We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles 2015) to calculate expectations with respect to the invariant measures of ergodic SDEs. In that context, we study the (over-damped) Langevin equations with strongly convex potential. We show that, when appropriate contracting couplings for the numerical integrators are available, one can obtain a time-uniform estimates of the MLMC variance in stark contrast to the majority of the results in the MLMC literature. As a consequence, one can approximate expectations with respect to the invariant measure in an unbiased way without the need of a Metropolis- Hastings step. In addition, a root mean square error of $\mathcal{O}(\epsilon)$ is achieved with $\mathcal{O}(\epsilon^{-2})$ complexity on par with Markov Chain Monte Carlo (MCMC) methods, which however can be computationally intensive when applied to large data sets. Finally, we present a multilevel version of the recently introduced Stochastic Gradient Langevin (SGLD) method (Welling and Teh, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $\mathcal{O}(\epsilon^{-2}|\log {\epsilon}|^{3})$, which is asymptotically an order $\epsilon$ lower than the $ \mathcal{O}(\epsilon^{-3})$ complexity of all stochastic gradient MCMC methods that are currently available. Numerical experiments confirm our theoretical findings.

This paper identifies a problem with the usual procedure for L2-regularization parameter estimation in a domain adaptation setting. In such a setting, there are differences between the distributions generating the training data (source domain) and the test data (target domain). The usual cross-validation procedure requires validation data, which can not be obtained from the unlabeled target data. The problem is that if one decides to use source validation data, the regularization parameter is underestimated. One possible solution is to scale the source validation data through importance weighting, but we show that this correction is not sufficient. We conclude the paper with an empirical analysis of the effect of several importance weight estimators on the estimation of the regularization parameter.

When faced with a supervised learning problem, we hope to have rich enough data to build a model that predicts future instances well. However, in practice, problems can exhibit predictive heterogeneity: most instances might be relatively easy to predict, while others might be predictive outliers for which a model trained on the entire dataset does not perform well. Identifying these can help focus future data collection. We present gLOP, the global and Local Penalty, a framework for capturing predictive heterogeneity and identifying predictive outliers. gLOP is based on penalized regression for multitask learning, which improves learning by leveraging training signal information from related tasks. We give two optimization algorithms for gLOP, one space-efficient, and another giving the full regularization path. We also characterize uniqueness in terms of the data and tuning parameters, and present empirical results on synthetic data and on two health research problems.

Scientific investigations that incorporate next generation sequencing involve analyses of high-dimensional data where the need to organize, collate and interpret the outcomes are pressingly important. Currently, data can be collected at the microbiome level leading to the possibility of personalized medicine whereby treatments can be tailored at this scale. In this paper, we lay down a statistical framework for this type of analysis with a view toward synthesis of products tailored to individual patients. Although the paper applies the technique to data for a particular infectious disease, the methodology is sufficiently rich to be expanded to other problems in medicine, especially those in which coincident `-omics' covariates and clinical responses are simultaneously captured.

Polynomial networks and factorization machines are two recently-proposed models that can efficiently use feature interactions in classification and regression tasks. In this paper, we revisit both models from a unified perspective. Based on this new view, we study the properties of both models and propose new efficient training algorithms. Key to our approach is to cast parameter learning as a low-rank symmetric tensor estimation problem, which we solve by multi-convex optimization. We demonstrate our approach on regression and recommender system tasks.

Support Vector Data Description (SVDD) is a machine learning technique used for single class classification and outlier detection. SVDD based K-chart was first introduced by Sun and Tsung for monitoring multivariate processes when underlying distribution of process parameters or quality characteristics depart from Normality. The method first trains a SVDD model on data obtained from stable or in-control operations of the process to obtain a threshold $R^2$ and kernel center a. For each new observation, its Kernel distance from the Kernel center a is calculated. The kernel distance is compared against the threshold $R^2$ to determine if the observation is within the control limits. The non-parametric K-chart provides an attractive alternative to the traditional control charts such as the Hotelling's $T^2$ charts when distribution of the underlying multivariate data is either non-normal or is unknown. But there are challenges when K-chart is deployed in practice. The K-chart requires calculating kernel distance of each new observation but there are no guidelines on how to interpret the kernel distance plot and infer about shifts in process mean or changes in process variation. This limits the application of K-charts in big-data applications such as equipment health monitoring, where observations are generated at a very high frequency. In this scenario, the analyst using the K-chart is inundated with kernel distance results at a very high frequency, generally without any recourse for detecting presence of any assignable causes of variation. We propose a new SVDD based control chart, called as $K_T$ chart, which addresses challenges encountered when using K-chart for big-data applications. The $K_T$ charts can be used to simultaneously track process variation and central tendency. We illustrate the successful use of $K_T$ chart using the Tennessee Eastman process data.

We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

This work addresses the issue of large covariance matrix estimation in high-dimensional statistical analysis. Recently, improved iterative algorithms with positive-definite guarantee have been developed. However, these algorithms cannot be directly extended to use a nonconvex penalty for sparsity inducing. Generally, a nonconvex penalty has the capability of ameliorating the bias problem of the popular convex lasso penalty, and thus is more advantageous. In this work, we propose a class of positive-definite covariance estimators using generalized nonconvex penalties. We develop a first-order algorithm based on the alternating direction method framework to solve the nonconvex optimization problem efficiently. The convergence of this algorithm has been proved. Further, the statistical properties of the new estimators have been analyzed for generalized nonconvex penalties. Moreover, extension of this algorithm to covariance estimation from sketched measurements has been considered. The performances of the new estimators have been demonstrated by both a simulation study and a gene clustering example for tumor tissues. Code for the proposed estimators is available at https://github.com/FWen/Nonconvex-PDLCE.git.

Non-convex optimization is ubiquitous in machine learning. The Majorization-Minimization (MM) procedure systematically optimizes non-convex functions through an iterative construction and optimization of upper bounds on the objective function. The bound at each iteration is required to \emph{touch} the objective function at the optimizer of the previous bound. We show that this touching constraint is unnecessary and overly restrictive. We generalize MM by relaxing this constraint, and propose a new framework for designing optimization algorithms, named Generalized Majorization-Minimization (G-MM). Compared to MM, G-MM is much more flexible. For instance, it can incorporate application-specific biases into the optimization procedure without changing the objective function. We derive G-MM algorithms for several latent variable models and show that they consistently outperform their MM counterparts in optimizing non-convex objectives. In particular, G-MM algorithms appear to be less sensitive to initialization.

With the development of high-throughput technologies, principal component analysis (PCA) in the high-dimensional regime is of great interest. Most of the existing theoretical and methodological results for high-dimensional PCA are based on the spiked population model in which all the population eigenvalues are equal except for a few large ones. Due to the presence of local correlation among features, however, this assumption may not be satisfied in many real-world datasets. To address this issue, we investigated the asymptotic behaviors of PCA under the generalized spiked population model. Based on the theoretical results, we proposed a series of methods for the consistent estimation of population eigenvalues, angles between the sample and population eigenvectors, correlation coefficients between the sample and population principal component (PC) scores, and the shrinkage bias adjustment for the predicted PC scores. Using numerical experiments and real data examples from the genetics literature, we showed that our methods can greatly reduce bias and improve prediction accuracy.