We study the estimation of risk-sensitive policies in reinforcement learning problems defined by a Markov Decision Process (MDPs) whose state and action spaces are countably finite. Prior efforts are predominately afflicted by computational challenges associated with the fact that risk-sensitive MDPs are time-inconsistent. To ameliorate this issue, we propose a new definition of risk, which we call caution, as a penalty function added to the dual objective of the linear programming (LP) formulation of reinforcement learning. The caution measures the distributional risk of a policy, which is a function of the policy's long-term state occupancy distribution. To solve this problem in an online model-free manner, we propose a stochastic variant of primal-dual method that uses Kullback-Lieber (KL) divergence as its proximal term. We establish that the number of iterations/samples required to attain approximately optimal solutions of this scheme matches tight dependencies on the cardinality of the state and action spaces, but differs in its dependence on the infinity norm of the gradient of the risk measure. Experiments demonstrate the merits of this approach for improving the reliability of reward accumulation without additional computational burdens.

We propose automated augmented conjugate inference, a new inference method for non-conjugate Gaussian processes (GP) models. Our method automatically constructs an auxiliary variable augmentation that renders the GP model conditionally conjugate. Building on the conjugate structure of the augmented model, we develop two inference methods. First, a fast and scalable stochastic variational inference method that uses efficient block coordinate ascent updates, which are computed in closed form. Second, an asymptotically correct Gibbs sampler that is useful for small datasets. Our experiments show that our method are up two orders of magnitude faster and more robust than existing state-of-the-art black-box methods.

Normalizing flows (NF) are a powerful framework for approximating posteriors. By mapping a simple base density through invertible transformations, flows provide an exact method of density evaluation and sampling. The trend in normalizing flow literature has been to devise deeper, more complex transformations to achieve greater flexibility. We propose an alternative: Gradient Boosted Flows (GBF) model a variational posterior by successively adding new NF components by gradient boosting so that each new NF component is fit to the residuals of the previously trained components. The GBF formulation results in a variational posterior that is a mixture model, whose flexibility increases as more components are added. Moreover, GBFs offer a wider, not deeper, approach that can be incorporated to improve the results of many existing NFs. We demonstrate the effectiveness of this technique for density estimation and, by coupling GBF with a variational autoencoder, generative modeling of images.

The profile of a sample is the multiset of its symbol frequencies. We show that for samples of discrete distributions, profile entropy is a fundamental measure unifying the concepts of estimation, inference, and compression. Specifically, profile entropy a) determines the speed of estimating the distribution relative to the best natural estimator; b) characterizes the rate of inferring all symmetric properties compared with the best estimator over any label-invariant distribution collection; c) serves as the limit of profile compression, for which we derive optimal near-linear-time block and sequential algorithms. To further our understanding of profile entropy, we investigate its attributes, provide algorithms for approximating its value, and determine its magnitude for numerous structural distribution families.

Bayesian learning rule is a recently proposed variational inference method, which not only contains many existing learning algorithms as special cases but also enables the design of new algorithms. Unfortunately, when posterior parameters lie in an open constraint set, the rule may not satisfy the constraints and requires line-searches which could slow down the algorithm. In this paper, we fix this issue for the positive-definite constraint by proposing an improved rule that naturally handles the constraint. Our modification is obtained using Riemannian gradient methods, and is valid when the approximation attains a \emph{block-coordinate natural parameterization} (e.g., Gaussian distributions and their mixtures). Our method outperforms existing methods without any significant increase in computation. Our work makes it easier to apply the learning rule in the presence of positive-definite constraints in parameter spaces.

