In this paper we develop a statistical theory and an implementation of deep learning (DL) models. We show that an elegant variable splitting scheme for the alternating direction method of multipliers (ADMM) optimises a deep learning objective. We allow for non-smooth non-convex regularisation penalties to induce sparsity in parameter weights. We provide a link between traditional shallow layer statistical models such as principal component and sliced inverse regression and deep layer models. We also define the degrees of freedom of a deep learning predictor and a predictive MSE criteria to perform model selection for comparing architecture designs. We focus on deep multiclass logistic learning although our methods apply more generally. Our results suggest an interesting and previously under-exploited relationship between deep learning and proximal splitting techniques. To illustrate our methodology, we provide a multi-class logit classification analysis of Fisher's Iris data where we illustrate the convergence of our algorithm. Finally, we conclude with directions for future research.

This paper covers the two approaches for sentiment analysis: i) lexicon based method; ii) machine learning method. We describe several techniques to implement these approaches and discuss how they can be adopted for sentiment classification of Twitter messages. We present a comparative study of different lexicon combinations and show that enhancing sentiment lexicons with emoticons, abbreviations and social-media slang expressions increases the accuracy of lexicon-based classification for Twitter. We discuss the importance of feature generation and feature selection processes for machine learning sentiment classification. To quantify the performance of the main sentiment analysis methods over Twitter we run these algorithms on a benchmark Twitter dataset from the SemEval-2013 competition, task 2-B. The results show that machine learning method based on SVM and Naive Bayes classifiers outperforms the lexicon method. We present a new ensemble method that uses a lexicon based sentiment score as input feature for the machine learning approach. The combined method proved to produce more precise classifications. We also show that employing a cost-sensitive classifier for highly unbalanced datasets yields an improvement of sentiment classification performance up to 7%.

Classification is an important tool with many useful applications. Among the many classification methods, Fisher's Linear Discriminant Analysis (LDA) is a traditional model-based approach which makes use of the covariance information. However, in the high-dimensional, low-sample size setting, LDA cannot be directly deployed because the sample covariance is not invertible. While there are modern methods designed to deal with high-dimensional data, they may not fully use the covariance information as LDA does. Hence in some situations, it is still desirable to use a model-based method such as LDA for classification. This article exploits the potential of LDA in more complicated data settings. In many real applications, it is costly to manually place labels on observations; hence it is often that only a small portion of labeled data is available while a large number of observations are left without a label. It is a great challenge to obtain good classification performance through the labeled data alone, especially when the dimension is greater than the size of the labeled data. In order to overcome this issue, we propose a semi-supervised sparse LDA classifier to take advantage of the seemingly useless unlabeled data. They provide additional information which helps to boost the classification performance in some situations. A direct estimation method is used to reconstruct LDA and achieve the sparsity; meanwhile we employ the difference-convex algorithm to handle the non-convex loss function associated with the unlabeled data. Theoretical properties of the proposed classifier are studied. Our simulated examples help to understand when and how the information extracted from the unlabeled data can be useful. A real data example further illustrates the usefulness of the proposed method.

Ensembles of climate models are commonly used to improve climate predictions and assess the uncertainties associated with them. Weighting the models according to their performances holds the promise of further improving their predictions. Here, we use an ensemble of decadal climate predictions to demonstrate the ability of sequential learning algorithms (SLAs) to reduce the forecast errors and reduce the uncertainties. Three different SLAs are considered, and their performances are compared with those of an equally weighted ensemble, a linear regression and the climatology. Predictions of four different variables--the surface temperature, the zonal and meridional wind, and pressure--are considered. The spatial distributions of the performances are presented, and the statistical significance of the improvements achieved by the SLAs is tested. Based on the performances of the SLAs, we propose one to be highly suitable for the improvement of decadal climate predictions.

Gathering the most information by picking the least amount of data is a common task in experimental design or when exploring an unknown environment in reinforcement learning and robotics. A widely used measure for quantifying the information contained in some distribution of interest is its entropy. Greedily minimizing the expected entropy is therefore a standard method for choosing samples in order to gain strong beliefs about the underlying random variables. We show that this approach is prone to temporally getting stuck in local optima corresponding to wrongly biased beliefs. We suggest instead maximizing the expected cross entropy between old and new belief, which aims at challenging refutable beliefs and thereby avoids these local optima. We show that both criteria are closely related and that their difference can be traced back to the asymmetry of the Kullback-Leibler divergence. In illustrative examples as well as simulated and real-world experiments we demonstrate the advantage of cross entropy over simple entropy for practical applications.

