The recently developed variational autoencoders (VAEs) have proved to be an effective confluence of the rich representational power of neural networks with Bayesian methods. However, most work on VAEs use a rather simple prior over the latent variables such as standard normal distribution, thereby restricting its applications to relatively simple phenomena. In this work, we propose hierarchical nonparametric variational autoencoders, which combines tree-structured Bayesian nonparametric priors with VAEs, to enable infinite flexibility of the latent representation space. Both the neural parameters and Bayesian priors are learned jointly using tailored variational inference. The resulting model induces a hierarchical structure of latent semantic concepts underlying the data corpus, and infers accurate representations of data instances. We apply our model in video representation learning. Our method is able to discover highly interpretable activity hierarchies, and obtain improved clustering accuracy and generalization capacity based on the learned rich representations.
Functional Distributional Semantics is a framework that aims to learn, from text, semantic representations which can be interpreted in terms of truth. Here we make two contributions to this framework. The first is to show how a type of logical inference can be performed by evaluating conditional probabilities. The second is to make these calculations tractable by means of a variational approximation. This approximation also enables faster convergence during training, allowing us to close the gap with state-of-the-art vector space models when evaluating on semantic similarity. We demonstrate promising performance on two tasks.
This paper makes two contributions to Bayesian machine learning algorithms. Firstly, we propose stochastic natural gradient expectation propagation (SNEP), a novel alternative to expectation propagation (EP), a popular variational inference algorithm. SNEP is a black box variational algorithm, in that it does not require any simplifying assumptions on the distribution of interest, beyond the existence of some Monte Carlo sampler for estimating the moments of the EP tilted distributions. Further, as opposed to EP which has no guarantee of convergence, SNEP can be shown to be convergent, even when using Monte Carlo moment estimates. Secondly, we propose a novel architecture for distributed Bayesian learning which we call the posterior server. The posterior server allows scalable and robust Bayesian learning in cases where a data set is stored in a distributed manner across a cluster, with each compute node containing a disjoint subset of data. An independent Monte Carlo sampler is run on each compute node, with direct access only to the local data subset, but which targets an approximation to the global posterior distribution given all data across the whole cluster. This is achieved by using a distributed asynchronous implementation of SNEP to pass messages across the cluster. We demonstrate SNEP and the posterior server on distributed Bayesian learning of logistic regression and neural networks. Keywords: Distributed Learning, Large Scale Learning, Deep Learning, Bayesian Learn- ing, Variational Inference, Expectation Propagation, Stochastic Approximation, Natural Gradient, Markov chain Monte Carlo, Parameter Server, Posterior Server.
In many biomedical, science, and engineering problems, one must sequentially decide which action to take next so as to maximize rewards. One general class of algorithms for optimizing interactions with the world, while simultaneously learning how the world operates, is the multi-armed bandit setting and, in particular, the contextual bandit case. In this setting, for each executed action, one observes rewards that are dependent on a given 'context', available at each interaction with the world. The Thompson sampling algorithm has recently been shown to enjoy provable optimality properties for this set of problems, and to perform well in real-world settings. It facilitates generative and interpretable modeling of the problem at hand. Nevertheless, the design and complexity of the model limit its application, since one must both sample from the distributions modeled and calculate their expected rewards. We here show how these limitations can be overcome using variational inference to approximate complex models, applying to the reinforcement learning case advances developed for the inference case in the machine learning community over the past two decades. We consider contextual multi-armed bandit applications where the true reward distribution is unknown and complex, which we approximate with a mixture model whose parameters are inferred via variational inference. We show how the proposed variational Thompson sampling approach is accurate in approximating the true distribution, and attains reduced regrets even with complex reward distributions. The proposed algorithm is valuable for practical scenarios where restrictive modeling assumptions are undesirable.
Computing partition function is the most important statistical inference task arising in applications of Graphical Models (GM). Since it is computationally intractable, approximate methods have been used to resolve the issue in practice, where mean-field (MF) and belief propagation (BP) are arguably the most popular and successful approaches of a variational type. In this paper, we propose two new variational schemes, coined Gauged-MF (G-MF) and Gauged-BP (G-BP), improving MF and BP, respectively. Both provide lower bounds for the partition function by utilizing the so-called gauge transformation which modifies factors of GM while keeping the partition function invariant. Moreover, we prove that both G-MF and G-BP are exact for GMs with a single loop of a special structure, even though the bare MF and BP perform badly in this case. Our extensive experiments, on complete GMs of relatively small size and on large GM (up-to 300 variables) confirm that the newly proposed algorithms outperform and generalize MF and BP.
