Generalized Bayesian learning algorithms are increasingly popular in machine learning, due to their PAC generalization properties and flexibility. The present paper aims at providing a self-contained survey on the resulting PAC-Bayes framework and some of its main theoretical and algorithmic developments.

Domain adaptation has become a prominent problem setting in machine learning and related fields. This review asks the questions: when and how a classifier can learn from a source domain and generalize to a target domain. As for when, we review conditions that allow for cross-domain generalization error bounds. As for how, we present a categorization of approaches, divided into, what we refer to as, sample-based, feature-based and inference-based methods. Sample-based methods focus on weighting individual observations during training based on their importance to the target domain. Feature-based methods focus on mapping, projecting and representing features such that a source classifier performs well on the target domain and inference-based methods focus on alternative estimators, such as robust, minimax or Bayesian. Our categorization highlights recurring ideas and raises a number of questions important to further research.

In binary classification framework, we are interested in making cost sensitive label predictions in the presence of uniform/symmetric label noise. We first observe that $0$-$1$ Bayes classifiers are not (uniform) noise robust in cost sensitive setting. To circumvent this impossibility result, we present two schemes; unlike the existing methods, our schemes do not require noise rate. The first one uses $\alpha$-weighted $\gamma$-uneven margin squared loss function, $l_{\alpha, usq}$, which can handle cost sensitivity arising due to domain requirement (using user given $\alpha$) or class imbalance (by tuning $\gamma$) or both. However, we observe that $l_{\alpha, usq}$ Bayes classifiers are also not cost sensitive and noise robust. We show that regularized ERM of this loss function over the class of linear classifiers yields a cost sensitive uniform noise robust classifier as a solution of a system of linear equations. We also provide a performance bound for this classifier. The second scheme that we propose is a re-sampling based scheme that exploits the special structure of the uniform noise models and uses in-class probability estimates. Our computational experiments on some UCI datasets with class imbalance show that classifiers of our two schemes are on par with the existing methods and in fact better in some cases w.r.t. Accuracy and Arithmetic Mean, without using/tuning noise rate. We also consider other cost sensitive performance measures viz., F measure and Weighted Cost for evaluation.

The great challenge was to generalize the reconstruction of the arm for any background without using segmentation. For that purpose, several background images were synthetically generated and were overlaid by automated labelled masks (i.e., boolean mask of the arm in the visual field) by means of background subtraction (Figure 1(b)). An example of the results of the generated arm given the a joint angle configuration is shown in Figure 1(c). The most right generated image shows difficulties of the model to properly reconstruct the robot arm when the majority of it is outside the field of view. Anyhow, the statistical evaluation of the network, over all experiments, showed an accuracy of 84.4% when comparing the matching between the original versus the generated image mask.

The Laplace approximation has been one of the workhorses of Bayesian inference. It often delivers good approximations in practice despite the fact that it does not strictly take into account where the volume of posterior density lies. Variational approaches avoid this issue by explicitly minimising the Kullback-Leibler divergence DKL between a postulated posterior and the true (unnormalised) logarithmic posterior. However, they rely on a closed form DKL in order to update the variational parameters. To address this, stochastic versions of variational inference have been devised that approximate the intractable DKL with a Monte Carlo average. This approximation allows calculating gradients with respect to the variational parameters. However, variational methods often postulate a factorised Gaussian approximating posterior. In doing so, they sacrifice a-posteriori correlations. In this work, we propose a method that combines the Laplace approximation with the variational approach. The advantages are that we maintain: applicability on non-conjugate models, posterior correlations and a reduced number of free variational parameters. Numerical experiments demonstrate improvement over the Laplace approximation and variational inference with factorised Gaussian posteriors.

Statistical inference involves estimation of parameters of a model based on observations. Building on the recently proposed Equilibrium Expectation approach and Persistent Contrastive Divergence, we derive a simple and fast Markov chain Monte Carlo algorithm for maximum likelihood estimation (MLE) of parameters of exponential family distributions. The algorithm has good scaling properties and is suitable for Monte Carlo inference on large network data with billions of tie variables. The performance of the algorithm is demonstrated on Markov random fields, conditional random fields, exponential random graph models and Boltzmann machines.

