We develop methods for parameter estimation in settings with large-scale data sets, where traditional methods are no longer tenable. Our methods rely on stochastic approximations, which are computationally efficient as they maintain one iterate as a parameter estimate, and successively update that iterate based on a single data point. When the update is based on a noisy gradient, the stochastic approximation is known as standard stochastic gradient descent, which has been fundamental in modern applications with large data sets. Additionally, our methods are numerically stable because they employ implicit updates of the iterates. Intuitively, an implicit update is a shrinked version of a standard one, where the shrinkage factor depends on the observed Fisher information at the corresponding data point. This shrinkage prevents numerical divergence of the iterates, which can be caused either by excess noise or outliers. Our sgd package in R offers the most extensive and robust implementation of stochastic gradient descent methods. We demonstrate that sgd dominates alternative software in runtime for several estimation problems with massive data sets. Our applications include the wide class of generalized linear models as well as M-estimation for robust regression.

The scarcity of data annotated at the desired level of granularity is a recurring issue in many applications. Significant amounts of effort have been devoted to developing weakly supervised methods tailored to each individual setting, which are often carefully designed to take advantage of the particular properties of weak supervision regimes, form of available data and prior knowledge of the task at hand. Unfortunately, it is difficult to adapt these methods to new tasks and/or forms of data, which often require different weak supervision regimes or models. We present a general-purpose method that can solve any weakly supervised learning problem irrespective of the weak supervision regime or the model. The proposed method turns any off-the-shelf strongly supervised classifier into a weakly supervised classifier and allows the user to specify any arbitrary weakly supervision regime via a loss function. We apply the method to several different weak supervision regimes and demonstrate competitive results compared to methods specifically engineered for those settings.

We propose a Conditional Density Filtering (C-DF) algorithm for efficient online Bayesian inference. C-DF adapts MCMC sampling to the online setting, sampling from approximations to conditional posterior distributions obtained by propagating surrogate conditional sufficient statistics (a function of data and parameter estimates) as new data arrive. These quantities eliminate the need to store or process the entire dataset simultaneously and offer a number of desirable features. Often, these include a reduction in memory requirements and runtime and improved mixing, along with state-of-the-art parameter inference and prediction. These improvements are demonstrated through several illustrative examples including an application to high dimensional compressed regression. Finally, we show that C-DF samples converge to the target posterior distribution asymptotically as sampling proceeds and more data arrives.

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally infeasible. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem in three ways: it generates proposed moves using only a subset of the data, it skips the Metropolis-Hastings accept-reject step, and it uses sequences of decreasing step sizes. In \cite{TehThierryVollmerSGLD2014}, we provided the mathematical foundations for the decreasing step size SGLD, including consistency and a central limit theorem. However, in practice the SGLD is run for a relatively small number of iterations, and its step size is not decreased to zero. The present article investigates the behaviour of the SGLD with fixed step size. In particular we characterise the asymptotic bias explicitly, along with its dependence on the step size and the variance of the stochastic gradient. On that basis a modified SGLD which removes the asymptotic bias due to the variance of the stochastic gradients up to first order in the step size is derived. Moreover, we are able to obtain bounds on the finite-time bias, variance and mean squared error (MSE). The theory is illustrated with a Gaussian toy model for which the bias and the MSE for the estimation of moments can be obtained explicitly. For this toy model we study the gain of the SGLD over the standard Euler method in the limit of large data sets.

This paper addresses the problem of neighborhood selection for Gaussian graphical models. We present two heuristic algorithms: a forward-backward greedy algorithm for general Gaussian graphical models based on mutual information test, and a threshold-based algorithm for walk summable Gaussian graphical models. Both algorithms are shown to be structurally consistent, and efficient. Numerical results show that both algorithms work very well.

Classification and clustering are both important topics in statistical learning. A natural question herein is whether predefined classes are really different from one another, or whether clusters are really there. Specifically, we may be interested in knowing whether the two classes defined by some class labels (when they are provided), or the two clusters tagged by a clustering algorithm (where class labels are not provided), are from the same underlying distribution. Although both are challenging questions for the high-dimensional, low-sample size data, there has been some recent development for both. However, when it is costly to manually place labels on observations, it is often that only a small portion of the class labels is available. In this article, we propose a significance analysis approach for such type of data, namely partially labeled data. Our method makes use of the whole data and tries to test the class difference as if all the labels were observed. Compared to a testing method that ignores the label information, our method provides a greater power, meanwhile, maintaining the size, illustrated by a comprehensive simulation study. Theoretical properties of the proposed method are studied with emphasis on the high-dimensional, low-sample size setting. Our simulated examples help to understand when and how the information extracted from the labeled data can be effective. A real data example further illustrates the usefulness of the proposed method.

Protein-protein interaction (PPI) prediction is an important problem in machine learning and computational biology. However, there is no data set for training or evaluation purposes, where all the instances are accurately labeled. Instead, what is available are instances of positive class (with possibly noisy labels) and no instances of negative class. The non-availability of negative class data is typically handled with the observation that randomly chosen protein-pairs have a nearly 100% chance of being negative class, as only 1 in 1,500 protein pairs expected is expected to be an interacting pair. In this paper, we focused on the problem that non-availability of accurately labeled testing data sets in the domain of protein-protein interaction (PPI) prediction may lead to biased evaluation results. We first showed that not acknowledging the inherent skew in the interactome (i.e. rare occurrence of positive instances) leads to an over-estimated accuracy of the predictor. Then we show that, with the belief that positive interactions are a rare category, sampling random pairs of proteins excluding known interacting proteins set as the negative testing data set could lead to an under-estimated evaluation result. We formalized those two problems to validate the above claim, and based on the formalization, we proposed a balancing method to cancel out the over-estimation with under-estimation. Finally, our experiments validated the theoretical aspects and showed that this balancing evaluation could evaluate the exact performance without availability of golden standard data sets.

Motivated by large-scale Collaborative-Filtering applications, we present a Non-Commuting Latent Factor (NCLF) tensor-completion approach for modeling three-way arrays, which is diagonal like the standard PARAFAC, but wherein different terms distinguish different kinds of three-way relations of co-clusters, as determined by permutations of latent factors. The first key component of the algebraic representation is the usage of two non-commutative real trilinear operations as the building blocks of the approximation. These operations are the standard three dimensional triple-product and a trilinear product on a two-dimensional real vector space, which is a representation of the real Clifford Algebra Cl(1,1) (a certain Majorana spinor). Both operations are purely ternary in that they cannot be decomposed into two group-operations on the relevant spaces. The second key component of the method is combining these operations using permutation-symmetry preserving linear combinations. We apply the model to the MovieLens and Fannie Mae datasets, and find that it outperforms the PARAFAC model. We propose some future directions, such as unsupervised-learning.

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.

Real-time estimation of destination and travel time for taxis is of great importance for existing electronic dispatch systems. We present an approach based on trip matching and ensemble learning, in which we leverage the patterns observed in a dataset of roughly 1.7 million taxi journeys to predict the corresponding final destination and travel time for ongoing taxi trips, as a solution for the ECML/PKDD Discovery Challenge 2015 competition. The results of our empirical evaluation show that our approach is effective and very robust, which led our team -- BlueTaxi -- to the 3rd and 7th position of the final rankings for the trip time and destination prediction tasks, respectively. Given the fact that the final rankings were computed using a very small test set (with only 320 trips) we believe that our approach is one of the most robust solutions for the challenge based on the consistency of our good results across the test sets.