We introduce overdispersed black-box variational inference, a method to reduce the variance of the Monte Carlo estimator of the gradient in black-box variational inference. Instead of taking samples from the variational distribution, we use importance sampling to take samples from an overdispersed distribution in the same exponential family as the variational approximation. Our approach is general since it can be readily applied to any exponential family distribution, which is the typical choice for the variational approximation. We run experiments on two non-conjugate probabilistic models to show that our method effectively reduces the variance, and the overhead introduced by the computation of the proposal parameters and the importance weights is negligible. We find that our overdispersed importance sampling scheme provides lower variance than black-box variational inference, even when the latter uses twice the number of samples.

Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient variational inference algorithm, freeing the scientist to refine and explore many models. ADVI supports a broad class of models-no conjugacy assumptions are required. We study ADVI across ten different models and apply it to a dataset with millions of observations. ADVI is integrated into Stan, a probabilistic programming system; it is available for immediate use.

Developing efficient and scalable algorithms for Latent Dirichlet Allocation (LDA) is of wide interest for many applications. Previous work has developed an O(1) Metropolis-Hastings sampling method for each token. However, the performance is far from being optimal due to random accesses to the parameter matrices and frequent cache misses. In this paper, we first carefully analyze the memory access efficiency of existing algorithms for LDA by the scope of random access, which is the size of the memory region in which random accesses fall, within a short period of time. We then develop WarpLDA, an LDA sampler which achieves both the best O(1) time complexity per token and the best O(K) scope of random access. Our empirical results in a wide range of testing conditions demonstrate that WarpLDA is consistently 5-15x faster than the state-of-the-art Metropolis-Hastings based LightLDA, and is comparable or faster than the sparsity aware F+LDA. With WarpLDA, users can learn up to one million topics from hundreds of millions of documents in a few hours, at an unprecedentedly throughput of 11G tokens per second.

Iterative methods for fitting a Gaussian Random Field (GRF) model to spatial data via maximum likelihood (ML) require $\mathcal{O}(n^3)$ floating point operations per iteration, where $n$ denotes the number of data locations. For large data sets, the $\mathcal{O}(n^3)$ complexity per iteration together with the non-convexity of the ML problem render traditional ML methods inefficient for GRF fitting. The problem is even more aggravated for anisotropic GRFs where the number of covariance function parameters increases with the process domain dimension. In this paper, we propose a new two-step GRF estimation procedure when the process is second-order stationary. First, a \emph{convex} likelihood problem regularized with a weighted $\ell_1$-norm, utilizing the available distance information between observation locations, is solved to fit a sparse \emph{{precision} (inverse covariance) matrix to the observed data using the Alternating Direction Method of Multipliers. Second, the parameters of the GRF spatial covariance function are estimated by solving a least squares problem. Theoretical error bounds for the proposed estimator are provided; moreover, convergence of the estimator is shown as the number of samples per location increases. The proposed method is numerically compared with state-of-the-art methods for big $n$. Data segmentation schemes are implemented to handle large data sets.

There is a long history in game theory on the topic of Bayesian or "rational" learning, in which each player maintains beliefs over a set of alternative behaviours, or types, for the other players. This idea has gained increasing interest in the artificial intelligence (AI) community, where it is used as a method to control a single agent in a system composed of multiple agents with unknown behaviours. The idea is to hypothesise a set of types, each specifying a possible behaviour for the other agents, and to plan our own actions with respect to those types which we believe are most likely, given the observed actions of the agents. The game theory literature studies this idea primarily in the context of equilibrium attainment. In contrast, many AI applications have a focus on task completion and payoff maximisation. With this perspective in mind, we identify and address a spectrum of questions pertaining to belief and truth in hypothesised types. We formulate three basic ways to incorporate evidence into posterior beliefs and show when the resulting beliefs are correct, and when they may fail to be correct. Moreover, we demonstrate that prior beliefs can have a significant impact on our ability to maximise payoffs in the long-term, and that they can be computed automatically with consistent performance effects. Furthermore, we analyse the conditions under which we are able complete our task optimally, despite inaccuracies in the hypothesised types. Finally, we show how the correctness of hypothesised types can be ascertained during the interaction via an automated statistical analysis.

