Variational inference is an umbrella term for algorithms which cast Bayesian inference as optimization. Classically, variational inference uses the Kullback-Leibler divergence to define the optimization. Though this divergence has been widely used, the resultant posterior approximation can suffer from undesirable statistical properties. To address this, we reexamine variational inference from its roots as an optimization problem. We use operators, or functions of functions, to design variational objectives. As one example, we design a variational objective with a Langevin-Stein operator. We develop a black box algorithm, operator variational inference (OPVI), for optimizing any operator objective. Importantly, operators enable us to make explicit the statistical and computational tradeoffs for variational inference. We can characterize different properties of variational objectives, such as objectives that admit data subsampling---allowing inference to scale to massive data---as well as objectives that admit variational programs---a rich class of posterior approximations that does not require a tractable density. We illustrate the benefits of OPVI on a mixture model and a generative model of images.
Dynamic topic models (DTMs) model the evolution of prevalent themes in literature, online media, and other forms of text over time. DTMs assume that word co-occurrence statistics change continuously and therefore impose continuous stochastic process priors on their model parameters. These dynamical priors make inference much harder than in regular topic models, and also limit scalability. In this paper, we present several new results around DTMs. First, we extend the class of tractable priors from Wiener processes to the generic class of Gaussian processes (GPs). This allows us to explore topics that develop smoothly over time, that have a long-term memory or are temporally concentrated (for event detection). Second, we show how to perform scalable approximate inference in these models based on ideas around stochastic variational inference and sparse Gaussian processes. This way we can train a rich family of DTMs to massive data. Our experiments on several large-scale datasets show that our generalized model allows us to find interesting patterns that were not accessible by previous approaches.
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 efficiently and automatically in any parameterization, using automatic differentiation. Our code is integrated into the GPflow package.
Inspired by the seminal work on Stein Variational Inference and Stein Variational Policy Gradient, we derived a method to generate samples from the posterior variational parameter distribution by \textit{explicitly} minimizing the KL divergence to match the target distribution in an amortize fashion. Consequently, we applied this varational inference technique into vanilla policy gradient, TRPO and PPO with Bayesian Neural Network parameterizations for reinforcement learning problems.
We introduce a Bayesian framework for inference with a supervised version of the Gaussian process latent variable model. The framework overcomes the high correlations between latent variables and hyperparameters by using an unbiased pseudo estimate for the marginal likelihood that approximately integrates over the latent variables. This is used to construct a Markov Chain to explore the posterior of the hyperparameters. We demonstrate the procedure on simulated and real examples, showing its ability to capture uncertainty and multimodality of the hyperparameters and improved uncertainty quantification in predictions when compared with variational inference.
Variational inference has experienced a recent surge in popularity owing to stochastic approaches, which have yielded practical tools for a wide range of model classes. A key benefit is that stochastic variational inference obviates the tedious process of deriving analytical expressions for closed-form variable updates. Instead, one simply needs to derive the gradient of the log-posterior, which is often much easier. Yet for certain model classes, the log-posterior itself is difficult to optimize using standard gradient techniques. One such example are random field models, where optimization based on gradient linearization has proven popular, since it speeds up convergence significantly and can avoid poor local optima. In this paper we propose stochastic variational inference with gradient linearization (SVIGL). It is similarly convenient as standard stochastic variational inference - all that is required is a local linearization of the energy gradient. Its benefit over stochastic variational inference with conventional gradient methods is a clear improvement in convergence speed, while yielding comparable or even better variational approximations in terms of KL divergence. We demonstrate the benefits of SVIGL in three applications: Optical flow estimation, Poisson-Gaussian denoising, and 3D surface reconstruction.
Well-established methods for the solution of stochastic partial differential equations (SPDEs) typically struggle in problems with high-dimensional inputs/outputs. Such difficulties are only amplified in large-scale applications where even a few tens of full-order model runs are impracticable. While dimensionality reduction can alleviate some of these issues, it is not known which and how many features of the (high-dimensional) input are actually predictive of the (high-dimensional) output. In this paper, we advocate a Bayesian formulation that is capable of performing simultaneous dimension and model-order reduction. It consists of a component that encodes the high-dimensional input into a low-dimensional set of feature functions by employing sparsity-enforcing priors and a decoding component that makes use of the solution of a coarse-grained model in order to reconstruct that of the full-order model. Both components are represented with latent variables in a probabilistic graphical model and are simultaneously trained using Stochastic Variational Inference methods. The model is capable of quantifying the predictive uncertainty due to the information loss that unavoidably takes place in any model-order/dimension reduction as well as the uncertainty arising from finite-sized training datasets. We demonstrate its capabilities in the context of random media where fine-scale fluctuations can give rise to random inputs with tens of thousands of variables. With a few tens of full-order model simulations, the proposed model is capable of identifying salient physical features and produce sharp predictions under different boundary conditions of the full output which itself consists of thousands of components.
