Deep neural networks have led to a series of breakthroughs, dramatically improving the state-of-the-art in many domains. The techniques driving these advances, however, lack a formal method to account for model uncertainty. While the Bayesian approach to learning provides a solid theoretical framework to handle uncertainty, inference in Bayesian-inspired deep neural networks is difficult. In this paper, we provide a practical approach to Bayesian learning that relies on a regularization technique found in nearly every modern network, \textit{batch normalization}. We show that training a deep network using batch normalization is equivalent to approximate inference in Bayesian models, and we demonstrate how this finding allows us to make useful estimates of the model uncertainty. With our approach, it is possible to make meaningful uncertainty estimates using conventional architectures without modifying the network or the training procedure. Our approach is thoroughly validated in a series of empirical experiments on different tasks and using various measures, outperforming baselines with strong statistical significance and displaying competitive performance with other recent Bayesian approaches.

Deep latent variable models combine the approximation abilities of deep neural networks and the statistical foundations of generative models. The induced data distribution is an infinite mixture model whose density is extremely delicate to compute. Variational methods are consequently used for inference, following the seminal work of Rezende et al. (2014) and Kingma and Welling (2014). We study the well-posedness of the exact problem (maximum likelihood) these techniques approximatively solve. In particular, we show that most unconstrained models used for continuous data have an unbounded likelihood. This ill-posedness and the problems it causes are illustrated on real data. We also show how to insure the existence of maximum likelihood estimates, and draw useful connections with nonparametric mixture models. Furthermore, we describe an algorithm that allows to perform missing data imputation using the exact conditional likelihood of a deep latent variable model. On several real data sets, our algorithm consistently and significantly outperforms the usual imputation scheme used within deep latent variable models.

A message passing algorithm is derived for recovering a dense subgraph within a graph generated by a variation of the Barab\'asi-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the vertices. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are the vertices that attached to it. Algorithm C significantly outperforms DT, showing it is beneficial to know the arrival times of the children, beyond simply knowing the number of them. For fixed fraction of vertices in the community, fixed number of new edges per arriving vertex, and fixed affinity between vertices in the community, the probability of error for recovering the label of a vertex is found as a function of the time of attachment, for either algorithm DT or C, in the large graph limit. By averaging over the time of attachment, the limit in probability of the fraction of label errors made over all vertices is identified, for either of the algorithms DT or C.

We propose a novel distributed inference algorithm for continuous graphical models, by extending Stein variational gradient descent (SVGD) (Liu & Wang, 2016) to leverage the Markov dependency structure of the distribution of interest. Our approach combines SVGD with a set of structured local kernel functions defined on the Markov blanket of each node, which alleviates the curse of high dimensionality and simultaneously yields a distributed algorithm for decentralized inference tasks. We justify our method with theoretical analysis and show that the use of local kernels can be viewed as a new type of localized approximation that matches the target distribution on the conditional distributions of each node over its Markov blanket. Our empirical results show that our method outperforms a variety of baselines including standard MCMC and particle message passing methods.

Of course as an introductory book, we can only leave it at that: an introductory book. For the mathematically trained, they may cure the curiosity this text generates with other texts designed with mathematical analysis in mind. For the enthusiast with less mathematical-background, or one who is not interested in the mathematics but simply the practice of Bayesian methods, this text should be sufficient and entertaining. The choice of PyMC as the probabilistic programming language is two-fold. As of this writing, there is currently no central resource for examples and explanations in the PyMC universe.

The purpose of deep learning is to learn a representation of high dimensional and noisy data using a sequence of differentiable functions, i.e., geometric transformations, that can perhaps be used for supervised learning tasks among other tasks. It has had great success in discriminative models while generative models have not fared perhaps quite as well due to the limitations of explicit maximum likelihood estimation (MLE). Adversarial learning as presented in the Generative Adversarial Network (GAN) aims to overcome these problems by using implicit MLE. We will use the MNIST computer vision dataset and a synthetic financial transactions dataset for an insurance task for these experiments using GANs. GANs are a remarkably different method of learning compared to explicit MLE. Our purpose will be to show that the representation learnt by a GAN can be used for supervised learning tasks such as image recognition and insurance loss risk prediction.

Hypernetworks are neural networks that transform a random input vector into weights for a specified target neural network. We formulate the hypernetwork training objective as a compromise between accuracy and diversity, where the diversity takes into account trivial symmetry transformations of the target network. We show that this formulation naturally arises as a relaxation of an optimistic probability distribution objective for the generated networks, and we explain how it is related to variational inference. We use multi-layered perceptrons to form the mapping from the low dimensional input random vector to the high dimensional weight space, and demonstrate how to reduce the number of parameters in this mapping by weight sharing. We perform experiments on a four layer convolutional target network which classifies MNIST images, and show that the generated weights are diverse and have interesting distributions.

The area of online machine learning in big data streams covers algorithms that are (1) distributed and (2) work from data streams with only a limited possibility to store past data. The first requirement mostly concerns software architectures and efficient algorithms. The second one also imposes nontrivial theoretical restrictions on the modeling methods: In the data stream model, older data is no longer available to revise earlier suboptimal modeling decisions as the fresh data arrives. In this article, we provide an overview of distributed software architectures and libraries as well as machine learning models for online learning. We highlight the most important ideas for classification, regression, recommendation, and unsupervised modeling from streaming data, and we show how they are implemented in various distributed data stream processing systems. This article is a reference material and not a survey. We do not attempt to be comprehensive in describing all existing methods and solutions; rather, we give pointers to the most important resources in the field. All related sub-fields, online algorithms, online learning, and distributed data processing are hugely dominant in current research and development with conceptually new research results and software components emerging at the time of writing. In this article, we refer to several survey results, both for distributed data processing and for online machine learning. Compared to past surveys, our article is different because we discuss recommender systems in extended detail.

In order to design and operate the process it is important to know which variables have the strongest influence on performance. To estimate this we compared each state with the length and diameter of the resulting fibers, performing second order polynomial fit using samples over all 9 experiments, and noting the value of the resulting correlation coefficients R. The correlations between process parameters and the Length and Diameter is contained in Supplementary Table S.1. The angle and position have very little influence on the characteristics of the produced fibers. Polymer flow has a moderate influence (0.37) on length, but only a weak influence on diameter. Thus the most significant influence on overall performance is solvent speed followed by channel width.

Machine learning models are vulnerable to Adversarial Examples: minor perturbations to input samples intended to deliberately cause misclassification. Current defenses against adversarial examples, especially for Deep Neural Networks (DNN), are primarily derived from empirical developments, and their security guarantees are often only justified retroactively. Many defenses therefore rely on hidden assumptions that are subsequently subverted by increasingly elaborate attacks. This is not surprising: deep learning notoriously lacks a comprehensive mathematical framework to provide meaningful guarantees. In this paper, we leverage Gaussian Processes to investigate adversarial examples in the framework of Bayesian inference. Across different models and datasets, we find deviating levels of uncertainty reflect the perturbation introduced to benign samples by state-of-the-art attacks, including novel white-box attacks on Gaussian Processes. Our experiments demonstrate that even unoptimized uncertainty thresholds already reject adversarial examples in many scenarios.