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 the Kalman gain. This modification is crucial to attaining good test performance where no measurements are available. Finally, we show experimentally that our learning algorithm performs well in stable and unstable real systems with hidden states.
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…
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…
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…
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 …
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
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…
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.…
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…
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…
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…
Reparameterization of variational auto-encoders with continuous random variables is an effective method for reducing the variance of their gradient estimates. In the discrete case, one can perform reparametrization using the Gumbel-Max trick, but the resulting objective relies on an $\arg \max$ operation and is non-differentiable. In contrast to previous works which resort to softmax-based relaxations, we propose to optimize it directly by applying the direct loss minimization approach. Our proposal extends naturally to structured discrete latent variable models when evaluating the $\arg \max$…
Variational Bayesian Methods for a Tree-Structured Stick-Breaking Process Mixture of Gaussians by Application of the Bayes Codes for Context Tree Models
Center for Data Science Waseda University（早稻田大学数据科学中心）
Although there is much recent work developing flexible variational methods for Bayesian computation, Gaussian approximations with structured covariance matrices are often preferred computationally in high-dimensional settings. This paper considers approximate inference methods for complex latent variable models where the posterior is close to Gaussian, but with some skewness in the posterior marginals. We consider skew decomposable graphical models (SDGMs), which are based on the closed skew normal family of distributions, as variational approximations. These approximations can reflect the tru…
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…
Columbia University（哥伦比亚大学）；Netflix Inc.（奈飞公司）；The University of Texas at Dallas（得克萨斯大学达拉斯分校）；Spotify Inc.（声田公司）
Variational Auto-Encoders (VAEs) have been widely applied for learning compact, low-dimensional latent representations of high-dimensional data. When the correlation structure among data points is available, previous work proposed Correlated Variational Auto-Encoders (CVAEs), which employ a structured mixture model as prior and a structured variational posterior for each mixture component to enforce that the learned latent representations follow the same correlation structure. However, as we demonstrate in this work, such a choice cannot guarantee that CVAEs capture all the correlations. Furth…
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…
We introduce a methodology for nonlinear inverse problems using a variational Bayesian approach where the unknown quantity is a spatial field. A structured Bayesian Gaussian process latent variable model is used both to construct a low-dimensional generative model of the sample-based stochastic prior as well as a surrogate for the forward evaluation. Its Bayesian formulation captures epistemic uncertainty introduced by the limited number of input and output examples, automatically selects an appropriate dimensionality for the learned latent representation of the data, and rigorously propagates…
Variational inference offers scalable and flexible tools to tackle intractable Bayesian inference of modern statistical models like Bayesian neural networks and Gaussian processes. For largely over-parameterized models, however, the over-regularization property of the variational objective makes the application of variational inference challenging. Inspired by the literature on kernel methods, and in particular on structured approximations of distributions of random matrices, this paper proposes Walsh-Hadamard Variational Inference, which uses Walsh-Hadamard-based factorization strategies to r…
We develop an automated variational inference method for Bayesian structured prediction problems with Gaussian process (GP) priors and linear-chain likelihoods. Our approach does not need to know the details of the structured likelihood model and can scale up to a large number of observations. Furthermore, we show that the required expected likelihood term and its gradients in the variational objective (ELBO) can be estimated efficiently by using expectations over very low-dimensional Gaussian distributions. Optimization of the ELBO is fully parallelizable over sequences and amenable to stocha…
In this note we consider setups in which variational objectives for Bayesian neural networks can be computed in closed form. In particular we focus on single-layer networks in which the activation function is piecewise polynomial (e.g. ReLU). In this case we show that for a Normal likelihood and structured Normal variational distributions one can compute a variational lower bound in closed form. In addition we compute the predictive mean and variance in closed form. Finally, we also show how to compute approximate lower bounds for other likelihoods (e.g. softmax classification). In experiments…
Aalto University（阿尔托大学）；HUS Helsinki University Hospital（赫尔辛基大学医院）；University of Helsinki（赫尔辛基大学）
Longitudinal datasets measured repeatedly over time from individual subjects, arise in many biomedical, psychological, social, and other studies. A common approach to analyse high-dimensional data that contains missing values is to learn a low-dimensional representation using variational autoencoders (VAEs). However, standard VAEs assume that the learnt representations are i.i.d., and fail to capture the correlations between the data samples. We propose the Longitudinal VAE (L-VAE), that uses a multi-output additive Gaussian process (GP) prior to extend the VAE's capability to learn structured…
Aalto University（阿尔托大学）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
Gaussian process (GP) regression with 1D inputs can often be performed in linear time via a stochastic differential equation formulation. However, for non-Gaussian likelihoods, this requires application of approximate inference methods which can make the implementation difficult, e.g., expectation propagation can be numerically unstable and variational inference can be computationally inefficient. In this paper, we propose a new method that removes such difficulties. Building upon an existing method called conjugate-computation variational inference, our approach enables linear-time inference …
