State-space models have been successfully used for more than fifty years in different areas of science and engineering. We present a procedure for efficient variational Bayesian learning of nonlinear state-space models based on sparse Gaussian processes. The result of learning is a tractable posterior over nonlinear dynamical systems. In comparison to conventional parametric models, we offer the possibility to straightforwardly trade off model capacity and computational cost whilst avoiding overfitting. Our main algorithm uses a hybrid inference approach combining variational Bayes and sequential Monte Carlo. We also present stochastic variational inference and online learning approaches for fast learning with long time series.
We describe \textit{deep exponential families} (DEFs), a class of latent variable models that are inspired by the hidden structures used in deep neural networks. DEFs capture a hierarchy of dependencies between latent variables, and are easily generalized to many settings through exponential families. We perform inference using recent "black box" variational inference techniques. We then evaluate various DEFs on text and combine multiple DEFs into a model for pairwise recommendation data. In an extensive study, we show that going beyond one layer improves predictions for DEFs. We demonstrate that DEFs find interesting exploratory structure in large data sets, and give better predictive performance than state-of-the-art models.
Stochastic variational inference (SVI) lets us scale up Bayesian computation to massive data. It uses stochastic optimization to fit a variational distribution, following easy-to-compute noisy natural gradients. As with most traditional stochastic optimization methods, SVI takes precautions to use unbiased stochastic gradients whose expectations are equal to the true gradients. In this paper, we explore the idea of following biased stochastic gradients in SVI. Our method replaces the natural gradient with a similarly constructed vector that uses a fixed-window moving average of some of its previous terms. We will demonstrate the many advantages of this technique. First, its computational cost is the same as for SVI and storage requirements only multiply by a constant factor. Second, it enjoys significant variance reduction over the unbiased estimates, smaller bias than averaged gradients, and leads to smaller mean-squared error against the full gradient. We test our method on latent Dirichlet allocation with three large corpora.
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 sensitivity to hyperparameters.
Deep Gaussian processes provide a flexible approach to probabilistic modelling of data using either supervised or unsupervised learning. For tractable inference approximations to the marginal likelihood of the model must be made. The original approach to approximate inference in these models used variational compression to allow for approximate variational marginalization of the hidden variables leading to a lower bound on the marginal likelihood of the model [Damianou and Lawrence, 2013]. In this paper we extend this idea with a nested variational compression. The resulting lower bound on the likelihood can be easily parallelized or adapted for stochastic variational inference.
Factor analysis provides linear factors that describe relationships between individual variables of a data set. We extend this classical formulation into linear factors that describe relationships between groups of variables, where each group represents either a set of related variables or a data set. The model also naturally extends canonical correlation analysis to more than two sets, in a way that is more flexible than previous extensions. Our solution is formulated as variational inference of a latent variable model with structural sparsity, and it consists of two hierarchical levels: The higher level models the relationships between the groups, whereas the lower models the observed variables given the higher level. We show that the resulting solution solves the group factor analysis problem accurately, outperforming alternative factor analysis based solutions as well as more straightforward implementations of group factor analysis. The method is demonstrated on two life science data sets, one on brain activation and the other on systems biology, illustrating its applicability to the analysis of different types of high-dimensional data sources.
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 the approximating variational distribution. Preliminary results on both synthetic and real examples demonstrate that the proposed inference algorithm can reasonably recover the hidden structure of the data.
Training of the neural autoregressive density estimator (NADE) can be viewed as doing one step of probabilistic inference on missing values in data. We propose a new model that extends this inference scheme to multiple steps, arguing that it is easier to learn to improve a reconstruction in $k$ steps rather than to learn to reconstruct in a single inference step. The proposed model is an unsupervised building block for deep learning that combines the desirable properties of NADE and multi-predictive training: (1) Its test likelihood can be computed analytically, (2) it is easy to generate independent samples from it, and (3) it uses an inference engine that is a superset of variational inference for Boltzmann machines. The proposed NADE-k is competitive with the state-of-the-art in density estimation on the two datasets tested.