Graphs are a fundamental abstraction for modeling relational data. However, graphs are discrete and combinatorial in nature, and learning representations suitable for machine learning tasks poses statistical and computational challenges. In this work, we propose Graphite, an algorithmic framework for unsupervised learning of representations over nodes in large graphs using deep latent variable generative models. Our model parameterizes variational autoencoders (VAE) with graph neural networks, and uses a novel iterative graph refinement strategy inspired by low-rank approximations for decoding. On a wide variety of synthetic and benchmark datasets, Graphite outperforms competing approaches for the tasks of density estimation, link prediction, and node classification. Finally, we derive a theoretical connection between message passing in graph neural networks and mean-field variational inference.
Deep generative models (DGMs) have brought about a major breakthrough, as well as renewed interest, in generative latent variable models. However, DGMs do not allow for performing data-driven inference of the number of latent features needed to represent the observed data. Traditional linear formulations address this issue by resorting to tools from the field of nonparametric statistics. Indeed, linear latent variable models imposed an Indian Buffet Process (IBP) prior have been extensively studied by the machine learning community; inference for such models can been performed either via exact sampling or via approximate variational techniques. Based on this inspiration, in this paper we examine whether similar ideas from the field of Bayesian nonparametrics can be utilized in the context of modern DGMs in order to address the latent variable dimensionality inference problem. To this end, we propose a novel DGM formulation, based on the imposition of an IBP prior. We devise an efficient Black-Box Variational inference algorithm for our model, and exhibit its efficacy in a number of semi-supervised classification experiments. In all cases, we use popular benchmark datasets, and compare to state-of-the-art DGMs.
Analyzing the underlying structure of multiple time-sequences provides insights into the understanding of social networks and human activities. In this work, we present the \emph{Bayesian nonparametric Poisson process allocation} (BaNPPA), a latent-function model for time-sequences, which automatically infers the number of latent functions. We model the intensity of each sequence as an infinite mixture of latent functions, each of which is obtained using a function drawn from a Gaussian process. We show that a technical challenge for the inference of such mixture models is the unidentifiability of the weights of the latent functions. We propose to cope with the issue by regulating the volume of each latent function within a variational inference algorithm. Our algorithm is computationally efficient and scales well to large data sets. We demonstrate the usefulness of our proposed model through experiments on both synthetic and real-world data sets.
Learning latent variable models with stochastic variational inference is challenging when the approximate posterior is far from the true posterior, due to high variance in the gradient estimates. We propose a novel rejection sampling step that discards samples from the variational posterior which are assigned low likelihoods by the model. Our approach provides an arbitrarily accurate approximation of the true posterior at the expense of extra computation. Using a new gradient estimator for the resulting unnormalized proposal distribution, we achieve average improvements of 3.71 nats and 0.21 nats over state-of-the-art single-sample and multi-sample alternatives respectively for estimating marginal log-likelihoods using sigmoid belief networks on the MNIST dataset.
Inferring missing links in knowledge graphs (KG) has attracted a lot of attention from the research community. In this paper, we tackle a practical query answering task involving predicting the relation of a given entity pair. We frame this prediction problem as an inference problem in a probabilistic graphical model and aim at resolving it from a variational inference perspective. In order to model the relation between the query entity pair, we assume that there exists an underlying latent variable (paths connecting two nodes) in the KG, which carries the equivalent semantics of their relations. However, due to the intractability of connections in large KGs, we propose to use variation inference to maximize the evidence lower bound. More specifically, our framework (\textsc{Diva}) is composed of three modules, i.e. a posterior approximator, a prior (path finder), and a likelihood (path reasoner). By using variational inference, we are able to incorporate them closely into a unified architecture and jointly optimize them to perform KG reasoning. With active interactions among these sub-modules, \textsc{Diva} is better at handling noise and coping with more complex reasoning scenarios. In order to evaluate our method, we conduct the experiment of the link prediction task on multiple datasets and achieve state-of-the-art performances on both datasets.
Hypernetworks are neural networks that generate weights for another neural network. We formulate the hypernetwork training objective as a compromise between accuracy and diversity, where the diversity takes into account trivial symmetry transformations of the target network. We explain how this simple formulation generalizes variational inference. We use multi-layered perceptrons to form the mapping from the low dimensional input random vector to the high dimensional weight space, and demonstrate how to reduce the number of parameters in this mapping by parameter sharing. We perform experiments and show that the generated weights are diverse and lie on a non-trivial manifold.