Aiming to augment generative models with external memory, we interpret the output of a memory module with stochastic addressing as a conditional mixture distribution, where a read operation corresponds to sampling a discrete memory address and retrieving the corresponding content from memory. This perspective allows us to apply variational inference to memory addressing, which enables effective training of the memory module by using the target information to guide memory lookups. Stochastic addressing is particularly well-suited for generative models as it naturally encourages multimodality which is a prominent aspect of most high-dimensional datasets. Treating the chosen address as a latent variable also allows us to quantify the amount of information gained with a memory lookup and measure the contribution of the memory module to the generative process. To illustrate the advantages of this approach we incorporate it into a variational autoencoder and apply the resulting model to the task of generative few-shot learning. The intuition behind this architecture is that the memory module can pick a relevant template from memory and the continuous part of the model can concentrate on modeling remaining variations. We demonstrate empirically that our model is able to identify and access the relevant memory contents even with hundreds of unseen Omniglot characters in memory
Advances in neural variational inference have facilitated the learning of powerful directed graphical models with continuous latent variables, such as variational autoencoders. The hope is that such models will learn to represent rich, multi-modal latent factors in real-world data, such as natural language text. However, current models often assume simplistic priors on the latent variables - such as the uni-modal Gaussian distribution - which are incapable of representing complex latent factors efficiently. To overcome this restriction, we propose the simple, but highly flexible, piecewise constant distribution. This distribution has the capacity to represent an exponential number of modes of a latent target distribution, while remaining mathematically tractable. Our results demonstrate that incorporating this new latent distribution into different models yields substantial improvements in natural language processing tasks such as document modeling and natural language generation for dialogue.
In this paper, we propose generative probabilistic models for label aggregation. We use Gibbs sampling and a novel variational inference algorithm to perform the posterior inference. Empirical results show that our methods consistently outperform state-of-the-art methods.
In this paper, we study two aspects of the variational autoencoder (VAE): the prior distribution over the latent variables and its corresponding posterior. First, we decompose the learning of VAEs into layerwise density estimation, and argue that having a flexible prior is beneficial to both sample generation and inference. Second, we analyze the family of inverse autoregressive flows (inverse AF) and show that with further improvement, inverse AF could be used as universal approximation to any complicated posterior. Our analysis results in a unified approach to parameterizing a VAE, without the need to restrict ourselves to use factorial Gaussians in the latent real space.
Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of recent advances in equilibrium computation for probabilistic inference. We present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of game-theoretic graphical models. We concretely establishes the precise connection between variational probabilistic inference in MRFs and correlated equilibria. No previous work exploits recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomial-time computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. We discuss how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. Also, inspired by a previously stated game-theoretic view of state-of-the-art tree-reweighed (TRW) message-passing techniques for belief inference as zero-sum game, we propose a different, general-sum potential game to design approximate fictitious-play techniques. We perform synthetic experiments evaluating our proposed approximation algorithms with standard methods and TRW on several classes of classical Ising models (i.e., with binary random variables). We also evaluate the algorithms using Ising models learned from the MNIST dataset. Our experiments show that our global approach is competitive, particularly shinning in a class of Ising models with constant, "highly attractive" edge-weights, in which it is often better than all other alternatives we evaluated. With a notable exception, our more local approach was not as effective. Yet, in fairness, almost all of the alternatives are often no better than a simple baseline: estimate 0.5.
We propose a dynamic edge exchangeable network model that can capture sparse connections observed in real temporal networks, in contrast to existing models which are dense. The model achieved superior link prediction accuracy on multiple data sets when compared to a dynamic variant of the blockmodel, and is able to extract interpretable time-varying community structures from the data. In addition to sparsity, the model accounts for the effect of social influence on vertices' future behaviours. Compared to the dynamic blockmodels, our model has a smaller latent space. The compact latent space requires a smaller number of parameters to be estimated in variational inference and results in a computationally friendly inference algorithm.
We study parameter estimation in Nonlinear Factor Analysis (NFA) where the generative model is parameterized by a deep neural network. Recent work has focused on learning such models using inference (or recognition) networks; we identify a crucial problem when modeling large, sparse, high-dimensional datasets -- underfitting. We study the extent of underfitting, highlighting that its severity increases with the sparsity of the data. We propose methods to tackle it via iterative optimization inspired by stochastic variational inference \citep{hoffman2013stochastic} and improvements in the sparse data representation used for inference. The proposed techniques drastically improve the ability of these powerful models to fit sparse data, achieving state-of-the-art results on a benchmark text-count dataset and excellent results on the task of top-N recommendation.
We propose a probabilistic model for interpreting gene expression levels that are observed through single-cell RNA sequencing. In the model, each cell has a low-dimensional latent representation. Additional latent variables account for technical effects that may erroneously set some observations of gene expression levels to zero. Conditional distributions are specified by neural networks, giving the proposed model enough flexibility to fit the data well. We use variational inference and stochastic optimization to approximate the posterior distribution. The inference procedure scales to over one million cells, whereas competing algorithms do not. Even for smaller datasets, for several tasks, the proposed procedure outperforms state-of-the-art methods like ZIFA and ZINB-WaVE. We also extend our framework to take into account batch effects and other confounding factors and propose a natural Bayesian hypothesis framework for differential expression that outperforms tradition DESeq2.