We consider the problem of approximate Bayesian inference in log-supermodular models. These models encompass regular pairwise MRFs with binary variables, but allow to capture high-order interactions, which are intractable for existing approximate inference techniques such as belief propagation, mean field, and variants. We show that a recently proposed variational approach to inference in log-supermodular models -L-FIELD- reduces to the widely-studied minimum norm problem for submodular minimization. This insight allows to leverage powerful existing tools, and hence to solve the variational problem orders of magnitude more efficiently than previously possible. We then provide another natural interpretation of L-FIELD, demonstrating that it exactly minimizes a specific type of R\'enyi divergence measure. This insight sheds light on the nature of the variational approximations produced by L-FIELD. Furthermore, we show how to perform parallel inference as message passing in a suitable factor graph at a linear convergence rate, without having to sum up over all the configurations of the factor. Finally, we apply our approach to a challenging image segmentation task. Our experiments confirm scalability of our approach, high quality of the marginals, and the benefit of incorporating higher-order potentials.
We introduce local expectation gradients which is a general purpose stochastic variational inference algorithm for constructing stochastic gradients through sampling from the variational distribution. This algorithm divides the problem of estimating the stochastic gradients over multiple variational parameters into smaller sub-tasks so that each sub-task exploits intelligently the information coming from the most relevant part of the variational distribution. This is achieved by performing an exact expectation over the single random variable that mostly correlates with the variational parameter of interest resulting in a Rao-Blackwellized estimate that has low variance and can work efficiently for both continuous and discrete random variables. Furthermore, the proposed algorithm has interesting similarities with Gibbs sampling but at the same time, unlike Gibbs sampling, it can be trivially parallelized.
Leveraging advances in variational inference, we propose to enhance recurrent neural networks with latent variables, resulting in Stochastic Recurrent Networks (STORNs). The model i) can be trained with stochastic gradient methods, ii) allows structured and multi-modal conditionals at each time step, iii) features a reliable estimator of the marginal likelihood and iv) is a generalisation of deterministic recurrent neural networks. We evaluate the method on four polyphonic musical data sets and motion capture data.
Multivariate categorical data occur in many applications of machine learning. One of the main difficulties with these vectors of categorical variables is sparsity. The number of possible observations grows exponentially with vector length, but dataset diversity might be poor in comparison. Recent models have gained significant improvement in supervised tasks with this data. These models embed observations in a continuous space to capture similarities between them. Building on these ideas we propose a Bayesian model for the unsupervised task of distribution estimation of multivariate categorical data. We model vectors of categorical variables as generated from a non-linear transformation of a continuous latent space. Non-linearity captures multi-modality in the distribution. The continuous representation addresses sparsity. Our model ties together many existing models, linking the linear categorical latent Gaussian model, the Gaussian process latent variable model, and Gaussian process classification. We derive inference for our model based on recent developments in sampling based variational inference. We show empirically that the model outperforms its linear and discrete counterparts in imputation tasks of sparse data.
Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance function with a finite Fourier series approximation and treat it as a random variable. The random covariance function has a posterior, on which a variational distribution is placed. The variational distribution transforms the random covariance function to fit the data. We study the properties of our approximate inference, compare it to alternative ones, and extend it to the distributed and stochastic domains. Our approximation captures complex functions better than standard approaches and avoids over-fitting.
Multi-output Gaussian processes have received increasing attention during the last few years as a natural mechanism to extend the powerful flexibility of Gaussian processes to the setup of multiple output variables. The key point here is the ability to design kernel functions that allow exploiting the correlations between the outputs while fulfilling the positive definiteness requisite for the covariance function. Alternatives to construct these covariance functions are the linear model of coregionalization and process convolutions. Each of these methods demand the specification of the number of latent Gaussian process used to build the covariance function for the outputs. We propose in this paper, the use of an Indian Buffet process as a way to perform model selection over the number of latent Gaussian processes. This type of model is particularly important in the context of latent force models, where the latent forces are associated to physical quantities like protein profiles or latent forces in mechanical systems. We use variational inference to estimate posterior distributions over the variables involved, and show examples of the model performance over artificial data, a motion capture dataset, and a gene expression dataset.