Likelihood-guided Regularization in Attention Based Models
Department of Statistics, Virginia Polytechnic Institute and State University（统计学系，弗吉尼亚理工大学和州立大学）；Department of Statistics, Virginia Tech.（统计学系，弗吉尼亚理工大学）
University of British Columbia（英属哥伦比亚大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
Uncertainty estimation in large deep-learning models is a computationally challenging task, where it is difficult to form even a Gaussian approximation to the posterior distribution. In such situations, existing methods usually resort to a diagonal approximation of the covariance matrix despite, the fact that these matrices are known to result in poor uncertainty estimates. To address this issue, we propose a new stochastic, low-rank, approximate natural-gradient (SLANG) method for variational inference in large, deep models. Our method estimates a "diagonal plus low-rank" structure based sole…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Tsinghua University（清华大学）
Variational Bayesian neural networks (BNNs) perform variational inference over weights, but it is difficult to specify meaningful priors and approximate posteriors in a high-dimensional weight space. We introduce functional variational Bayesian neural networks (fBNNs), which maximize an Evidence Lower BOund (ELBO) defined directly on stochastic processes, i.e. distributions over functions. We prove that the KL divergence between stochastic processes equals the supremum of marginal KL divergences over all finite sets of inputs. Based on this, we introduce a practical training objective which ap…
Multiparameter Uncertainty Mapping in Quantitative Molecular MRI using a Physics-Structured Variational Autoencoder (PS-VAE)
School of Biomedical Engineering, Tel-Aviv University（生物医学工程学院，特拉维夫大学）；School of Biochemistry, Neurobiology, and Biophysics, Tel-Aviv University（生物化学、神经生物学与生物物理学学院，特拉维夫大学）；Institute of Neuroradiology, University Hospital Erlangen（神经放射学研究所，埃朗根大学医院）；Department of Artificial Intelligence in Biomedical Engineering, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU)（生物医学工程人工智能系，弗赖堡-艾尔朗根-纽伦堡大学（FAU））；Sagol School of Neuroscience, Tel-Aviv University（萨戈尔神经科学学院，特拉维夫大学）
We propose a general modeling and inference framework that composes probabilistic graphical models with deep learning methods and combines their respective strengths. Our model family augments graphical structure in latent variables with neural network observation models. For inference, we extend variational autoencoders to use graphical model approximating distributions with recognition networks that output conjugate potentials. All components of these models are learned simultaneously with a single objective, giving a scalable algorithm that leverages stochastic variational inference, natura…
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…
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…
Radboud University（拉德堡德大学）；Donders Institute for Brain, Cognition and Behaviour（唐德斯脑、认知与行为研究所）；Google Research（谷歌研究院）
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 dependencie…
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）
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 …
Stochastic variational inference makes it possible to approximate posterior distributions induced by large datasets quickly using stochastic optimization. The algorithm relies on the use of fully factorized variational distributions. However, this "mean-field" independence approximation limits the fidelity of the posterior approximation, and introduces local optima. We show how to relax the mean-field approximation to allow arbitrary dependencies between global parameters and local hidden variables, producing better parameter estimates by reducing bias, sensitivity to local optima, and sensiti…
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…
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…
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…
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…
Several numerical approximation strategies for the expectation-propagation algorithm are studied in the context of large-scale learning: the Laplace method, a faster variant of it, Gaussian quadrature, and a deterministic version of variational sampling (i.e., combining quadrature with variational approximation). Experiments in training linear binary classifiers show that the expectation-propagation algorithm converges best using variational sampling, while it also converges well using Laplace-style methods with smooth factors but tends to be unstable with non-differentiable ones. Gaussian qua…
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…
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 …
University of Edinburgh（爱丁堡大学）；University of Amsterdam（阿姆斯特丹大学）
Human annotation for syntactic parsing is expensive, and large resources are available only for a fraction of languages. A question we ask is whether one can leverage abundant unlabeled texts to improve syntactic parsers, beyond just using the texts to obtain more generalisable lexical features (i.e. beyond word embeddings). To this end, we propose a novel latent-variable generative model for semi-supervised syntactic dependency parsing. As exact inference is intractable, we introduce a differentiable relaxation to obtain approximate samples and compute gradients with respect to the parser par…
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…
Rutgers University（罗格斯大学）；Toyota Technological Institute at Chicago（芝加哥丰田工业大学）
We propose learning discrete structured representations from unlabeled data by maximizing the mutual information between a structured latent variable and a target variable. Calculating mutual information is intractable in this setting. Our key technical contribution is an adversarial objective that can be used to tractably estimate mutual information assuming only the feasibility of cross entropy calculation. We develop a concrete realization of this general formulation with Markov distributions over binary encodings. We report critical and unexpected findings on practical aspects of the objec…
Georgia Institute of Technology（佐治亚理工学院）；Stony Brook University（石溪大学）；Ant Financial（蚂蚁金服）
Deep generative models have been enjoying success in modeling continuous data. However it remains challenging to capture the representations for discrete structures with formal grammars and semantics, e.g., computer programs and molecular structures. How to generate both syntactically and semantically correct data still remains largely an open problem. Inspired by the theory of compiler where the syntax and semantics check is done via syntax-directed translation (SDT), we propose a novel syntax-directed variational autoencoder (SD-VAE) by introducing stochastic lazy attributes. This approach c…
