This paper introduces the $f$-divergence variational inference ($f$-VI) that generalizes variational inference to all $f$-divergences. Initiated from minimizing a crafty surrogate $f$-divergence that shares the statistical consistency with the $f$-divergence, the $f$-VI framework not only unifies a number of existing VI methods, e.g. Kullback-Leibler VI, R\'{e}nyi's $\alpha$-VI, and $\chi$-VI, but offers a standardized toolkit for VI subject to arbitrary divergences from $f$-divergence family. A general $f$-variational bound is derived and provides a sandwich estimate of marginal likelihood (or evidence). The development of the $f$-VI unfolds with a stochastic optimization scheme that utilizes the reparameterization trick, importance weighting and Monte Carlo approximation; a mean-field approximation scheme that generalizes the well-known coordinate ascent variational inference (CAVI) is also proposed for $f$-VI. Empirical examples, including variational autoencoders and Bayesian neural networks, are provided to demonstrate the effectiveness and the wide applicability of $f$-VI.
We develop a recurrent gamma belief network (rGBN) for radar automatic target recognition (RATR) based on high-resolution range profile (HRRP), which characterizes the temporal dependence across the range cells of HRRP. The proposed rGBN adopts a hierarchy of gamma distributions to build its temporal deep generative model. For scalable training and fast out-of-sample prediction, we propose the hybrid of a stochastic-gradient Markov chain Monte Carlo (MCMC) and a recurrent variational inference model to perform posterior inference. To utilize the label information to extract more discriminative latent representations, we further propose supervised rGBN to jointly model the HRRP samples and their corresponding labels. Experimental results on synthetic and measured HRRP data show that the proposed models are efficient in computation, have good classification accuracy and generalization ability, and provide highly interpretable multi-stochastic-layer latent structure.
We show how to use Stein variational gradient descent (SVGD) to carry out inference in Gaussian process (GP) models with non-Gaussian likelihoods and large data volumes. Markov chain Monte Carlo (MCMC) is extremely computationally intensive for these situations, but the parametric assumptions required for efficient variational inference (VI) result in incorrect inference when they encounter the multi-modal posterior distributions that are common for such models. SVGD provides a non-parametric alternative to variational inference which is substantially faster than MCMC. We prove that for GP models with Lipschitz gradients the SVGD algorithm monotonically decreases the Kullback-Leibler divergence from the sampling distribution to the true posterior. Our method is demonstrated on benchmark problems in both regression and classification, a multimodal posterior, and an air quality example with 550,134 spatiotemporal observations, showing substantial performance improvements over MCMC and VI.
Trajectory data generation is an important domain that characterizes the generative process of mobility data. Traditional methods heavily rely on predefined heuristics and distributions and are weak in learning unknown mechanisms. Inspired by the success of deep generative neural networks for images and texts, a fast-developing research topic is deep generative models for trajectory data which can learn expressively explanatory models for sophisticated latent patterns. This is a nascent yet promising domain for many applications. We first propose novel deep generative models factorizing time-variant and time-invariant latent variables that characterize global and local semantics, respectively. We then develop new inference strategies based on variational inference and constrained optimization to encapsulate the spatiotemporal validity. New deep neural network architectures have been developed to implement the inference and generation models with newly-generalized latent variable priors. The proposed methods achieved significant improvements in quantitative and qualitative evaluations in extensive experiments.
The linear-chain Conditional Random Field (CRF) model is one of the most widely-used neural sequence labeling approaches. Exact probabilistic inference algorithms such as the forward-backward and Viterbi algorithms are typically applied in training and prediction stages of the CRF model. However, these algorithms require sequential computation that makes parallelization impossible. In this paper, we propose to employ a parallelizable approximate variational inference algorithm for the CRF model. Based on this algorithm, we design an approximate inference network that can be connected with the encoder of the neural CRF model to form an end-to-end network, which is amenable to parallelization for faster training and prediction. The empirical results show that our proposed approaches achieve a 12.7-fold improvement in decoding speed with long sentences and a competitive accuracy compared with the traditional CRF approach.
Estimating time-varying graphical models are of paramount importance in various social, financial, biological, and engineering systems, since the evolution of such networks can be utilized for example to spot trends, detect anomalies, predict vulnerability, and evaluate the impact of interventions. Existing methods require extensive tuning of parameters that control the graph sparsity and temporal smoothness. Furthermore, these methods are computationally burdensome with time complexity $O(NP^3)$ for $P$ variables and $N$ time points. As a remedy, we propose a low-complexity tuning-free Bayesian approach, named BASS. Specifically, we impose temporally-dependent spike-and-slab priors on the graphs such that they are sparse and varying smoothly across time. A variational inference algorithm is then derived to learn the graph structures from the data automatically. Owning to the pseudo-likelihood and the mean-field approximation, the time complexity of BASS is only $O(NP^2)$. Additionally, by identifying the frequency-domain resemblance to the time-varying graphical models, we show that BASS can be extended to learning frequency-varying inverse spectral density matrices, and yields graphical models for multivariate stationary time series. Numerical results on both synthetic and real data show that that BASS can better recover the underlying true graphs, while being more efficient than the existing methods, especially for high-dimensional cases.