We show that unconverged stochastic gradient descent can be interpreted as a procedure that samples from a nonparametric variational approximate posterior distribution. This distribution is implicitly defined as the transformation of an initial distribution by a sequence of optimization updates. By tracking the change in entropy over this sequence of transformations during optimization, we form a scalable, unbiased estimate of the variational lower bound on the log marginal likelihood. We can use this bound to optimize hyperparameters instead of using cross-validation. This Bayesian interpretation of SGD suggests improved, overfitting-resistant optimization procedures, and gives a theoretical foundation for popular tricks such as early stopping and ensembling. We investigate the properties of this marginal likelihood estimator on neural network models.
We propose a generative model for robust tensor factorization in the presence of both missing data and outliers. The objective is to explicitly infer the underlying low-CP-rank tensor capturing the global information and a sparse tensor capturing the local information (also considered as outliers), thus providing the robust predictive distribution over missing entries. The low-CP-rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior, while the sparse tensor is modeled by a hierarchical view of Student-$t$ distribution that associates an individual hyperparameter with each element independently. For model learning, we develop an efficient closed-form variational inference under a fully Bayesian treatment, which can effectively prevent the overfitting problem and scales linearly with data size. In contrast to existing related works, our method can perform model selection automatically and implicitly without need of tuning parameters. More specifically, it can discover the groundtruth of CP rank and automatically adapt the sparsity inducing priors to various types of outliers. In addition, the tradeoff between the low-rank approximation and the sparse representation can be optimized in the sense of maximum model evidence. The extensive experiments and comparisons with many state-of-the-art algorithms on both synthetic and real-world datasets demonstrate the superiorities of our method from several perspectives.
Access to web-scale corpora is gradually bringing robust automatic knowledge base creation and extension within reach. To exploit these large unannotated---and extremely difficult to annotate---corpora, unsupervised machine learning methods are required. Probabilistic models of text have recently found some success as such a tool, but scalability remains an obstacle in their application, with standard approaches relying on sampling schemes that are known to be difficult to scale. In this report, we therefore present an empirical assessment of the sublinear time sparse stochastic variational inference (SSVI) scheme applied to RelLDA. We demonstrate that online inference leads to relatively strong qualitative results but also identify some of its pathologies---and those of the model---which will need to be overcome if SSVI is to be used for large-scale relation extraction.
In theory, Bayesian nonparametric (BNP) models are well suited to streaming data scenarios due to their ability to adapt model complexity with the observed data. Unfortunately, such benefits have not been fully realized in practice; existing inference algorithms are either not applicable to streaming applications or not extensible to BNP models. For the special case of Dirichlet processes, streaming inference has been considered. However, there is growing interest in more flexible BNP models building on the class of normalized random measures (NRMs). We work within this general framework and present a streaming variational inference algorithm for NRM mixture models. Our algorithm is based on assumed density filtering (ADF), leading straightforwardly to expectation propagation (EP) for large-scale batch inference as well. We demonstrate the efficacy of the algorithm on clustering documents in large, streaming text corpora.
The Gaussian process latent variable model (GP-LVM) is a popular approach to non-linear probabilistic dimensionality reduction. One design choice for the model is the number of latent variables. We present a spike and slab prior for the GP-LVM and propose an efficient variational inference procedure that gives a lower bound of the log marginal likelihood. The new model provides a more principled approach for selecting latent dimensions than the standard way of thresholding the length-scale parameters. The effectiveness of our approach is demonstrated through experiments on real and simulated data. Further, we extend multi-view Gaussian processes that rely on sharing latent dimensions (known as manifold relevance determination) with spike and slab priors. This allows a more principled approach for selecting a subset of the latent space for each view of data. The extended model outperforms the previous state-of-the-art when applied to a cross-modal multimedia retrieval task.
In this paper, we develop the continuous time dynamic topic model (cDTM). The cDTM is a dynamic topic model that uses Brownian motion to model the latent topics through a sequential collection of documents, where a "topic" is a pattern of word use that we expect to evolve over the course of the collection. We derive an efficient variational approximate inference algorithm that takes advantage of the sparsity of observations in text, a property that lets us easily handle many time points. In contrast to the cDTM, the original discrete-time dynamic topic model (dDTM) requires that time be discretized. Moreover, the complexity of variational inference for the dDTM grows quickly as time granularity increases, a drawback which limits fine-grained discretization. We demonstrate the cDTM on two news corpora, reporting both predictive perplexity and the novel task of time stamp prediction.