Recent work in variational inference (VI) uses ideas from Monte Carlo estimation to tighten the lower bounds on the log-likelihood that are used as objectives. However, there is no systematic understanding of how optimizing different objectives relates to approximating the posterior distribution. Developing such a connection is important if the ideas are to be applied to inference-i.e., applications that require an approximate posterior and not just an approximation of the log-likelihood. Given a VI objective defined by a Monte Carlo estimator of the likelihood, we use a "divide and couple" procedure to identify augmented proposal and target distributions. The divergence between these is equal to the gap between the VI objective and the log-likelihood. Thus, after maximizing the VI objective, the augmented variational distribution may be used to approximate the posterior distribution.
We propose a novel framework for multi-task reinforcement learning (MTRL). Using a variational inference formulation, we learn policies that generalize across both changing dynamics and goals. The resulting policies are parametrized by shared parameters that allow for transfer between different dynamics and goal conditions, and by task-specific latent-space embeddings that allow for specialization to particular tasks. We show how the latent-spaces enable generalization to unseen dynamics and goals conditions. Additionally, policies equipped with such embeddings serve as a space of skills (or options) for hierarchical reinforcement learning. Since we can change task dynamics and goals independently, we name our framework Disentangled Skill Embeddings (DSE).
Latent force models are systems whereby there is a mechanistic model describing the dynamics of the system state, with some unknown forcing term that is approximated with a Gaussian process. If such dynamics are non-linear, it can be difficult to estimate the posterior state and forcing term jointly, particularly when there are system parameters that also need estimating. This paper uses black-box variational inference to jointly estimate the posterior, designing a multivariate extension to local inverse autoregressive flows as a flexible approximater of the system. We compare estimates on systems where the posterior is known, demonstrating the effectiveness of the approximation, and apply to problems with non-linear dynamics, multi-output systems and models with non-Gaussian likelihoods.
Recent variational inference methods use stochastic gradient estimators whose variance is not well understood. Theoretical guarantees for these estimators are important to understand when these methods will or will not work. This paper gives bounds for the common "reparameterization" estimators when the target is smooth and the variational family is a location-scale distribution. These bounds are unimprovable and thus provide the best possible guarantees under the stated assumptions.
Semantic dependency parsing aims to identify semantic relationships between words in a sentence that form a graph. In this paper, we propose a second-order semantic dependency parser, which takes into consideration not only individual dependency edges but also interactions between pairs of edges. We show that second-order parsing can be approximated using mean field (MF) variational inference or loopy belief propagation (LBP). We can unfold both algorithms as recurrent layers of a neural network and therefore can train the parser in an end-to-end manner. Our experiments show that our approach achieves state-of-the-art performance.
This paper focuses on two fundamental tasks of graph analysis: community detection and node representation learning, which capture the global and local structures of graphs, respectively. In the current literature, these two tasks are usually independently studied while they are actually highly correlated. We propose a probabilistic generative model called vGraph to learn community membership and node representation collaboratively. Specifically, we assume that each node can be represented as a mixture of communities, and each community is defined as a multinomial distribution over nodes. Both the mixing coefficients and the community distribution are parameterized by the low-dimensional representations of the nodes and communities. We designed an effective variational inference algorithm which regularizes the community membership of neighboring nodes to be similar in the latent space. Experimental results on multiple real-world graphs show that vGraph is very effective in both community detection and node representation learning, outperforming many competitive baselines in both tasks. We show that the framework of vGraph is quite flexible and can be easily extended to detect hierarchical communities.
The Black Box Variational Inference (Ranganath et al. (2014)) algorithm provides a universal method for Variational Inference, but taking advantage of special properties of the approximation family or of the target can improve the convergence speed significantly. For example, if the approximation family is a transformation family, such as a Gaussian, then switching to the reparameterization gradient (Kingma and Welling (2014)) often yields a major reduction in gradient variance. Ultimately, reducing the variance can reduce the computational cost and yield better approximations. We present a new method to extend the reparameterization trick to more general exponential families including the Wishart, Gamma, and Student distributions. Variational Inference with Numerical Derivatives (VIND) approximates the gradient with numerical derivatives and reduces its variance using a tight coupling of the approximation family. The resulting algorithm is simple to implement and can profit from widely known couplings. Our experiments confirm that VIND effectively decreases the gradient variance and therefore improves the posterior approximation in relevant cases. It thus provides an efficient yet simple Variational Inference method for computing non-Gaussian approximations.
The unsupervised training of GANs and VAEs has enabled them to generate realistic images mimicking real-world distributions and perform image-based unsupervised clustering or semi-supervised classification. Combining the power of these two generative models, we introduce Multi-Adversarial Variational autoEncoder Networks (MAVENs), a novel network architecture that incorporates an ensemble of discriminators in a VAE-GAN network, with simultaneous adversarial learning and variational inference. We apply MAVENs to the generation of synthetic images and propose a new distribution measure to quantify the quality of the generated images. Our experimental results using datasets from the computer vision and medical imaging domains---Street View House Numbers, CIFAR-10, and Chest X-Ray datasets---demonstrate competitive performance against state-of-the-art semi-supervised models both in image generation and classification tasks.
