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.
Topic modeling analyzes documents to learn meaningful patterns of words. For documents collected in sequence, dynamic topic models capture how these patterns vary over time. We develop the dynamic embedded topic model (D-ETM), a generative model of documents that combines dynamic latent Dirichlet allocation (D-LDA) and word embeddings. The D-ETM models each word with a categorical distribution parameterized by the inner product between the word embedding and a per-time-step embedding representation of its assigned topic. The D-ETM learns smooth topic trajectories by defining a random walk prior over the embedding representations of the topics. We fit the D-ETM using structured amortized variational inference with a recurrent neural network. On three different corpora---a collection of United Nations debates, a set of ACL abstracts, and a dataset of Science Magazine articles---we found that the D-ETM outperforms D-LDA on a document completion task. We further found that the D-ETM learns more diverse and coherent topics than D-LDA while requiring significantly less time to fit.
Topic modeling analyzes documents to learn meaningful patterns of words. However, existing topic models fail to learn interpretable topics when working with large and heavy-tailed vocabularies. To this end, we develop the Embedded Topic Model (ETM), a generative model of documents that marries traditional topic models with word embeddings. In particular, it models each word with a categorical distribution whose natural parameter is the inner product between a word embedding and an embedding of its assigned topic. To fit the ETM, we develop an efficient amortized variational inference algorithm. The ETM discovers interpretable topics even with large vocabularies that include rare words and stop words. It outperforms existing document models, such as latent Dirichlet allocation (LDA), in terms of both topic quality and predictive performance.
In this paper, we present a Bayesian view on model-based reinforcement learning. We use expert knowledge to impose structure on the transition model and present an efficient learning scheme based on variational inference. This scheme is applied to a heteroskedastic and bimodal benchmark problem on which we compare our results to NFQ and show how our approach yields human-interpretable insight about the underlying dynamics while also increasing data-efficiency.
Datasets in engineering applications are often limited and contaminated, mainly due to unavoidable measurement noise and signal distortion. Thus, using conventional data-driven approaches to build a reliable discriminative model, and further applying this identified surrogate to uncertainty analysis remains to be very challenging. A deep learning approach is presented to provide predictions based on limited and noisy data. To address noise perturbation, the Bayesian learning method that naturally facilitates an automatic updating mechanism is considered to quantify and propagate model uncertainties into predictive quantities. Specifically, hierarchical Bayesian modeling (HBM) is first adopted to describe model uncertainties, which allows the prior assumption to be less subjective, while also makes the proposed surrogate more robust. Next, the Bayesian inference is seamlessly integrated into the DL framework, which in turn supports probabilistic programming by yielding a probability distribution of the quantities of interest rather than their point estimates. Variational inference (VI) is implemented for the posterior distribution analysis where the intractable marginalization of the likelihood function over parameter space is framed in an optimization format, and stochastic gradient descent method is applied to solve this optimization problem. Finally, Monte Carlo simulation is used to obtain an unbiased estimator in the predictive phase of Bayesian inference, where the proposed Bayesian deep learning (BDL) scheme is able to offer confidence bounds for the output estimation by analyzing propagated uncertainties. The effectiveness of Bayesian shrinkage is demonstrated in improving predictive performance using contaminated data, and various examples are provided to illustrate concepts, methodologies, and algorithms of this proposed BDL modeling technique.
Many methods for machine learning rely on approximate inference from intractable probability distributions. Variational inference approximates such distributions by tractable models that can be subsequently used for approximate inference. Learning sufficiently accurate approximations requires a rich model family and careful exploration of the relevant modes of the target distribution. We propose a method for learning accurate GMM approximations of intractable probability distributions based on insights from policy search by using information-geometric trust regions for principled exploration. For efficient improvement of the GMM approximation, we derive a lower bound on the corresponding optimization objective enabling us to update the components independently. Our use of the lower bound ensures convergence to a stationary point of the original objective. The number of components is adapted online by adding new components in promising regions and by deleting components with negligible weight. We demonstrate on several domains that we can learn approximations of complex, multimodal distributions with a quality that is unmet by previous variational inference methods, and that the GMM approximation can be used for drawing samples that are on par with samples created by state-of-the-art MCMC samplers while requiring up to three orders of magnitude less computational resources.
