, this paper suggests an amortized approach where shared parameters simultaneously represent all local distributions. This approach is similarly accurate as using a given joint distribution (e.g., a full-ra…
We present a new algorithm for approximate inference in probabilistic programs, based on a stochastic gradient for variational programs. This method is efficient without restrictions on the probabilistic program; it is particularly practical for distributions which are not analytically tractable, including highly structured distributions that arise in probabilistic programs. We show how to automatically derive mean-field probabilistic programs and optimize them, and demonstrate that our perspective improves inference efficiency over other algorithms.
A recurring problem when building probabilistic latent variable models is regularization and model selection, for instance, the choice of the dimensionality of the latent space. In the context of belief networks with latent variables, this problem has been adressed with Automatic Relevance Determination (ARD) employing Monte Carlo inference. We present a variational inference approach to ARD for Deep Generative Models using doubly stochastic variational inference to provide fast and scalable learning. We show empirical results on a standard dataset illustrating the effects of contracting the l…
We consider the problem of joint modelling of metabolic signals and gene expression in systems biology applications. We propose an approach based on input-output factorial hidden Markov models and propose a structured variational inference approach to infer the structure and states of the model. We start from the classical free form structured variational mean field approach and use a expectation propagation to approximate the expectations needed in the variational loop. We show that this corresponds to a factored expectation constrained approximate inference. We validate our model through ext…
Variational inference provides a powerful tool for approximate probabilistic in- ference on complex, structured models. Typical variational inference methods, however, require to use inference networks with computationally tractable proba- bility density functions. This largely limits the design and implementation of vari- ational inference methods. We consider wild variational inference methods that do not require tractable density functions on the inference networks, and hence can be applied in more challenging cases. As an example of application, we treat stochastic gradient Langevin dynami…
Latent variable time-series models are among the most heavily used tools from machine learning and applied statistics. These models have the advantage of learning latent structure both from noisy observations and from the temporal ordering in the data, where it is assumed that meaningful correlation structure exists across time. A few highly-structured models, such as the linear dynamical system with linear-Gaussian observations, have closed-form inference procedures (e.g. the Kalman Filter), but this case is an exception to the general rule that exact posterior inference in more complex gener…
Continuous latent time series models are prevalent in Bayesian modeling; examples include the Kalman filter, dynamic collaborative filtering, or dynamic topic models. These models often benefit from structured, non mean field variational approximations that capture correlations between time steps. Black box variational inference with reparameterization gradients (BBVI) allows us to explore a rich new class of Bayesian non-conjugate latent time series models; however, a naive application of BBVI to such structured variational models would scale quadratically in the number of time steps. We desc…
VinAI Research（VinAI研究院）；University of Texas, Austin（德克萨斯大学奥斯汀分校）；Hanoi University of Science and Technology（河内科技大学）
Approximate inference in Bayesian deep networks exhibits a dilemma of how to yield high fidelity posterior approximations while maintaining computational efficiency and scalability. We tackle this challenge by introducing a novel variational structured approximation inspired by the Bayesian interpretation of Dropout regularization. Concretely, we focus on the inflexibility of the factorized structure in Dropout posterior and then propose an improved method called Variational Structured Dropout (VSD). VSD employs an orthogonal transformation to learn a structured representation on the variation…
Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the EM algorithm. The conventional approach to determining a suitable number of components is to compare different mixture models using penalized log-likelihood criteria such as BIC.We propose fitting MLMMs with variational methods which can perform parameter estimation and model selection simultaneously. A variational approximation is described where the variational lower bound and parameter updates are in closed form, allowing fast evaluation. A new variation…
Gatsby Computational Neuroscience Unit（盖茨比计算神经科学单元）；University College London（伦敦大学学院）
Sparse variational approximations allow for principled and scalable inference in Gaussian Process (GP) models. In settings where several GPs are part of the generative model, theses GPs are a posteriori coupled. For many applications such as regression where predictive accuracy is the quantity of interest, this coupling is not crucial. Howewer if one is interested in posterior uncertainty, it cannot be ignored. A key element of variational inference schemes is the choice of the approximate posterior parameterization. When the number of latent variables is large, mean field (MF) methods provide…
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Santa Cruz（加州大学圣克鲁兹分校）；UC Berkeley（加州大学伯克利分校）