Herding defines a deterministic dynamical system at the edge of chaos. It generates a sequence of model states and parameters by alternating parameter perturbations with state maximizations, where the sequence of states can be interpreted as "samples" from an associated MRF model. Herding differs from maximum likelihood estimation in that the sequence of parameters does not converge to a fixed point and differs from an MCMC posterior sampling approach in that the sequence of states is generated deterministically. Herding may be interpreted as a"perturb and map" method where the parameter perturbations are generated using a deterministic nonlinear dynamical system rather than randomly from a Gumbel distribution. This chapter studies the distinct statistical characteristics of the herding algorithm and shows that the fast convergence rate of the controlled moments may be attributed to edge of chaos dynamics. The herding algorithm can also be generalized to models with latent variables and to a discriminative learning setting. The perceptron cycling theorem ensures that the fast moment matching property is preserved in the more general framework.

In a multi-class classification problem, it is standard to model the output of a neural network as a categorical distribution conditioned on the inputs. The output must therefore be positive and sum to one, which is traditionally enforced by a softmax. This probabilistic mapping allows to use the maximum likelihood principle, which leads to the well-known log-softmax loss. However the choice of the softmax function seems somehow arbitrary as there are many other possible normalizing functions. It is thus unclear why the log-softmax loss would perform better than other loss alternatives. In particular Vincent et al. (2015) recently introduced a class of loss functions, called the spherical family, for which there exists an efficient algorithm to compute the updates of the output weights irrespective of the output size. In this paper, we explore several loss functions from this family as possible alternatives to the traditional log-softmax. In particular, we focus our investigation on spherical bounds of the log-softmax loss and on two spherical log-likelihood losses, namely the log-Spherical Softmax suggested by Vincent et al. (2015) and the log-Taylor Softmax that we introduce. Although these alternatives do not yield as good results as the log-softmax loss on two language modeling tasks, they surprisingly outperform it in our experiments on MNIST and CIFAR-10, suggesting that they might be relevant in a broad range of applications.

The plausibility of uncommon events and miracles based on testimony of such an event has been much discussed. When analyzing the probabilities involved, it has mostly been assumed that the common events can be taken as data in the calculations. However, we usually have only testimonies for the common events. While this difference does not have a significant effect on the inductive part of the inference, it has a large influence on how one should view the reliability of testimonies. In this work, a full Bayesian solution is given for the more realistic case, where one has a large number of testimonies for a common event and one testimony for an uncommon event. It is seen that, in order for there to be a large amount of testimonies for a common event, the testimonies will probably be quite reliable. For this reason, because the testimonies are quite reliable based on the testimonies for the common events, the probability for the uncommon event, given a testimony for it, is also higher. Hence, one should be more open-minded when considering the plausibility of uncommon events.

Hyper-parameters play a major role in the learning and inference process of latent Dirichlet allocation (LDA). In order to begin the LDA latent variables learning process, these hyper-parameters values need to be pre-determined. We propose an extension for LDA that we call 'Latent Dirichlet allocation Gibbs Newton' (LDA-GN), which places non-informative priors over these hyper-parameters and uses Gibbs sampling to learn appropriate values for them. At the heart of LDA-GN is our proposed 'Gibbs-Newton' algorithm, which is a new technique for learning the parameters of multivariate Polya distributions. We report Gibbs-Newton performance results compared with two prominent existing approaches to the latter task: Minka's fixed-point iteration method and the Moments method. We then evaluate LDA-GN in two ways: (i) by comparing it with standard LDA in terms of the ability of the resulting topic models to generalize to unseen documents; (ii) by comparing it with standard LDA in its performance on a binary classification task.

Human language is used to express logical and mathematical computations whose cognitive bases still remain unexplained. We ask ourselves if language acquisition involves a kind of implicit logical and mathematical programming that could explain such performances. Examples of these performances are some logical propositions, transferred by natural language, valid for different languages and for large populations of humans sharing similar cultural traditions. In the present work we choose -as a largely accepted logical statement-one of the most cited expressions that Arthur Conan Doyle attributed to Sherlock Holmes, the "old maxim" mentioned by the character in the story "The Adventure of the Beryl Coronet". For an allusion to this maxim in a scientific context see Cairns-Smith (1990).