Recently much attention has been paid to deep generative models, since they have been used to great success for variational inference, generation of complex data types, and more. In most all of these settings, the goal has been to find a particular member of that model family: optimized parameters index a distribution that is close (via a divergence or classification metric) to a target distribution. Much less attention, however, has been paid to the problem of learning a model itself. Here we introduce a two-network architecture and optimization procedure for learning intractable exponential family models (not a single distribution from those models). These exponential families are learned accurately, allowing operations like posterior inference to be executed directly and generically with an input choice of natural parameters, rather than performing inference via optimization for each particular distribution within that model.
Multi-fidelity methods are prominently used when cheaply-obtained, but possibly biased and noisy, observations must be effectively combined with limited or expensive true data in order to construct reliable models. This arises in both fundamental machine learning procedures such as Bayesian optimization, as well as more practical science and engineering applications. In this paper we develop a novel multi-fidelity model which treats layers of a deep Gaussian process as fidelity levels, and uses a variational inference scheme to propagate uncertainty across them. This allows for capturing nonlinear correlations between fidelities with lower risk of overfitting than existing methods exploiting compositional structure, which are conversely burdened by structural assumptions and constraints. We show that the proposed approach makes substantial improvements in quantifying and propagating uncertainty in multi-fidelity set-ups, which in turn improves their effectiveness in decision making pipelines.
Bayesian neural networks (BNNs) have recently regained a significant amount of attention in the deep learning community due to the development of scalable approximate Bayesian inference techniques. There are several advantages of using Bayesian approach: Parameter and prediction uncertainty become easily available, facilitating rigid statistical analysis. Furthermore, prior knowledge can be incorporated. However so far there have been no scalable techniques capable of combining both model (structural) and parameter uncertainty. In this paper we introduce the concept of model uncertainty in BNNs and hence make inference in the joint space of models and parameters. Moreover, we suggest an adaptation of a scalable variational inference approach with reparametrization of marginal inclusion probabilities to incorporate the model space constraints. Finally, we show that incorporating model uncertainty via Bayesian model averaging and Bayesian model selection allows to drastically sparsify the structure of BNNs.
Variational Bayesian neural networks (BNNs) perform variational inference over weights, but it is difficult to specify meaningful priors and approximate posteriors in a high-dimensional weight space. We introduce functional variational Bayesian neural networks (fBNNs), which maximize an Evidence Lower BOund (ELBO) defined directly on stochastic processes, i.e. distributions over functions. We prove that the KL divergence between stochastic processes equals the supremum of marginal KL divergences over all finite sets of inputs. Based on this, we introduce a practical training objective which approximates the functional ELBO using finite measurement sets and the spectral Stein gradient estimator. With fBNNs, we can specify priors entailing rich structures, including Gaussian processes and implicit stochastic processes. Empirically, we find fBNNs extrapolate well using various structured priors, provide reliable uncertainty estimates, and scale to large datasets.
Bayesian optimal experimental design (BOED) is a principled framework for making efficient use of limited experimental resources. Unfortunately, its applicability is hampered by the difficulty of obtaining accurate estimates of the expected information gain (EIG) of an experiment. To address this, we introduce several classes of fast EIG estimators by building on ideas from amortized variational inference. We show theoretically and empirically that these estimators can provide significant gains in speed and accuracy over previous approaches. We further demonstrate the practicality of our approach on a number of end-to-end experiments.
A core problem in statistics and probabilistic machine learning is to compute probability distributions and expectations. This is the fundamental problem of Bayesian statistics and machine learning, which frames all inference as expectations with respect to the posterior distribution. The key challenge is to approximate these intractable expectations. In this tutorial, we review sequential Monte Carlo (SMC), a random-sampling-based class of methods for approximate inference. First, we explain the basics of SMC, discuss practical issues, and review theoretical results. We then examine two of the main user design choices: the proposal distributions and the so called intermediate target distributions. We review recent results on how variational inference and amortization can be used to learn efficient proposals and target distributions. Next, we discuss the SMC estimate of the normalizing constant, how this can be used for pseudo-marginal inference and inference evaluation. Throughout the tutorial we illustrate the use of SMC on various models commonly used in machine learning, such as stochastic recurrent neural networks, probabilistic graphical models, and probabilistic programs.
We consider multi-task regression models where observations are assumed to be a linear combination of several latent node and weight functions, all drawn from Gaussian process (GP) priors that allow nonzero covariance between grouped latent functions. We show that when these grouped functions are conditionally independent given a group-dependent pivot, it is possible to parameterize the prior through sparse Cholesky factors directly, hence avoiding their computation during inference. Furthermore, we establish that kernels that are multiplicatively separable over input points give rise to such sparse parameterizations naturally without any additional assumptions. Finally, we extend the use of these sparse structures to approximate posteriors within variational inference, further improving scalability on the number of functions. We test our approach on multi-task datasets concerning distributed solar forecasting and show that it outperforms several multi-task GP baselines and that our sparse specifications achieve the same or better accuracy than non-sparse counterparts.