This paper presents an efficient variational inference framework for deriving a family of structured gaussian process regression network (SGPRN) models. The key idea is to incorporate auxiliary inducing variables in latent functions and jointly treats both the distributions of the inducing variables and hyper-parameters as variational parameters. Then we propose structured variable distributions and marginalize latent variables, which enables the decomposability of a tractable variational lower bound and leads to stochastic optimization. Our inference approach is able to model data in which ou…
Amortized Structured Stochastic Variational Inference for Gaussian Process Latent Variable Models
LMU Munich（慕尼黑大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Imperial College London（帝国理工学院）；University of Technology Nuremberg (UTN)（纽伦堡工业大学）；Helmholtz AI（亥姆霍兹人工智能中心）；Xyme
Imperial College London（帝国理工学院）；Microsoft Research（微软研究院）；University College London（伦敦大学学院）
Learning workable representations of dynamical systems is becoming an increasingly important problem in a number of application areas. By leveraging recent work connecting deep neural networks to systems of differential equations, we propose \emph{variational integrator networks}, a class of neural network architectures designed to preserve the geometric structure of physical systems. This class of network architectures facilitates accurate long-term prediction, interpretability, and data-efficient learning, while still remaining highly flexible and capable of modeling complex behavior. We dem…
Carnegie Mellon University（卡内基梅隆大学）；Google Research（谷歌研究院）；DeepMind；Duke University（杜克大学）；Norwegian University of Science and Technology（挪威科技大学）
We propose a novel framework for structured bandits, which we call an influence diagram bandit. Our framework captures complex statistical dependencies between actions, latent variables, and observations; and thus unifies and extends many existing models, such as combinatorial semi-bandits, cascading bandits, and low-rank bandits. We develop novel online learning algorithms that learn to act efficiently in our models. The key idea is to track a structured posterior distribution of model parameters, either exactly or approximately. To act, we sample model parameters from their posterior and the…
Recent efforts on combining deep models with probabilistic graphical models are promising in providing flexible models that are also easy to interpret. We propose a variational message-passing algorithm for variational inference in such models. We make three contributions. First, we propose structured inference networks that incorporate the structure of the graphical model in the inference network of variational auto-encoders (VAE). Second, we establish conditions under which such inference networks enable fast amortized inference similar to VAE. Finally, we derive a variational message passin…
IBM research（IBM研究院）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；Harvard University（哈佛大学）
Bayesian Neural Networks (BNNs) have recently received increasing attention for their ability to provide well-calibrated posterior uncertainties. However, model selection---even choosing the number of nodes---remains an open question. Recent work has proposed the use of a horseshoe prior over node pre-activations of a Bayesian neural network, which effectively turns off nodes that do not help explain the data. In this work, we propose several modeling and inference advances that consistently improve the compactness of the model learned while maintaining predictive performance, especially in sm…
Natural image statistics exhibit hierarchical dependencies across multiple scales. Representing such prior knowledge in non-factorial latent tree models can boost performance of image denoising, inpainting, deconvolution or reconstruction substantially, beyond standard factorial "sparse" methodology. We derive a large scale approximate Bayesian inference algorithm for linear models with non-factorial (latent tree-structured) scale mixture priors. Experimental results on a range of denoising and inpainting problems demonstrate substantially improved performance compared to MAP estimation or to …
Technion, IIT（以色列理工学院）；Meta AI Research（Meta人工智能研究院）
Discrete variational auto-encoders (VAEs) are able to represent semantic latent spaces in generative learning. In many real-life settings, the discrete latent space consists of high-dimensional structures, and propagating gradients through the relevant structures often requires enumerating over an exponentially large latent space. Recently, various approaches were devised to propagate approximated gradients without enumerating over the space of possible structures. In this work, we use Natural Evolution Strategies (NES), a class of gradient-free black-box optimization algorithms, to learn disc…
Dynamic trees are mixtures of tree structured belief networks. They solve some of the problems of fixed tree networks at the cost of making exact inference intractable. For this reason approximate methods such as sampling or mean field approaches have been used. However, mean field approximations assume a factorized distribution over node states. Such a distribution seems unlickely in the posterior, as nodes are highly correlated in the prior. Here a structured variational approach is used, where the posterior distribution over the non-evidential nodes is itself approximated by a dynamic tree.…
University of Amsterdam（阿姆斯特丹大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
We introduce a variational Bayesian neural network where the parameters are governed via a probability distribution on random matrices. Specifically, we employ a matrix variate Gaussian \cite{gupta1999matrix} parameter posterior distribution where we explicitly model the covariance among the input and output dimensions of each layer. Furthermore, with approximate covariance matrices we can achieve a more efficient way to represent those correlations that is also cheaper than fully factorized parameter posteriors. We further show that with the "local reprarametrization trick" \cite{kingma2015va…