In recent studies on model-based reinforcement learning (MBRL), incorporating uncertainty in forward dynamics is a state-of-the-art strategy to enhance learning performance, making MBRLs competitive to cutting-edge model free methods, especially in simulated robotics tasks. Probabilistic ensembles with trajectory sampling (PETS) is a leading type of MBRL, which employs Bayesian inference to dynamics modeling and model predictive control (MPC) with stochastic optimization via the cross entropy method (CEM). In this paper, we propose a novel extension to the uncertainty-aware MBRL. Our main contributions are twofold: Firstly, we introduce a variational inference MPC, which reformulates various stochastic methods, including CEM, in a Bayesian fashion. Secondly, we propose a novel instance of the framework, called probabilistic action ensembles with trajectory sampling (PaETS). As a result, our Bayesian MBRL can involve multimodal uncertainties both in dynamics and optimal trajectories. In comparison to PETS, our method consistently improves asymptotic performance on several challenging locomotion tasks.
Convolutional neural networks (CNNs) in recent years have made a dramatic impact in science, technology and industry, yet the theoretical mechanism of CNN architecture design remains surprisingly vague. The CNN neurons, including its distinctive element, convolutional filters, are known to be learnable features, yet their individual role in producing the output is rather unclear. The thesis of this work is that not all neurons are equally important and some of them contain more useful information to perform a given task . Consequently, we quantify the significance of each filter and rank its importance in describing input to produce the desired output. This work presents two different methods: (1) a game theoretical approach based on Shapley value which computes the marginal contribution of each filter; and (2) a probabilistic approach based on what-we-call, the Importance switch using variational inference. Strikingly, these two vastly different methods produce similar experimental results, confirming the general theory that some of the filters are inherently more important that the others. The learned ranks can be readily useable for network compression and interpretability.
As electronically stored data grow in daily life, obtaining novel and relevant information becomes challenging in text mining. Thus people have sought statistical methods based on term frequency, matrix algebra, or topic modeling for text mining. Popular topic models have centered on one single text collection, which is deficient for comparative text analyses. We consider a setting where one can partition the corpus into subcollections. Each subcollection shares a common set of topics, but there exists relative variation in topic proportions among collections. Including any prior knowledge about the corpus (e.g. organization structure), we propose the compound latent Dirichlet allocation (cLDA) model, improving on previous work, encouraging generalizability, and depending less on user-input parameters. To identify the parameters of interest in cLDA, we study Markov chain Monte Carlo (MCMC) and variational inference approaches extensively, and suggest an efficient MCMC method. We evaluate cLDA qualitatively and quantitatively using both synthetic and real-world corpora. The usability study on some real-world corpora illustrates the superiority of cLDA to explore the underlying topics automatically but also model their connections and variations across multiple collections.
Gaussian Process Latent Variable Model (GPLVM) is a flexible framework to handle uncertain inputs in Gaussian Processes (GPs) and incorporate GPs as components of larger graphical models. Nonetheless, the standard GPLVM variational inference approach is tractable only for a narrow family of kernel functions. The most popular implementations of GPLVM circumvent this limitation using quadrature methods, which may become a computational bottleneck even for relatively low dimensions. For instance, the widely employed Gauss-Hermite quadrature has exponential complexity on the number of dimensions. In this work, we propose using the unscented transformation instead. Overall, this method presents comparable, if not better, performance than offthe-shelf solutions to GPLVM and its computational complexity scales only linearly on dimension. In contrast to Monte Carlo methods, our approach is deterministic and works well with quasi-Newton methods, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. We illustrate the applicability of our method with experiments on dimensionality reduction and multistep-ahead prediction with uncertainty propagation.