The COVID-19 pandemic provides new motivation for a classic problem in epidemiology: estimating the empirical rate of transmission during an outbreak (formally, the time-varying reproduction number) from case counts. While standard methods exist, they work best at coarse-grained national or state scales with abundant data, and struggle to accommodate the partial observability and sparse data common at finer scales (e.g., individual schools or towns). For example, case counts may be sparse when only a small fraction of infections are caught by a testing program. Or, whether an infected individual tests positive may depend on the kind of test and the point in time when they are tested. We propose a Bayesian framework which accommodates partial observability in a principled manner. Our model places a Gaussian process prior over the unknown reproduction number at each time step and models observations sampled from the distribution of a specific testing program. For example, our framework can accommodate a variety of kinds of tests (viral RNA, antibody, antigen, etc.) and sampling schemes (e.g., longitudinal or cross-sectional screening). Inference in this framework is complicated by the presence of tens or hundreds of thousands of discrete latent variables. To address this challenge, we propose an efficient stochastic variational inference method which relies on a novel gradient estimator for the variational objective. Experimental results for an example motivated by COVID-19 show that our method produces an accurate and well-calibrated posterior, while standard methods for estimating the reproduction number can fail badly.
We introduce deep switching auto-regressive factorization (DSARF), a deep generative model for spatio-temporal data with the capability to unravel recurring patterns in the data and perform robust short- and long-term predictions. Similar to other factor analysis methods, DSARF approximates high dimensional data by a product between time dependent weights and spatially dependent factors. These weights and factors are in turn represented in terms of lower dimensional latent variables that are inferred using stochastic variational inference. DSARF is different from the state-of-the-art techniques in that it parameterizes the weights in terms of a deep switching vector auto-regressive likelihood governed with a Markovian prior, which is able to capture the non-linear inter-dependencies among weights to characterize multimodal temporal dynamics. This results in a flexible hierarchical deep generative factor analysis model that can be extended to (i) provide a collection of potentially interpretable states abstracted from the process dynamics, and (ii) perform short- and long-term vector time series prediction in a complex multi-relational setting. Our extensive experiments, which include simulated data and real data from a wide range of applications such as climate change, weather forecasting, traffic, infectious disease spread and nonlinear physical systems attest the superior performance of DSARF in terms of long- and short-term prediction error, when compared with the state-of-the-art methods.
Data often are formed of multiple modalities, which jointly describe the observed phenomena. Modeling the joint distribution of multimodal data requires larger expressive power to capture high-level concepts and provide better data representations. However, multimodal generative models based on variational inference are limited due to the lack of flexibility of the approximate posterior, which is obtained by searching within a known parametric family of distributions. We introduce a method that improves the representational capacity of multimodal variational methods using normalizing flows. It approximates the joint posterior with a simple parametric distribution and subsequently transforms into a more complex one. Through several experiments, we demonstrate that the model improves on state-of-the-art multimodal methods based on variational inference on various computer vision tasks such as colorization, edge and mask detection, and weakly supervised learning. We also show that learning more powerful approximate joint distributions improves the quality of the generated samples. The code of our model is publicly available at https://github.com/SashoNedelkoski/BPFDMVM.
Tensor rank learning for canonical polyadic decomposition (CPD) has long been deemed as an essential yet challenging problem. In particular, since the tensor rank controls the complexity of the CPD model, its inaccurate learning would cause overfitting to noise or underfitting to the signal sources, and even destroy the interpretability of model parameters. However, the optimal determination of a tensor rank is known to be a non-deterministic polynomial-time hard (NP-hard) task. Rather than exhaustively searching for the best tensor rank via trial-and-error experiments, Bayesian inference under the Gaussian-gamma prior was introduced in the context of probabilistic CPD modeling, and it was shown to be an effective strategy for automatic tensor rank determination. This triggered flourishing research on other structured tensor CPDs with automatic tensor rank learning. On the other side of the coin, these research works also reveal that the Gaussian-gamma model does not perform well for high-rank tensors and/or low signal-to-noise ratios (SNRs). To overcome these drawbacks, in this paper, we introduce a more advanced generalized hyperbolic (GH) prior to the probabilistic CPD model, which not only includes the Gaussian-gamma model as a special case, but also is more flexible to adapt to different levels of sparsity. Based on this novel probabilistic model, an algorithm is developed under the framework of variational inference, where each update is obtained in a closed-form. Extensive numerical results, using synthetic data and real-world datasets, demonstrate the significantly improved performance of the proposed method in learning both low as well as high tensor ranks even for low SNR cases.