In federated learning, a central server coordinates the training of a single model on a massively distributed network of devices. This setting can be naturally extended to a multi-task learning framework, to handle real-world federated datasets that typically show strong statistical heterogeneity among devices. Despite federated multi-task learning being shown to be an effective paradigm for real-world datasets, it has been applied only on convex models. In this work, we introduce VIRTUAL, an algorithm for federated multi-task learning for general non-convex models. In VIRTUAL the federated network of the server and the clients is treated as a star-shaped Bayesian network, and learning is performed on the network using approximated variational inference. We show that this method is effective on real-world federated datasets, outperforming the current state-of-the-art for federated learning, and concurrently allowing sparser gradient updates.
Stochastic variational inference for Bayesian deep neural network (DNN) requires specifying priors and approximate posterior distributions over neural network weights. Specifying meaningful weight priors is a challenging problem, particularly for scaling variational inference to deeper architectures involving high dimensional weight space. We propose MOdel Priors with Empirical Bayes using DNN (MOPED) method to choose informed weight priors in Bayesian neural networks. We formulate a two-stage hierarchical modeling, first find the maximum likelihood estimates of weights with DNN, and then set the weight priors using empirical Bayes approach to infer the posterior with variational inference. We empirically evaluate the proposed approach on real-world tasks including image classification, video activity recognition and audio classification with varying complex neural network architectures. We also evaluate our proposed approach on diabetic retinopathy diagnosis task and benchmark with the state-of-the-art Bayesian deep learning techniques. We demonstrate MOPED method enables scalable variational inference and provides reliable uncertainty quantification.
Training deep generative models with maximum likelihood remains a challenge. The typical workaround is to use variational inference (VI) and maximize a lower bound to the log marginal likelihood of the data. Variational auto-encoders (VAEs) adopt this approach. They further amortize the cost of inference by using a recognition network to parameterize the variational family. Amortized VI scales approximate posterior inference in deep generative models to large datasets. However it introduces an amortization gap and leads to approximate posteriors of reduced expressivity due to the problem known as posterior collapse. In this paper, we consider expectation maximization (EM) as a paradigm for fitting deep generative models. Unlike VI, EM directly maximizes the log marginal likelihood of the data. We rediscover the importance weighted auto-encoder (IWAE) as an instance of EM and propose a new EM-based algorithm for fitting deep generative models called reweighted expectation maximization (REM). REM learns better generative models than the IWAE by decoupling the learning dynamics of the generative model and the recognition network using a separate expressive proposal found by moment matching. We compared REM to the VAE and the IWAE on several density estimation benchmarks and found it leads to significantly better performance as measured by log-likelihood.
We identify a new variational inference scheme for dynamical systems whose transition function is modelled by a Gaussian process. Inference in this setting has either employed computationally intensive MCMC methods, or relied on factorisations of the variational posterior. As we demonstrate in our experiments, the factorisation between latent system states and transition function can lead to a miscalibrated posterior and to learning unnecessarily large noise terms. We eliminate this factorisation by explicitly modelling the dependence between state trajectories and the Gaussian process posterior. Samples of the latent states can then be tractably generated by conditioning on this representation. The method we obtain (VCDT: variationally coupled dynamics and trajectories) gives better predictive performance and more calibrated estimates of the transition function, yet maintains the same time and space complexities as mean-field methods. Code is available at: github.com/ialong/GPt.
Recent advances in Neural Variational Inference allowed for a renaissance in latent variable models in a variety of domains involving high-dimensional data. While traditional variational methods derive an analytical approximation for the intractable distribution over the latent variables, here we construct an inference network conditioned on the symbolic representation of entities and relation types in the Knowledge Graph, to provide the variational distributions. The new framework results in a highly-scalable method. Under a Bernoulli sampling framework, we provide an alternative justification for commonly used techniques in large-scale stochastic variational inference, which drastically reduce training time at a cost of an additional approximation to the variational lower bound. We introduce two models from this highly scalable probabilistic framework, namely the Latent Information and Latent Fact models, for reasoning over knowledge graph-based representations. Our Latent Information and Latent Fact models improve upon baseline performance under certain conditions. We use the learnt embedding variance to estimate predictive uncertainty during link prediction, and discuss the quality of these learnt uncertainty estimates. Our source code and datasets are publicly available online at https://github.com/alexanderimanicowenrivers/Neural-Variational-Knowledge-Graphs.
Variational inference (VI) and Markov chain Monte Carlo (MCMC) are two main approximate approaches for learning deep generative models by maximizing marginal likelihood. In this paper, we propose using annealed importance sampling for learning deep generative models. Our proposed approach bridges VI with MCMC. It generalizes VI methods such as variational auto-encoders and importance weighted auto-encoders (IWAE) and the MCMC method proposed in (Hoffman, 2017). It also provides insights into why running multiple short MCMC chains can help learning deep generative models. Through experiments, we show that our approach yields better density models than IWAE and can effectively trade computation for model accuracy without increasing memory cost.