We propose Radial Bayesian Neural Networks (BNNs): a variational approximate posterior for BNNs which scales well to large models while maintaining a distribution over weight-space with full support. Other scalable Bayesian deep learning methods, like MC dropout or deep ensembles, have discrete support-they assign zero probability to almost all of the weight-space. Unlike these discrete support methods, Radial BNNs' full support makes them suitable for use as a prior for sequential inference. In addition, they solve the conceptual challenges with the a priori implausibility of weight distributions with discrete support. The Radial BNN is motivated by avoiding a sampling problem in 'mean-field' variational inference (MFVI) caused by the so-called 'soap-bubble' pathology of multivariate Gaussians. We show that, unlike MFVI, Radial BNNs are robust to hyperparameters and can be efficiently applied to a challenging real-world medical application without needing ad-hoc tweaks and intensive tuning. In fact, in this setting Radial BNNs out-perform discrete-support methods like MC dropout. Lastly, by using Radial BNNs as a theoretically principled, robust alternative to MFVI we make significant strides in a Bayesian continual learning evaluation.
One of the most surprising and exciting discoveries in supervised learning was the benefit of overparameterization (i.e. training a very large model) to improving the optimization landscape of a problem, with minimal effect on statistical performance (i.e. generalization). In contrast, unsupervised settings have been under-explored, despite the fact that it was observed that overparameterization can be helpful as early as Dasgupta & Schulman (2007). We perform an empirical study of different aspects of overparameterization in unsupervised learning of latent variable models via synthetic and semi-synthetic experiments. We discuss benefits to different metrics of success (recovering the parameters of the ground-truth model, held-out log-likelihood), sensitivity to variations of the training algorithm, and behavior as the amount of overparameterization increases. We find that across a variety of models (noisy-OR networks, sparse coding, probabilistic context-free grammars) and training algorithms (variational inference, alternating minimization, expectation-maximization), overparameterization can significantly increase the number of ground truth latent variables recovered.
We introduce the thermodynamic variational objective (TVO) for learning in both continuous and discrete deep generative models. The TVO arises from a key connection between variational inference and thermodynamic integration that results in a tighter lower bound to the log marginal likelihood than the standard variational variational evidence lower bound (ELBO) while remaining as broadly applicable. We provide a computationally efficient gradient estimator for the TVO that applies to continuous, discrete, and non-reparameterizable distributions and show that the objective functions used in variational inference, variational autoencoders, wake sleep, and inference compilation are all special cases of the TVO. We use the TVO to learn both discrete and continuous deep generative models and empirically demonstrate state of the art model and inference network learning.
As 3D point clouds become the representation of choice for multiple vision and graphics applications, the ability to synthesize or reconstruct high-resolution, high-fidelity point clouds becomes crucial. Despite the recent success of deep learning models in discriminative tasks of point clouds, generating point clouds remains challenging. This paper proposes a principled probabilistic framework to generate 3D point clouds by modeling them as a distribution of distributions. Specifically, we learn a two-level hierarchy of distributions where the first level is the distribution of shapes and the second level is the distribution of points given a shape. This formulation allows us to both sample shapes and sample an arbitrary number of points from a shape. Our generative model, named PointFlow, learns each level of the distribution with a continuous normalizing flow. The invertibility of normalizing flows enables the computation of the likelihood during training and allows us to train our model in the variational inference framework. Empirically, we demonstrate that PointFlow achieves state-of-the-art performance in point cloud generation. We additionally show that our model can faithfully reconstruct point clouds and learn useful representations in an unsupervised manner. The code will be available at https://github.com/stevenygd/PointFlow.
We describe a limitation in the expressiveness of the predictive uncertainty estimate given by mean-field variational inference (MFVI), a popular approximate inference method for Bayesian neural networks. In particular, MFVI fails to give calibrated uncertainty estimates in between separated regions of observations. This can lead to catastrophically overconfident predictions when testing on out-of-distribution data. Avoiding such overconfidence is critical for active learning, Bayesian optimisation and out-of-distribution robustness. We instead find that a classical technique, the linearised Laplace approximation, can handle 'in-between' uncertainty much better for small network architectures.