University of Amsterdam（阿姆斯特丹大学）；Canadian Institute for Advanced Research (CIFAR)（加拿大高级研究院）
We introduce the variational graph auto-encoder (VGAE), a framework for unsupervised learning on graph-structured data based on the variational auto-encoder (VAE). This model makes use of latent variables and is capable of learning interpretable latent representations for undirected graphs. We demonstrate this model using a graph convolutional network (GCN) encoder and a simple inner product decoder. Our model achieves competitive results on a link prediction task in citation networks. In contrast to most existing models for unsupervised learning on graph-structured data and link prediction, o…
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Bosch Center for Artificial Intelligence（博世人工智能中心）；CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）
Prediction of future states of the environment and interacting agents is a key competence required for autonomous agents to operate successfully in the real world. Prior work for structured sequence prediction based on latent variable models imposes a uni-modal standard Gaussian prior on the latent variables. This induces a strong model bias which makes it challenging to fully capture the multi-modality of the distribution of the future states. In this work, we introduce Conditional Flow Variational Autoencoders (CF-VAE) using our novel conditional normalizing flow based prior to capture compl…
Institute of Mathematical Sciences (ICMAT)（数学科学研究所（ICMAT））；SAMSI, Duke University（杜克大学SAMSI）
A framework to boost the efficiency of Bayesian inference in probabilistic programs is introduced by embedding a sampler inside a variational posterior approximation. We call it the refined variational approximation. Its strength lies both in ease of implementation and automatically tuning of the sampler parameters to speed up mixing time using automatic differentiation. Several strategies to approximate \emph{evidence lower bound} (ELBO) computation are introduced. Experimental evidence of its efficient performance is shown solving an influence diagram in a high-dimensional space using a cond…
We propose a new variational inference algorithm for learning in Gaussian Process State-Space Models (GPSSMs). Our algorithm enables learning of unstable and partially observable systems, where previous algorithms fail. Our main algorithmic contribution is a novel approximate posterior that can be calculated efficiently using a single forward and backward pass along the training trajectories. The forward-backward pass is inspired on Kalman smoothing for linear dynamical systems but generalizes to GPSSMs. Our second contribution is a modification of the conditioning step that effectively lowers…
University of Warwick（华威大学）；The Alan Turing Institute（艾伦·图灵研究所）
Specifying a governing physical model in the presence of missing physics and recovering its parameters are two intertwined and fundamental problems in science. Modern machine learning allows one to circumvent these, via emulators and surrogates, but in doing so disregards prior knowledge and physical laws that are especially important for small data regimes, interpretability, and decision making. In this work we fold the mechanistic model into a flexible data-driven surrogate to arrive at a physically structured decoder network. This provides accelerated inference for the Bayesian inverse prob…
We introduce a technique that can automatically tune the parameters of a rule-based computer vision system comprised of thresholds, combinational logic, and time constants. This lets us retain the flexibility and perspicacity of a conventionally structured system while allowing us to perform approximate gradient descent using labeled data. While this is only a heuristic procedure, as far as we are aware there is no other efficient technique for tuning such systems. We describe the components of the system and the associated supervised learning mechanism. We also demonstrate the utility of the …
University of Sheffield（谢菲尔德大学）；University of Manchester（曼彻斯特大学）
In this publication, we combine two Bayesian non-parametric models: the Gaussian Process (GP) and the Dirichlet Process (DP). Our innovation in the GP model is to introduce a variation on the GP prior which enables us to model structured time-series data, i.e. data containing groups where we wish to model inter- and intra-group variability. Our innovation in the DP model is an implementation of a new fast collapsed variational inference procedure which enables us to optimize our variationala pproximation significantly faster than standard VB approaches. In a biological time series application …
In the classic sparsity-driven problems, the fundamental L-1 penalty method has been shown to have good performance in reconstructing signals for a wide range of problems. However this performance relies on a good choice of penalty weight which is often found from empirical experiments. We propose an algorithm called the Laplacian variational automatic relevance determination (Lap-VARD) that takes this penalty weight as a parameter of a prior Laplace distribution. Optimization of this parameter using an automatic relevance determination framework results in a balance between the sparsity and a…
The natural gradient method has been used effectively in conjugate Gaussian process models, but the non-conjugate case has been largely unexplored. We examine how natural gradients can be used in non-conjugate stochastic settings, together with hyperparameter learning. We conclude that the natural gradient can significantly improve performance in terms of wall-clock time. For ill-conditioned posteriors the benefit of the natural gradient method is especially pronounced, and we demonstrate a practical setting where ordinary gradients are unusable. We show how natural gradients can be computed e…
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…