We focus on the problem of streaming recommender system and explore novel collaborative filtering algorithms to handle the data dynamicity and complexity in a streaming manner. Although deep neural networks have demonstrated the effectiveness of recommendation tasks, it is lack of explorations on integrating probabilistic models and deep architectures under streaming recommendation settings. Conjoining the complementary advantages of probabilistic models and deep neural networks could enhance both model effectiveness and the understanding of inference uncertainties. To bridge the gap, in this paper, we propose a Coupled Variational Recurrent Collaborative Filtering (CVRCF) framework based on the idea of Deep Bayesian Learning to handle the streaming recommendation problem. The framework jointly combines stochastic processes and deep factorization models under a Bayesian paradigm to model the generation and evolution of users' preferences and items' popularities. To ensure efficient optimization and streaming update, we further propose a sequential variational inference algorithm based on a cross variational recurrent neural network structure. Experimental results on three benchmark datasets demonstrate that the proposed framework performs favorably against the state-of-the-art methods in terms of both temporal dependency modeling and predictive accuracy. The learned latent variables also provide visualized interpretations for the evolution of temporal dynamics.
Recent advances in variational inference enable the modelling of highly structured joint distributions, but are limited in their capacity to scale to the high-dimensional setting of stochastic neural networks. This limitation motivates a need for scalable parameterizations of the noise generation process, in a manner that adequately captures the dependencies among the various parameters. In this work, we address this need and present the Kronecker Flow, a generalization of the Kronecker product to invertible mappings designed for stochastic neural networks. We apply our method to variational Bayesian neural networks on predictive tasks, PAC-Bayes generalization bound estimation, and approximate Thompson sampling in contextual bandits. In all setups, our methods prove to be competitive with existing methods and better than the baselines.
Variational methods are employed in situations where exact Bayesian inference becomes intractable due to the difficulty in performing certain integrals. Typically, variational methods postulate a tractable posterior and formulate a lower bound on the desired integral to be approximated, e.g. marginal likelihood. The lower bound is then optimised with respect to its free parameters, the so called variational parameters. However, this is not always possible as for certain integrals it is very challenging (or tedious) to come up with a suitable lower bound. Here we propose a simple scheme that overcomes some of the awkward cases where the usual variational treatment becomes difficult. The scheme relies on a rewriting of the lower bound on the model log-likelihood. We demonstrate the proposed scheme on a number of synthetic and real examples, as well as on a real geophysical model for which the standard variational approaches are inapplicable.
The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in extracting the Koopman operator from a data-driven perspective, several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural networks in a measure-theoretic framework. Our approach induces two types of models: differential and recurrent form, the choice of which depends on the availability of the governing equations and data. We then enforce a structural parameterization that renders the realization of the Koopman operator provably stable. A new autoencoder architecture is constructed, such that only the residual of the dynamic mode decomposition is learned. Finally, we employ mean-field variational inference (MFVI) on the aforementioned framework in a hierarchical Bayesian setting to quantify uncertainties in the characterization and prediction of the dynamics of observables. The framework is evaluated on a simple polynomial system, the Duffing oscillator, and an unstable cylinder wake flow with noisy measurements.
In order to better model high-dimensional sequential data, we propose a collaborative multi-output Gaussian process dynamical system (CGPDS), which is a novel variant of GPDSs. The proposed model assumes that the output on each dimension is controlled by a shared global latent process and a private local latent process. Thus, the dependence among different dimensions of the sequences can be captured, and the unique characteristics of each dimension of the sequences can be maintained. For training models and making prediction, we introduce inducing points and adopt stochastic variational inference methods.
The proliferation of automated inference algorithms in Bayesian statistics has provided practitioners newfound access to fast, reproducible data analysis and powerful statistical models. Designing automated methods that are also both computationally scalable and theoretically sound, however, remains a significant challenge. Recent work on Bayesian coresets takes the approach of compressing the dataset before running a standard inference algorithm, providing both scalability and guarantees on posterior approximation error. But the automation of past coreset methods is limited because they depend on the availability of a reasonable coarse posterior approximation, which is difficult to specify in practice. In the present work we remove this requirement by formulating coreset construction as sparsity-constrained variational inference within an exponential family. This perspective leads to a novel construction via greedy optimization, and also provides a unifying information-geometric view of present and past methods. The proposed Riemannian coreset construction algorithm is fully automated, requiring no problem-specific inputs aside from the probabilistic model and dataset. In addition to being significantly easier to use than past methods, experiments demonstrate that past coreset constructions are fundamentally limited by the fixed coarse posterior approximation; in contrast, the proposed algorithm is able to continually improve the coreset, providing state-of-the-art Bayesian dataset summarization with orders-of-magnitude reduction in KL divergence to the exact posterior.