Gradient matching is a promising tool for learning parameters and state dynamics of ordinary differential equations. It is a grid free inference approach, which, for fully observable systems is at times competitive with numerical integration. However, for many real-world applications, only sparse observations are available or even unobserved variables are included in the model description. In these cases most gradient matching methods are difficult to apply or simply do not provide satisfactory results. That is why, despite the high computational cost, numerical integration is still the gold standard in many applications. Using an existing gradient matching approach, we propose a scalable variational inference framework which can infer states and parameters simultaneously, offers computational speedups, improved accuracy and works well even under model misspecifications in a partially observable system.
Variational inference is a general approach for approximating complex density functions, such as those arising in latent variable models, popular in machine learning. It has been applied to approximate the maximum likelihood estimator and to carry out Bayesian inference, however, quantification of uncertainty with variational inference remains challenging from both theoretical and practical perspectives. This paper is concerned with developing uncertainty measures for variational inference by using bootstrap procedures. We first develop two general bootstrap approaches for assessing the uncertainty of a variational estimate and the study the underlying bootstrap theory in both fixed- and increasing-dimension settings. We then use the bootstrap approach and our theoretical results in the context of mixed membership modeling with multivariate binary data on functional disability from the National Long Term Care Survey. We carry out a two-sample approach to test for changes in the repeated measures of functional disability for the subset of individuals present in 1989 and 1994 waves.
Dropout Variational Inference, or Dropout Sampling, has been recently proposed as an approximation technique for Bayesian Deep Learning and evaluated for image classification and regression tasks. This paper investigates the utility of Dropout Sampling for object detection for the first time. We demonstrate how label uncertainty can be extracted from a state-of-the-art object detection system via Dropout Sampling. We evaluate this approach on a large synthetic dataset of 30,000 images, and a real-world dataset captured by a mobile robot in a versatile campus environment. We show that this uncertainty can be utilized to increase object detection performance under the open-set conditions that are typically encountered in robotic vision. A Dropout Sampling network is shown to achieve a 12.3% increase in recall (for the same precision score as a standard network) and a 15.1% increase in precision (for the same recall score as the standard network).
The Pachinko Allocation Machine (PAM) is a deep topic model that allows representing rich correlation structures among topics by a directed acyclic graph over topics. Because of the flexibility of the model, however, approximate inference is very difficult. Perhaps for this reason, only a small number of potential PAM architectures have been explored in the literature. In this paper we present an efficient and flexible amortized variational inference method for PAM, using a deep inference network to parameterize the approximate posterior distribution in a manner similar to the variational autoencoder. Our inference method produces more coherent topics than state-of-art inference methods for PAM while being an order of magnitude faster, which allows exploration of a wider range of PAM architectures than have previously been studied.
We study Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks. A Bayesian hypernetwork $\h$ is a neural network which learns to transform a simple noise distribution, $p(\vec\epsilon) = \N(\vec 0,\mat I)$, to a distribution $q(\pp) := q(h(\vec\epsilon))$ over the parameters $\pp$ of another neural network (the "primary network")\@. We train $q$ with variational inference, using an invertible $\h$ to enable efficient estimation of the variational lower bound on the posterior $p(\pp | \D)$ via sampling. In contrast to most methods for Bayesian deep learning, Bayesian hypernets can represent a complex multimodal approximate posterior with correlations between parameters, while enabling cheap iid sampling of~$q(\pp)$. In practice, Bayesian hypernets can provide a better defense against adversarial examples than dropout, and also exhibit competitive performance on a suite of tasks which evaluate model uncertainty, including regularization, active learning, and anomaly detection.
One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this paper, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find the member of that family which is close to the target. Closeness is measured by Kullback-Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data. We discuss modern research in VI and highlight important open problems. VI is powerful, but it is not yet well understood. Our hope in writing this paper is to catalyze statistical research on this class of algorithms.
Semantic hashing has become a powerful paradigm for fast similarity search in many information retrieval systems. While fairly successful, previous techniques generally require two-stage training, and the binary constraints are handled ad-hoc. In this paper, we present an end-to-end Neural Architecture for Semantic Hashing (NASH), where the binary hashing codes are treated as Bernoulli latent variables. A neural variational inference framework is proposed for training, where gradients are directly back-propagated through the discrete latent variable to optimize the hash function. We also draw connections between proposed method and rate-distortion theory, which provides a theoretical foundation for the effectiveness of the proposed framework. Experimental results on three public datasets demonstrate that our method significantly outperforms several state-of-the-art models on both unsupervised and supervised scenarios.