While all kinds of mixed data---from personal data, over panel and scientific data, to public and commercial data---are collected and stored, building probabilistic graphical models for these hybrid domains becomes more difficult. Users spend significant amounts of time in identifying the parametric form of the random variables (Gaussian, Poisson, Logit, etc.) involved and learning the mixed models. To make this difficult task easier, we propose the first trainable probabilistic deep architecture for hybrid domains that features tractable queries. It is based on Sum-Product Networks (SPNs) with piecewise polynomial leaf distributions together with novel nonparametric decomposition and conditioning steps using the Hirschfeld-Gebelein-Renyi Maximum Correlation Coefficient. This relieves the user from deciding a-priori the parametric form of the random variables but is still expressive enough to effectively approximate any distribution and permits efficient learning and inference.Our experiments show that the architecture, called Mixed SPNs, can indeed capture complex distributions across a wide range of hybrid domains.

Many applications infer the structure of a probabilistic graphical model from data to elucidate the relationships between variables. But how can we train graphical models on a massive data set? In this paper, we show how to construct coresets---compressed data sets which can be used as proxy for the original data and have provably bounded worst case error---for Gaussian dependency networks (DNs), i.e., cyclic directed graphical models over Gaussians, where the parents of each variable are its Markov blanket. Specifically, we prove that Gaussian DNs admit coresets of size independent of the size of the data set. Unfortunately, this does not extend to DNs over members of the exponential family in general. As we will prove, Poisson DNs do not admit small coresets. Despite this worst-case result, we will provide an argument why our coreset construction for DNs can still work well in practice on count data.To corroborate our theoretical results, we empirically evaluated the resulting Core DNs on real data sets. The results demonstrate significant gains over no or naive sub-sampling, even in the case of count data.

Temporal point processes are a statistical framework for modelling the times at which events of interest occur. The Hawkes process is a well-studied instance of this framework that captures self-exciting behaviour, wherein the occurrence of one event increases the likelihood of future events. Such processes have been successfully applied to model phenomena ranging from earthquakes to behaviour in a social network. We propose a framework to design new loss functions to train linear and nonlinear Hawkes processes. This captures standard maximum likelihood as a special case, but allows for other losses that guarantee convex objective functions (for certain types of kernel), and admit simpler optimisation. We illustrate these points with three concrete examples: for linear Hawkes processes, we provide a least-squares style loss potentially admitting closed-form optimisation; for exponential Hawkes processes, we reduce training to a weighted logistic regression; and for sigmoidal Hawkes processes, we propose an asymmetric form of logistic regression.

A key challenge in many reinforcement learning problems is delayed rewards, which can significantly slow down learning. Although reward shaping has previously been introduced to accelerate learning by bootstrapping an agent with additional information, this can lead to problems with convergence. We present a novel Bayesian reward shaping framework that augments the reward distribution with prior beliefs that decay with experience. Formally, we prove that under suitable conditions a Markov decision process augmented with our framework is consistent with the optimal policy of the original MDP when using the Q-learning algorithm. However, in general our method integrates seamlessly with any reinforcement learning algorithm that learns a value or action-value function through experience. Experiments are run on a gridworld and a more complex backgammon domain that show that we can learn tasks significantly faster when we specify intuitive priors on the reward distribution.

We develop Riemannian Stein Variational Gradient Descent (RSVGD), a Bayesian inference method that generalizes Stein Variational Gradient Descent (SVGD) to Riemann manifold. The benefits are two-folds: (i) for inference tasks in Euclidean spaces, RSVGD has the advantage over SVGD of utilizing information geometry, and (ii) for inference tasks on Riemann manifolds, RSVGD brings the unique advantages of SVGD to the Riemannian world. To appropriately transfer to Riemann manifolds, we conceive novel and non-trivial techniques for RSVGD, which are required by the intrinsically different characteristics of general Riemann manifolds from Euclidean spaces. We also discover Riemannian Stein's Identity and Riemannian Kernelized Stein Discrepancy. Experimental results show the advantages over SVGD of exploring distribution geometry and the advantages of particle-efficiency, iteration-effectiveness and approximation flexibility over other inference methods on Riemann manifolds.

We present a novel method for frequentist statistical inference in M-estimation problems, based on stochastic gradient descent (SGD) with a fixed step size: we demonstrate that the average of such SGD sequences can be used for statistical inference, after proper scaling. An intuitive analysis using the Ornstein-Uhlenbeck process suggests that such averages are asymptotically normal. To show the merits of our scheme, we apply it to both synthetic and real data sets, and demonstrate that its accuracy is comparable to classical statistical methods, while requiring potentially far less computation.

We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer models with semi-random features, we prove with no unrealistic assumptions that the model classes contain an arbitrarily good function as the width increases (universality), and despite non-convexity, we can find such a good function (optimization theory) that generalizes to unseen new data (generalization bound). For deep models, with no unrealistic assumptions, we prove universal approximation ability, a lower bound on approximation error, a partial optimization guarantee, and a generalization bound. Depending on the problems, the generalization bound of deep semi-random features can be exponentially better than the known bounds of deep ReLU nets; our generalization error bound can be independent of the depth, the number of trainable weights as well as the input dimensionality. In experiments, we show that semi-random features can match the performance of neural networks by using slightly more units, and it outperforms random features by using significantly fewer units. Moreover, we introduce a new implicit ensemble method by using semi-random features.

We propose a framework for modeling and estimating the state of controlled dynamical systems, where an agent can affect the system through actions and receives partial observations. Based on this framework, we propose Predictive State Representation with Random Fourier Features (RFF-PSR). A key property in RFF-PSRs is that the state estimate is represented by a conditional distribution of future observations given future actions. RFFPSRs combine this representation with moment-matching, kernel embedding, and local optimization to achieve a method that enjoys several favorable qualities: It can represent controlled environments which can be affected by actions, it has an efficient and theoretically justified learning algorithm, it uses a non-parametric representation that has expressive power to represent continuous non-linear dynamics. We provide a detailed formulation, a theoretical analysis and an experimental evaluation that demonstrates the effectiveness of our method.

We propose a novel approach for using unsupervised boosting to create an ensemble of generative models, where models are trained in sequence to correct earlier mistakes. Our meta-algorithmic framework can leverage any existing base learner that permits likelihood evaluation, including recent deep expressive models. Further, our approach allows the ensemble to include discriminative models trained to distinguish real data from model-generated data. We show theoretical conditions under which incorporating a new model in the ensemble will improve the fit and empirically demonstrate the effectiveness of our black-box boosting algorithms on density estimation, classification, and sample generation on benchmark datasets for a wide range of generative models.

Adversarial learning of probabilistic models has recently emerged as a promising alternative to maximum likelihood. Implicit models such as generative adversarial networks (GAN) often generate better samples compared to explicit models trained by maximum likelihood. Yet, GANs sidestep the characterization of an explicit density which makes quantitative evaluations challenging. To bridge this gap, we propose Flow-GANs, a generative adversarial network for which we can perform exact likelihood evaluation, thus supporting both adversarial and maximum likelihood training. When trained adversarially, Flow-GANs generate high-quality samples but attain extremely poor log-likelihood scores, inferior even to a mixture model memorizing the training data; the opposite is true when trained by maximum likelihood. Results on MNIST and CIFAR-10 demonstrate that hybrid training can attain high held-out likelihoods while retaining visual fidelity in the generated samples.