We present a non-parametric prognostic framework for individualized event prediction based on joint modeling of both longitudinal and time-to-event data. Our approach exploits a multivariate Gaussian convolution process (MGCP) to model the evolution of longitudinal signals and a Cox model to map time-to-event data with longitudinal data modeled through the MGCP. Taking advantage of the unique structure imposed by convolved processes, we provide a variational inference framework to simultaneously estimate parameters in the joint MGCP-Cox model. This significantly reduces computational complexity and safeguards against model overfitting. Experiments on synthetic and real world data show that the proposed framework outperforms state-of-the art approaches built on two-stage inference and strong parametric assumptions.
Hamiltonian Monte Carlo is a powerful algorithm for sampling from difficult-to-normalize posterior distributions. However, when the geometry of the posterior is unfavorable, it may take many expensive evaluations of the target distribution and its gradient to converge and mix. We propose neural transport (NeuTra) HMC, a technique for learning to correct this sort of unfavorable geometry using inverse autoregressive flows (IAF), a powerful neural variational inference technique. The IAF is trained to minimize the KL divergence from an isotropic Gaussian to the warped posterior, and then HMC sampling is performed in the warped space. We evaluate NeuTra HMC on a variety of synthetic and real problems, and find that it significantly outperforms vanilla HMC both in time to reach the stationary distribution and asymptotic effective-sample-size rates.
A wide range of systems exhibit high dimensional incomplete data. Accurate estimation of the missing data is often desired, and is crucial for many downstream analyses. Many state-of-the-art recovery methods involve supervised learning using datasets containing full observations. In contrast, we focus on unsupervised estimation of missing image data, where no full observations are available - a common situation in practice. Unsupervised imputation methods for images often employ a simple linear subspace to capture correlations between data dimensions, omitting more complex relationships. In this work, we introduce a general probabilistic model that describes sparse high dimensional imaging data as being generated by a deep non-linear embedding. We derive a learning algorithm using a variational approximation based on convolutional neural networks and discuss its relationship to linear imputation models, the variational auto encoder, and deep image priors. We introduce sparsity-aware network building blocks that explicitly model observed and missing data. We analyze proposed sparsity-aware network building blocks, evaluate our method on public domain imaging datasets, and conclude by showing that our method enables imputation in an important real-world problem involving medical images. The code is freely available as part of the \verb|neuron| library at http://github.com/adalca/neuron.
Variational inference transforms posterior inference into parametric optimization thereby enabling the use of latent variable models where otherwise impractical. However, variational inference can be finicky when different variational parameters control variables that are strongly correlated under the model. Traditional natural gradients based on the variational approximation fail to correct for correlations when the approximation is not the true posterior. To address this, we construct a new natural gradient called the Variational Predictive Natural Gradient (VPNG). Unlike traditional natural gradients for variational inference, this natural gradient accounts for the relationship between model parameters and variational parameters. We demonstrate the insight with a simple example as well as the empirical value on a classification task, a deep generative model of images, and probabilistic matrix factorization for recommendation.
In model-based reinforcement learning, the agent interleaves between model learning and planning. These two components are inextricably intertwined. If the model is not able to provide sensible long-term prediction, the executed planner would exploit model flaws, which can yield catastrophic failures. This paper focuses on building a model that reasons about the long-term future and demonstrates how to use this for efficient planning and exploration. To this end, we build a latent-variable autoregressive model by leveraging recent ideas in variational inference. We argue that forcing latent variables to carry future information through an auxiliary task substantially improves long-term predictions. Moreover, by planning in the latent space, the planner's solution is ensured to be within regions where the model is valid. An exploration strategy can be devised by searching for unlikely trajectories under the model. Our method achieves higher reward faster compared to baselines on a variety of tasks and environments in both the imitation learning and model-based reinforcement learning settings.
We introduce and study a class of probabilistic generative models, where the latent object is a finite-dimensional diffusion process on a finite time interval and the observed variable is drawn conditionally on the terminal point of the diffusion. We make the following contributions: We provide a unified viewpoint on both sampling and variational inference in such generative models through the lens of stochastic control. We quantify the expressiveness of diffusion-based generative models. Specifically, we show that one can efficiently sample from a wide class of terminal target distributions by choosing the drift of the latent diffusion from the class of multilayer feedforward neural nets, with the accuracy of sampling measured by the Kullback-Leibler divergence to the target distribution. Finally, we present and analyze a scheme for unbiased simulation of generative models with latent diffusions and provide bounds on the variance of the resulting estimators. This scheme can be implemented as a deep generative model with a random number of layers.
