Variational methods with dependence structure

It is a common practice among humans to deduce, to explain and to make predictions based on concepts that are not directly observable. In Bayesian statistics, the underlying propositions of the unobserved latent variables are summarized in the posterior distribution. With the increasing complexity of real-world data and statistical models, fast and accurate inference for the posterior becomes essential. Variational methods, by casting the posterior inference problem in the optimization framework, are widely used for their flexibility and computational efficiency. In this thesis, we develop new variational methods, studying their theoretical properties and applications. In the first part of the thesis, we utilize dependence structures towards addressing fundamental problems in variational inference (VI): posterior uncertainty estimation, convergence properties, and discrete optimization. Though it is flexible, variational inference often underestimates the posterior uncertainty. This is a consequence of the over-simplified variational family. Mean-field variational inference (MFVI), for example, uses a product of independent distributions as a coarse approximation to the posterior. As a remedy, we propose a hierarchical variational distribution with flexible parameterization that can model the dependence structure between latent variables. With a newly derived objective, we show that the proposed variational method can achieve accurate and efficient uncertainty estimation. We further theoretically study the structured variational inference in the setting of the Stochastic Blockmodel (SBM). The variational distribution is constructed with a pairwise structure among the nodes of a graph. We prove that, in a broad density regime and for general random initializations, the estimated class labels by structured VI converge to the ground truth with high probability. Empirically, we demonstrate structured VI is more robust compared with MFVI when the graph is sparse and the signal to noise ratio is low. When the latent variables are discrete, gradient descent based VI often suffers from bias and high variance in the gradient estimation. With correlated random samples, we propose a novel unbiased, low-variance gradient estimator. We demonstrate that under certain constraints, such correlated sampling gives an optimal control variates for the variance reduction. The efficient gradient estimation can be applied to solve a wide range of problems such as the variable selection, reinforcement learning, natural language processing, among others. For the second part of the thesis, we apply variational methods to the study of generalization problems in the meta-learning. When trained over multiple-tasks, we identify that a variety of the meta-learning algorithms implicitly require the tasks to have a mutually-exclusive dependence structure. This prevents the task-level overfitting problem and ensures the fast adaptation of the algorithm in the face of a new task. However, such dependence structure may not exist for general tasks. When the tasks are non-mutually exclusive, we develop new meta-learning algorithms with variational regularization to prevent the task-level overfitting. Consequently, we can expand the meta-learning to the domains which it cannot be effective on before.Statistic

Hierarchical Implicit Models and Likelihood-Free Variational Inference

Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent structure in them, and in performing inferences in such models with large data sets. In this paper, we first introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data with rich hidden structure. Next, we develop likelihood-free variational inference (LFVI), a scalable variational inference algorithm for HIMs. Key to LFVI is specifying a variational family that is also implicit. This matches the model's flexibility and allows for accurate approximation of the posterior. We demonstrate diverse applications: a large-scale physical simulator for predator-prey populations in ecology; a Bayesian generative adversarial network for discrete data; and a deep implicit model for text generation.Comment: Appears in Neural Information Processing Systems, 201

Accuracy of Latent-Variable Estimation in Bayesian Semi-Supervised Learning

Hierarchical probabilistic models, such as Gaussian mixture models, are widely used for unsupervised learning tasks. These models consist of observable and latent variables, which represent the observable data and the underlying data-generation process, respectively. Unsupervised learning tasks, such as cluster analysis, are regarded as estimations of latent variables based on the observable ones. The estimation of latent variables in semi-supervised learning, where some labels are observed, will be more precise than that in unsupervised, and one of the concerns is to clarify the effect of the labeled data. However, there has not been sufficient theoretical analysis of the accuracy of the estimation of latent variables. In a previous study, a distribution-based error function was formulated, and its asymptotic form was calculated for unsupervised learning with generative models. It has been shown that, for the estimation of latent variables, the Bayes method is more accurate than the maximum-likelihood method. The present paper reveals the asymptotic forms of the error function in Bayesian semi-supervised learning for both discriminative and generative models. The results show that the generative model, which uses all of the given data, performs better when the model is well specified.Comment: 25 pages, 4 figure