Bayesian optimal design of experiments (BODE) has been successful in acquiring information about a quantity of interest (QoI) which depends on a black-box function. BODE is characterized by sequentially querying the function at specific designs selected by an infill-sampling criterion. However, most current BODE methods operate in specific contexts like optimization, or learning a universal representation of the black-box function. The objective of this paper is to design a BODE for estimating the statistical expectation of a physical response surface. This QoI is omnipresent in uncertainty propagation and design under uncertainty problems. Our hypothesis is that an optimal BODE should be maximizing the expected information gain in the QoI. We represent the information gain from a hypothetical experiment as the Kullback-Liebler (KL) divergence between the prior and the posterior probability distributions of the QoI. The prior distribution of the QoI is conditioned on the observed data and the posterior distribution of the QoI is conditioned on the observed data and a hypothetical experiment. The main contribution of this paper is the derivation of a semi-analytic mathematical formula for the expected information gain about the statistical expectation of a physical response. The developed BODE is validated on synthetic functions with varying number of input-dimensions. We demonstrate the performance of the methodology on a steel wire manufacturing problem.

We present a probabilistic deep learning methodology that enables the construction of predictive data-driven surrogates for stochastic systems. Leveraging recent advances in variational inference with implicit distributions, we put forth a statistical inference framework that enables the end-to-end training of surrogate models on paired input-output observations that may be stochastic in nature, originate from different information sources of variable fidelity, or be corrupted by complex noise processes. The resulting surrogates can accommodate high-dimensional inputs and outputs and are able to return predictions with quantified uncertainty. The effectiveness our approach is demonstrated through a series of canonical studies, including the regression of noisy data, multi-fidelity modeling of stochastic processes, and uncertainty propagation in high-dimensional dynamical systems.

Statistical inference of analytically non-tractable posteriors is a difficult problem because of marginalization of correlated variables and stochastic methods such as MCMC and VI are commonly used. We argue that stochastic KL divergence minimization used by MCMC and VI is noisy, and we propose instead EL_2O, expectation optimization of L_2 distance squared between the approximate log posterior q and the un-normalized log posterior of p. When sampling from q the solutions agree with stochastic KL divergence minimization based VI in the large sample limit, however EL_2O method is free of sampling noise, has better optimization properties, and requires only as many sample evaluations as the number of parameters we are optimizing if q covers p. As a consequence, increasing the expressivity of q improves both the quality of results and the convergence rate, allowing EL_2O to approach exact inference. Use of automatic differentiation methods enables us to develop Hessian, gradient and gradient free versions of the method, which can determine M(M+2)/2+1, M+1 and 1 parameter(s) of q with a single sample, respectively. EL_2O provides a reliable estimate of the quality of the approximating posterior, and converges rapidly on full rank gaussian approximation for q and extensions beyond it, such as nonlinear transformations and gaussian mixtures. These can handle general posteriors, while still allowing fast analytic marginalizations. We test it on several examples, including a realistic 13 dimensional galaxy clustering analysis, showing that it is several orders of magnitude faster than MCMC, while giving smooth and accurate non-gaussian posteriors, often requiring a few to a few dozen of iterations only.

This paper presents a non-trivial reconstruction of a previous joint topic-sentiment-preference review model TSPRA with stick-breaking representation under the framework of variational inference (VI) and stochastic variational inference (SVI). TSPRA is a Gibbs Sampling based model that solves topics, word sentiments and user preferences altogether and has been shown to achieve good performance, but for large data set it can only learn from a relatively small sample. We develop the variational models vTSPRA and svTSPRA to improve the time use, and our new approach is capable of processing millions of reviews. We rebuild the generative process, improve the rating regression, solve and present the coordinate-ascent updates of variational parameters, and show the time complexity of each iteration is theoretically linear to the corpus size, and the experiments on Amazon data sets show it converges faster than TSPRA and attains better results given the same amount of time. In addition, we tune svTSPRA into an online algorithm ovTSPRA that can monitor oscillations of sentiment and preference overtime. Some interesting fluctuations are captured and possible explanations are provided. The results give strong visual evidence that user preference is better treated as an independent factor from sentiment.