Approaches based on computing with words find good applicability in decision making systems. Predominantly finding their basis in type-1 fuzzy sets, computing with words approaches employ type-1 fuzzy sets as semantics of the linguistic terms. However, type-2 fuzzy sets have been proven to be scientifically more appropriate to represent linguistic information in practical systems. They take into account both the intra-uncertainty as well as the inter-uncertainty in cases where the linguistic information comes from a group of experts. Hence in this paper, we propose to introduce linguistic terms whose semantics are denoted by interval type-2 fuzzy sets within the hesitant fuzzy linguistic term set framework, resulting in type-2 fuzzy sets based hesitant fuzzy linguistic term sets. We also introduce a novel method of computing type-2 fuzzy envelopes out of multiple interval type-2 fuzzy sets with trapezoidal membership functions. Furthermore, the proposed framework with interval type-2 fuzzy sets is applied on a supplier performance evaluation scenario. Since humans are predominantly involved in the entire process of supply chain, their feedback is crucial while deciding many factors. Towards the end of the paper, we compare our presented model with various existing models and demonstrate the advantages of the former.

Modelling exchangeable relational data can be described by \textit{graphon theory}. Most Bayesian methods for modelling exchangeable relational data can be attributed to this framework by exploiting different forms of graphons. However, the graphons adopted by existing Bayesian methods are either piecewise-constant functions, which are insufficiently flexible for accurate modelling of the relational data, or are complicated continuous functions, which incur heavy computational costs for inference. In this work, we introduce a smoothing procedure to piecewise-constant graphons to form {\em smoothing graphons}, which permit continuous intensity values for describing relations, but without impractically increasing computational costs. In particular, we focus on the Bayesian Stochastic Block Model (SBM) and demonstrate how to adapt the piecewise-constant SBM graphon to the smoothed version. We initially propose the Integrated Smoothing Graphon (ISG) which introduces one smoothing parameter to the SBM graphon to generate continuous relational intensity values. We then develop the Latent Feature Smoothing Graphon (LFSG), which improves on the ISG by introducing auxiliary hidden labels to decompose the calculation of the ISG intensity and enable efficient inference. Experimental results on real-world data sets validate the advantages of applying smoothing strategies to the Stochastic Block Model, demonstrating that smoothing graphons can greatly improve AUC and precision for link prediction without increasing computational complexity.

Artificial Neural Networks (ANNs) implement a specific form of multi-variate extrapolation and will generate an output for any input pattern, even when there is no similar training pattern. Extrapolations are not necessarily to be trusted, and in order to support safety critical systems, we require such systems to give an indication of the training sample related uncertainty associated with their output. Some readers may think that this is a well known issue which is already covered by the basic principles of pattern recognition. We will explain below how this is not the case and how the conventional (Likelihood estimate of) conditional probability of classification does not correctly assess this uncertainty. We provide a discussion of the standard interpretations of this problem and show how a quantitative approach based upon long standing methods can be practically applied. The methods are illustrated on the task of early diagnosis of dementing diseases using Magnetic Resonance Imaging.

Normalizing flows (Rezende & Mohamed, 2015) are probabilistic models constructed as a sequence of successive transformations applied to some initial distribution. A key strength of normalizing flows is their expressive power as generative models, while enjoying an explicitly computable form of the likelihood function evaluated on the transformed space. This makes them especially well-equipped for variational inference (VI). Neural networks are often used as inspiration for finding effective transformations (Dinh et al., 2015; van den Berg et al., 2018). Continuous normalizing flows were later developed in Chen et al. (2018) as a means to perform maximum likelihood estimation and VI for large-scale probabilistic models derived from ordinary differential equations (ODEs).

Neural networks with binary weights are computation-efficient and hardware-friendly, but their training is challenging because it involves a discrete optimization problem. Surprisingly, ignoring the discrete nature of the problem and using gradient-based methods, such as Straight-Through Estimator, still works well in practice. This raises the question: are there principled approaches which justify such methods? In this paper, we propose such an approach using the Bayesian learning rule. The rule, when applied to estimate a Bernoulli distribution over the binary weights, results in an algorithm which justifies some of the algorithmic choices made by the previous approaches. The algorithm not only obtains state-of-the-art performance, but also enables uncertainty estimation for continual learning to avoid catastrophic forgetting. Our work provides a principled approach for training binary neural networks which justifies and extends existing approaches.