Human perception is structured around objects which form the basis for our higher-level cognition and impressive systematic generalization abilities. Yet most work on representation learning focuses on feature learning without even considering multiple objects, or treats segmentation as an (often supervised) preprocessing step. Instead, we argue for the importance of learning to segment and represent objects jointly. We demonstrate that, starting from the simple assumption that a scene is composed of multiple entities, it is possible to learn to segment images into interpretable objects with disentangled representations. Our method learns -- without supervision -- to inpaint occluded parts, and extrapolates to scenes with more objects and to unseen objects with novel feature combinations. We also show that, due to the use of iterative variational inference, our system is able to learn multi-modal posteriors for ambiguous inputs and extends naturally to sequences.
We present a novel approach for training deep neural networks in a Bayesian way. Classical, i.e. non-Bayesian, deep learning has two major drawbacks both originating from the fact that network parameters are considered to be deterministic. First, model uncertainty cannot be measured thus limiting the use of deep learning in many fields of application and second, training of deep neural networks is often hampered by overfitting. The proposed approach uses variational inference to approximate the intractable a posteriori distribution on basis of a normal prior. The variational density is designed in such a way that the a posteriori uncertainty of the network parameters is represented per network layer and depending on the estimated parameter expectation values. This way, only a few additional parameters need to be optimized compared to a non-Bayesian network. We apply this Bayesian approach to train and test the LeNet architecture on the MNIST dataset. Compared to classical deep learning, the test error is reduced by 15%. In addition, the trained model contains information about the parameter uncertainty in each layer. We show that this information can be used to calculate credible intervals for the prediction and to optimize the network architecture for a given training data set.
The recognition network in deep latent variable models such as variational autoencoders (VAEs) relies on amortized inference for efficient posterior approximation that can scale up to large datasets. However, this technique has also been demonstrated to select suboptimal variational parameters, often resulting in considerable additional error called the amortization gap. To close the amortization gap and improve the training of the generative model, recent works have introduced an additional refinement step that applies stochastic variational inference (SVI) to improve upon the variational parameters returned by the amortized inference model. In this paper, we propose the Buffered Stochastic Variational Inference (BSVI), a new refinement procedure that makes use of SVI's sequence of intermediate variational proposal distributions and their corresponding importance weights to construct a new generalized importance-weighted lower bound. We demonstrate empirically that training the variational autoencoders with BSVI consistently out-performs SVI, yielding an improved training procedure for VAEs.
While Bayesian neural networks (BNNs) have drawn increasing attention, their posterior inference remains challenging, due to the high-dimensional and over-parameterized nature. To address this issue, several highly flexible and scalable variational inference procedures based on the idea of particle optimization have been proposed. These methods directly optimize a set of particles to approximate the target posterior. However, their application to BNNs often yields sub-optimal performance, as such methods have a particular failure mode on over-parameterized models. In this paper, we propose to solve this issue by performing particle optimization directly in the space of regression functions. We demonstrate through extensive experiments that our method successfully overcomes this issue, and outperforms strong baselines in a variety of tasks including prediction, defense against adversarial examples, and reinforcement learning.
Banded matrices can be used as precision matrices in several models including linear state-space models, some Gaussian processes, and Gaussian Markov random fields. The aim of the paper is to make modern inference methods (such as variational inference or gradient-based sampling) available for Gaussian models with banded precision. We show that this can efficiently be achieved by equipping an automatic differentiation framework, such as TensorFlow or PyTorch, with some linear algebra operators dedicated to banded matrices. This paper studies the algorithmic aspects of the required operators, details their reverse-mode derivatives, and show that their complexity is linear in the number of observations.
Defining action spaces for conversational agents and optimizing their decision-making process with reinforcement learning is an enduring challenge. Common practice has been to use handcrafted dialog acts, or the output vocabulary, e.g. in neural encoder decoders, as the action spaces. Both have their own limitations. This paper proposes a novel latent action framework that treats the action spaces of an end-to-end dialog agent as latent variables and develops unsupervised methods in order to induce its own action space from the data. Comprehensive experiments are conducted examining both continuous and discrete action types and two different optimization methods based on stochastic variational inference. Results show that the proposed latent actions achieve superior empirical performance improvement over previous word-level policy gradient methods on both DealOrNoDeal and MultiWoz dialogs. Our detailed analysis also provides insights about various latent variable approaches for policy learning and can serve as a foundation for developing better latent actions in future research.
Human decision-making deviates from the optimal solution, that maximizes cumulative rewards, in many situations. Here we approach this discrepancy from the perspective of bounded rationality and our goal is to provide a justification for such seemingly sub-optimal strategies. More specifically we investigate the hypothesis, that humans do not know optimal decision-making algorithms in advance, but instead employ a learned, resource-bounded approximation. The idea is formalized through combining a recently proposed meta-learning model based on Recurrent Neural Networks with a resource-bounded objective. The resulting approach is closely connected to variational inference and the Minimum Description Length principle. Empirical evidence is obtained from a two-armed bandit task. Here we observe patterns in our family of models that resemble differences between individual human participants.