The problem of estimating event truths from conflicting agent opinions in a social network is investigated. An autoencoder learns the complex relationships between event truths, agent reliabilities and agent observations. A Bayesian network model is proposed to guide the learning process by modeling the relationship of the autoencoder's outputs with different variables. At the same time, it also models the social relationships between agents in the network. The proposed approach is unsupervised and is applicable when ground truth labels of events are unavailable. A variational inference method is used to jointly estimate the hidden variables in the Bayesian network and the parameters in the autoencoder. Experiments on three real datasets demonstrate that our proposed approach is competitive with, and in most cases better than, several state-of-the-art benchmark methods.
We study a formalization of the grammar induction problem that models sentences as being generated by a compound probabilistic context-free grammar. In contrast to traditional formulations which learn a single stochastic grammar, our grammar's rule probabilities are modulated by a per-sentence continuous latent variable, which induces marginal dependencies beyond the traditional context-free assumptions. Inference in this grammar is performed by collapsed variational inference, in which an amortized variational posterior is placed on the continuous variable, and the latent trees are marginalized out with dynamic programming. Experiments on English and Chinese show the effectiveness of our approach compared to recent state-of-the-art methods when evaluated on unsupervised parsing.
Supervised Deep Learning has been highly successful in recent years, achieving state-of-the-art results in most tasks. However, with the ongoing uptake of such methods in industrial applications, the requirement for large amounts of annotated data is often a challenge. In most real world problems, manual annotation is practically intractable due to time/labour constraints, thus the development of automated and adaptive data annotation systems is highly sought after. In this paper, we propose both a (i) Deep Bayesian Self-Training methodology for automatic data annotation, by leveraging predictive uncertainty estimates using variational inference and modern Neural Network architectures, as well as (ii) a practical adaptation procedure for handling high label variability between different dataset distributions through clustering of Neural Network latent variable representations. An experimental study on both public and private datasets is presented illustrating the superior performance of the proposed approach over standard Self-Training baselines, highlighting the importance of predictive uncertainty estimates in safety-critical domains.
We investigate the potential of stochastic neural networks for learning effective waveform-based acoustic models. The waveform-based setting, inherent to fully end-to-end speech recognition systems, is motivated by several comparative studies of automatic and human speech recognition that associate standard non-adaptive feature extraction techniques with information loss which can adversely affect robustness. Stochastic neural networks, on the other hand, are a class of models capable of incorporating rich regularization mechanisms into the learning process. We consider a deep convolutional neural network that first decomposes speech into frequency sub-bands via an adaptive parametric convolutional block where filters are specified by cosine modulations of compactly supported windows. The network then employs standard non-parametric 1D convolutions to extract relevant spectro-temporal patterns while gradually compressing the structured high dimensional representation generated by the parametric block. We rely on a probabilistic parametrization of the proposed neural architecture and learn the model using stochastic variational inference. This requires evaluation of an analytically intractable integral defining the Kullback-Leibler divergence term responsible for regularization, for which we propose an effective approximation based on the Gauss-Hermite quadrature. Our empirical results demonstrate a superior performance of the proposed approach over comparable waveform-based baselines and indicate that it could lead to robustness. Moreover, the approach outperforms a recently proposed deep convolutional neural network for learning of robust acoustic models with standard FBANK features.
We implement gradient-based variational inference routines for Wishart and inverse Wishart processes, which we apply as Bayesian models for the dynamic, heteroskedastic covariance matrix of a multivariate time series. The Wishart and inverse Wishart processes are constructed from i.i.d. Gaussian processes, existing variational inference algorithms for which form the basis of our approach. These methods are easy to implement as a black-box and scale favorably with the length of the time series, however, they fail in the case of the Wishart process, an issue we resolve with a simple modification into an additive white noise parameterization of the model. This modification is also key to implementing a factored variant of the construction, allowing inference to additionally scale to high-dimensional covariance matrices. Through experimentation, we demonstrate that some (but not all) model variants outperform multivariate GARCH when forecasting the covariances of returns on financial instruments.