Models for recommender systems use latent factors to explain the preferences and behaviors of users with respect to a set of items (e.g., movies, books, academic papers). Typically, the latent factors are assumed to be static and, given these factors, the observed preferences and behaviors of users are assumed to be generated without order. These assumptions limit the explorative and predictive capabilities of such models, since users' interests and item popularity may evolve over time. To address this, we propose dPF, a dynamic matrix factorization model based on the recent Poisson factorization model for recommendations. dPF models the time evolving latent factors with a Kalman filter and the actions with Poisson distributions. We derive a scalable variational inference algorithm to infer the latent factors. Finally, we demonstrate dPF on 10 years of user click data from arXiv.org, one of the largest repository of scientific papers and a formidable source of information about the behavior of scientists. Empirically we show performance improvement over both static and, more recently proposed, dynamic recommendation models. We also provide a thorough exploration of the inferred posteriors over the latent variables.

Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in high-dimensional settings. All existing methods for learning under the assumption of structured sparsity rely on prior knowledge on how to weight (or how to penalize) individual subsets of variables during the subset selection process, which is not available in general. Inferring group weights from data is a key open research problem in structured sparsity.In this paper, we propose a Bayesian approach to the problem of group weight learning. We model the group weights as hyperparameters of heavy-tailed priors on groups of variables and derive an approximate inference scheme to infer these hyperparameters. We empirically show that we are able to recover the model hyperparameters when the data are generated from the model, and we demonstrate the utility of learning weights in synthetic and real denoising problems.

In the Bayesian Reinforcement Learning (BRL) setting, agents try to maximise the collected rewards while interacting with their environment while using some prior knowledge that is accessed beforehand. Many BRL algorithms have already been proposed, but even though a few toy examples exist in the literature, there are still no extensive or rigorous benchmarks to compare them. The paper addresses this problem, and provides a new BRL comparison methodology along with the corresponding open source library. In this methodology, a comparison criterion that measures the performance of algorithms on large sets of Markov Decision Processes (MDPs) drawn from some probability distributions is defined. In order to enable the comparison of non-anytime algorithms, our methodology also includes a detailed analysis of the computation time requirement of each algorithm. Our library is released with all source code and documentation: it includes three test problems, each of which has two different prior distributions, and seven state-of-the-art RL algorithms. Finally, our library is illustrated by comparing all the available algorithms and the results are discussed.

We consider a network of agents that aim to learn some unknown state of the world using private observations and exchange of beliefs. At each time, agents observe private signals generated based on the true unknown state. Each agent might not be able to distinguish the true state based only on her private observations. This occurs when some other states are observationally equivalent to the true state from the agent's perspective. To overcome this shortcoming, agents must communicate with each other to benefit from local observations. We propose a model where each agent selects one of her neighbors randomly at each time. Then, she refines her opinion using her private signal and the prior of that particular neighbor. The proposed rule can be thought of as a Bayesian agent who cannot recall the priors based on which other agents make inferences. This learning without recall approach preserves some aspects of the Bayesian inference while being computationally tractable. By establishing a correspondence with a random walk on the network graph, we prove that under the described protocol, agents learn the truth exponentially fast in the almost sure sense. The asymptotic rate is expressed as the sum of the relative entropies between the signal structures of every agent weighted by the stationary distribution of the random walk.

Ridge regularized linear models (RRLMs), such as ridge regression and the SVM, are a popular group of methods that are used in conjunction with coefficient hypothesis testing to discover explanatory variables with a significant multivariate association to a response. However, many investigators are reluctant to draw causal interpretations of the selected variables due to the incomplete knowledge of the capabilities of RRLMs in causal inference. Under reasonable assumptions, we show that a modified form of RRLMs can get very close to identifying a subset of the Markov boundary by providing a worst-case bound on the space of possible solutions. The results hold for any convex loss, even when the underlying functional relationship is nonlinear, and the solution is not unique. Our approach combines ideas in Markov boundary and sufficient dimension reduction theory. Experimental results show that the modified RRLMs are competitive against state-of-the-art algorithms in discovering part of the Markov boundary from gene expression data.