Auto-encoding generative adversarial networks (GANs) combine the standard GAN algorithm, which discriminates between real and model-generated data, with a reconstruction loss given by an auto-encoder. Such models aim to prevent mode collapse in the learned generative model by ensuring that it is grounded in all the available training data. In this paper, we develop a principle upon which auto-encoders can be combined with generative adversarial networks by exploiting the hierarchical structure of the generative model. The underlying principle shows that variational inference can be used a basic tool for learning, but with the in- tractable likelihood replaced by a synthetic likelihood, and the unknown posterior distribution replaced by an implicit distribution; both synthetic likelihoods and implicit posterior distributions can be learned using discriminators. This allows us to develop a natural fusion of variational auto-encoders and generative adversarial networks, combining the best of both these methods. We describe a unified objective for optimization, discuss the constraints needed to guide learning, connect to the wide range of existing work, and use a battery of tests to systematically and quantitatively assess the performance of our method.
Many matching, tracking, sorting, and ranking problems require probabilistic reasoning about possible permutations, a set that grows factorially with dimension. Combinatorial optimization algorithms may enable efficient point estimation, but fully Bayesian inference poses a severe challenge in this high-dimensional, discrete space. To surmount this challenge, we start with the usual step of relaxing a discrete set (here, of permutation matrices) to its convex hull, which here is the Birkhoff polytope: the set of all doubly-stochastic matrices. We then introduce two novel transformations: first, an invertible and differentiable stick-breaking procedure that maps unconstrained space to the Birkhoff polytope; second, a map that rounds points toward the vertices of the polytope. Both transformations include a temperature parameter that, in the limit, concentrates the densities on permutation matrices. We then exploit these transformations and reparameterization gradients to introduce variational inference over permutation matrices, and we demonstrate its utility in a series of experiments.
Variational inference provides a powerful tool for approximate probabilistic in- ference on complex, structured models. Typical variational inference methods, however, require to use inference networks with computationally tractable proba- bility density functions. This largely limits the design and implementation of vari- ational inference methods. We consider wild variational inference methods that do not require tractable density functions on the inference networks, and hence can be applied in more challenging cases. As an example of application, we treat stochastic gradient Langevin dynamics (SGLD) as an inference network, and use our methods to automatically adjust the step sizes of SGLD, yielding significant improvement over the hand-designed step size schemes
This paper presents a novel variational inference framework for deriving a family of Bayesian sparse Gaussian process regression (SGPR) models whose approximations are variationally optimal with respect to the full-rank GPR model enriched with various corresponding correlation structures of the observation noises. Our variational Bayesian SGPR (VBSGPR) models jointly treat both the distributions of the inducing variables and hyperparameters as variational parameters, which enables the decomposability of the variational lower bound that in turn can be exploited for stochastic optimization. Such a stochastic optimization involves iteratively following the stochastic gradient of the variational lower bound to improve its estimates of the optimal variational distributions of the inducing variables and hyperparameters (and hence the predictive distribution) of our VBSGPR models and is guaranteed to achieve asymptotic convergence to them. We show that the stochastic gradient is an unbiased estimator of the exact gradient and can be computed in constant time per iteration, hence achieving scalability to big data. We empirically evaluate the performance of our proposed framework on two real-world, massive datasets.
In this paper, we use variational recurrent neural network to investigate the anomaly detection problem on graph time series. The temporal correlation is modeled by the combination of recurrent neural network (RNN) and variational inference (VI), while the spatial information is captured by the graph convolutional network. In order to incorporate external factors, we use feature extractor to augment the transition of latent variables, which can learn the influence of external factors. With the target function as accumulative ELBO, it is easy to extend this model to on-line method. The experimental study on traffic flow data shows the detection capability of the proposed method.
There has recently been significant interest in hard attention models for tasks such as object recognition, visual captioning and speech recognition. Hard attention can offer benefits over soft attention such as decreased computational cost, but training hard attention models can be difficult because of the discrete latent variables they introduce. Previous work used REINFORCE and Q-learning to approach these issues, but those methods can provide high-variance gradient estimates and be slow to train. In this paper, we tackle the problem of learning hard attention for a sequential task using variational inference methods, specifically the recently introduced VIMCO and NVIL. Furthermore, we propose a novel baseline that adapts VIMCO to this setting. We demonstrate our method on a phoneme recognition task in clean and noisy environments and show that our method outperforms REINFORCE, with the difference being greater for a more complicated task.
We introduce TrustVI, a fast second-order algorithm for black-box variational inference based on trust-region optimization and the reparameterization trick. At each iteration, TrustVI proposes and assesses a step based on minibatches of draws from the variational distribution. The algorithm provably converges to a stationary point. We implemented TrustVI in the Stan framework and compared it to two alternatives: Automatic Differentiation Variational Inference (ADVI) and Hessian-free Stochastic Gradient Variational Inference (HFSGVI). The former is based on stochastic first-order optimization. The latter uses second-order information, but lacks convergence guarantees. TrustVI typically converged at least one order of magnitude faster than ADVI, demonstrating the value of stochastic second-order information. TrustVI often found substantially better variational distributions than HFSGVI, demonstrating that our convergence theory can matter in practice.