Variational autoencoder (VAE) is a very popular and well-investigated generative model in neural learning research. To leverage VAE in practical tasks dealing with a massive dataset of large dimensions, it is required to deal with the difficulty of building low variance evidence lower bounds (ELBO). Markov Chain Monte Carlo (MCMC) is an effective approach to tighten the ELBO for approximating the posterior distribution and Hamiltonian Variational Autoencoder (HVAE) is an effective MCMC inspired approach for constructing a low-variance ELBO that is amenable to the reparameterization trick. The HVAE adapted the Hamiltonian dynamic flow into variational inference that significantly improves the performance of the posterior estimation. We propose in this work a Langevin dynamic flow-based inference approach by incorporating the gradients information in the inference process through the Langevin dynamic which is a kind of MCMC based method similar to HVAE. Specifically, we employ a quasi-symplectic integrator to cope with the prohibit problem of the Hessian computing in naive Langevin flow. We show the theoretical and practical effectiveness of the proposed framework with other gradient flow-based methods.
A common goal in network modeling is to uncover the latent community structure present among nodes. For many real-world networks, the true connections consist of events arriving as streams, which are then aggregated to form edges, ignoring the dynamic temporal component. A natural way to take account of these temporal dynamics of interactions is to use point processes as the foundation of network models for community detection. Computational complexity hampers the scalability of such approaches to large sparse networks. To circumvent this challenge, we propose a fast online variational inference algorithm for estimating the latent structure underlying dynamic event arrivals on a network, using continuous-time point process latent network models. We describe this procedure for networks models capturing community structure. This structure can be learned as new events are observed on the network, updating the inferred community assignments. We investigate the theoretical properties of such an inference scheme, and provide regret bounds on the loss function of this procedure. The proposed inference procedure is then thoroughly compared, using both simulation studies and real data, to non-online variants. We demonstrate that online inference can obtain comparable performance, in terms of community recovery, to non-online variants, while realising computational gains. Our proposed inference framework can also be readily modified to incorporate other popular network structures.
This paper proposes to generalize the variational recurrent neural network (RNN) with variational inference (VI)-based dropout regularization employed for the long short-term memory (LSTM) cells to more advanced RNN architectures like gated recurrent unit (GRU) and bi-directional LSTM/GRU. The new variational RNNs are employed for slot filling, which is an intriguing but challenging task in spoken language understanding. The experiments on the ATIS dataset suggest that the variational RNNs with the VI-based dropout regularization can significantly improve the naive dropout regularization RNNs-based baseline systems in terms of F-measure. Particularly, the variational RNN with bi-directional LSTM/GRU obtains the best F-measure score.
Representation learning over graph structure data has been widely studied due to its wide application prospects. However, previous methods mainly focus on static graphs while many real-world graphs evolve over time. Modeling such evolution is important for predicting properties of unseen networks. To resolve this challenge, we propose SGRNN, a novel neural architecture that applies stochastic latent variables to simultaneously capture the evolution in node attributes and topology. Specifically, deterministic states are separated from stochastic states in the iterative process to suppress mutual interference. With semi-implicit variational inference integrated to SGRNN, a non-Gaussian variational distribution is proposed to help further improve the performance. In addition, to alleviate KL-vanishing problem in SGRNN, a simple and interpretable structure is proposed based on the lower bound of KL-divergence. Extensive experiments on real-world datasets demonstrate the effectiveness of the proposed model. Code is available at https://github.com/StochasticGRNN/SGRNN.
We consider the problem of fitting variational posterior approximations using stochastic optimization methods. The performance of these approximations depends on (1) how well the variational family matches the true posterior distribution,(2) the choice of divergence, and (3) the optimization of the variational objective. We show that even in the best-case scenario when the exact posterior belongs to the assumed variational family, common stochastic optimization methods lead to poor variational approximations if the problem dimension is moderately large. We also demonstrate that these methods are not robust across diverse model types. Motivated by these findings, we develop a more robust and accurate stochastic optimization framework by viewing the underlying optimization algorithm as producing a Markov chain. Our approach is theoretically motivated and includes a diagnostic for convergence and a novel stopping rule, both of which are robust to noisy evaluations of the objective function. We show empirically that the proposed framework works well on a diverse set of models: it can automatically detect stochastic optimization failure or inaccurate variational approximation
In the past few years, deep generative models, such as generative adversarial networks \autocite{GAN}, variational autoencoders \autocite{vaepaper}, and their variants, have seen wide adoption for the task of modelling complex data distributions. In spite of the outstanding sample quality achieved by those early methods, they model the target distributions \emph{implicitly}, in the sense that the probability density functions induced by them are not explicitly accessible. This fact renders those methods unfit for tasks that require, for example, scoring new instances of data with the learned distributions. Normalizing flows have overcome this limitation by leveraging the change-of-variables formula for probability density functions, and by using transformations designed to have tractable and cheaply computable Jacobians. Although flexible, this framework lacked (until recently \autocites{semisuplearning_nflows, RAD}) a way to introduce discrete structure (such as the one found in mixtures) in the models it allows to construct, in an unsupervised scenario. The present work overcomes this by using normalizing flows as components in a mixture model and devising an end-to-end training procedure for such a model. This procedure is based on variational inference, and uses a variational posterior parameterized by a neural network. As will become clear, this model naturally lends itself to (multimodal) density estimation, semi-supervised learning, and clustering. The proposed model is illustrated on two synthetic datasets, as well as on a real-world dataset. Keywords: Deep generative models, normalizing flows, variational inference, probabilistic modelling, mixture models.
We introduce a non-parametric hierarchical Bayesian approach for open-ended 3D object categorization, named the Local Hierarchical Dirichlet Process (Local-HDP). This method allows an agent to learn independent topics for each category incrementally and to adapt to the environment in time. Hierarchical Bayesian approaches like Latent Dirichlet Allocation (LDA) can transform low-level features to high-level conceptual topics for 3D object categorization. However, the efficiency and accuracy of LDA-based approaches depend on the number of topics that is chosen manually. Moreover, fixing the number of topics for all categories can lead to overfitting or underfitting of the model. In contrast, the proposed Local-HDP can autonomously determine the number of topics for each category. Furthermore, the online variational inference method has been adapted for fast posterior approximation in the Local-HDP model. Experiments show that the proposed Local-HDP method outperforms other state-of-the-art approaches in terms of accuracy, scalability, and memory efficiency by a large margin. Moreover, two robotic experiments have been conducted to show the applicability of the proposed approach in real-time applications.
For a learning task, Gaussian process (GP) is interested in learning the statistical relationship between inputs and outputs, since it offers not only the prediction mean but also the associated variability. The vanilla GP however struggles to learn complicated distribution with the property of, e.g., heteroscedastic noise, multi-modality and non-stationarity, from massive data due to the Gaussian marginal and the cubic complexity. To this end, this article studies new scalable GP paradigms including the non-stationary heteroscedastic GP, the mixture of GPs and the latent GP, which introduce additional latent variables to modulate the outputs or inputs in order to learn richer, non-Gaussian statistical representation. We further resort to different variational inference strategies to arrive at analytical or tighter evidence lower bounds (ELBOs) of the marginal likelihood for efficient and effective model training. Extensive numerical experiments against state-of-the-art GP and neural network (NN) counterparts on various tasks verify the superiority of these scalable modulated GPs, especially the scalable latent GP, for learning diverse data distributions.
We extend the empirical results from the structural equation model (SEM) published in the paper Assortment Planning for Retail Buying, Retail Store Operations, and Firm Performance [1] by implementing the directed acyclic graph as a causal Bayesian neural network. Neural network convergence is shown to improve with the removal of the node with the weakest SEM path when variational inference is provided by perturbing weights with Flipout layers, while results from perturbing weights at the output with the Vadam optimizer are inconclusive.
Modern machine learning applications should be able to address the intrinsic challenges arising over inference on massive real-world datasets, including scalability and robustness to outliers. Despite the multiple benefits of Bayesian methods (such as uncertainty-aware predictions, incorporation of experts knowledge, and hierarchical modeling), the quality of classic Bayesian inference depends critically on whether observations conform with the assumed data generating model, which is impossible to guarantee in practice. In this work, we propose a variational inference method that, in a principled way, can simultaneously scale to large datasets, and robustify the inferred posterior with respect to the existence of outliers in the observed data. Reformulating Bayes theorem via the $\beta$-divergence, we posit a robustified pseudo-Bayesian posterior as the target of inference. Moreover, relying on the recent formulations of Riemannian coresets for scalable Bayesian inference, we propose a sparse variational approximation of the robustified posterior and an efficient stochastic black-box algorithm to construct it. Overall our method allows releasing cleansed data summaries that can be applied broadly in scenarios including structured data corruption. We illustrate the applicability of our approach in diverse simulated and real datasets, and various statistical models, including Gaussian mean inference, logistic and neural linear regression, demonstrating its superiority to existing Bayesian summarization methods in the presence of outliers.