Modeling item--item similarities for personalized recommendations on Yahoo! front page

We consider the problem of algorithmically recommending items to users on a Yahoo! front page module. Our approach is based on a novel multilevel hierarchical model that we refer to as a User Profile Model with Graphical Lasso (UPG). The UPG provides a personalized recommendation to users by simultaneously incorporating both user covariates and historical user interactions with items in a model based way. In fact, we build a per-item regression model based on a rich set of user covariates and estimate individual user affinity to items by introducing a latent random vector for each user. The vector random effects are assumed to be drawn from a prior with a precision matrix that measures residual partial associations among items. To ensure better estimates of a precision matrix in high-dimensions, the matrix elements are constrained through a Lasso penalty. Our model is fitted through a penalized-quasi likelihood procedure coupled with a scalable EM algorithm. We employ several computational strategies like multi-threading, conjugate gradients and heavily exploit problem structure to scale our computations in the E-step. For the M-step we take recourse to a scalable variant of the Graphical Lasso algorithm for covariance selection. Through extensive experiments on a new data set obtained from Yahoo! front page and a benchmark data set from a movie recommender application, we show that our UPG model significantly improves performance compared to several state-of-the-art methods in the literature, especially those based on a bilinear random effects model (BIRE). In particular, we show that the gains of UPG are significant compared to BIRE when the number of users is large and the number of items to select from is small. For large item sets and relatively small user sets the results of UPG and BIRE are comparable. The UPG leads to faster model building and produces outputs which are interpretable.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS475 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Factorial graphical lasso for dynamic networks

Dynamic networks models describe a growing number of important scientific processes, from cell biology and epidemiology to sociology and finance. There are many aspects of dynamical networks that require statistical considerations. In this paper we focus on determining network structure. Estimating dynamic networks is a difficult task since the number of components involved in the system is very large. As a result, the number of parameters to be estimated is bigger than the number of observations. However, a characteristic of many networks is that they are sparse. For example, the molecular structure of genes make interactions with other components a highly-structured and therefore sparse process. Penalized Gaussian graphical models have been used to estimate sparse networks. However, the literature has focussed on static networks, which lack specific temporal constraints. We propose a structured Gaussian dynamical graphical model, where structures can consist of specific time dynamics, known presence or absence of links and block equality constraints on the parameters. Thus, the number of parameters to be estimated is reduced and accuracy of the estimates, including the identification of the network, can be tuned up. Here, we show that the constrained optimization problem can be solved by taking advantage of an efficient solver, logdetPPA, developed in convex optimization. Moreover, model selection methods for checking the sensitivity of the inferred networks are described. Finally, synthetic and real data illustrate the proposed methodologies.Comment: 30 pp, 5 figure

Multinomial Inverse Regression for Text Analysis

Text data, including speeches, stories, and other document forms, are often connected to sentiment variables that are of interest for research in marketing, economics, and elsewhere. It is also very high dimensional and difficult to incorporate into statistical analyses. This article introduces a straightforward framework of sentiment-preserving dimension reduction for text data. Multinomial inverse regression is introduced as a general tool for simplifying predictor sets that can be represented as draws from a multinomial distribution, and we show that logistic regression of phrase counts onto document annotations can be used to obtain low dimension document representations that are rich in sentiment information. To facilitate this modeling, a novel estimation technique is developed for multinomial logistic regression with very high-dimension response. In particular, independent Laplace priors with unknown variance are assigned to each regression coefficient, and we detail an efficient routine for maximization of the joint posterior over coefficients and their prior scale. This "gamma-lasso" scheme yields stable and effective estimation for general high-dimension logistic regression, and we argue that it will be superior to current methods in many settings. Guidelines for prior specification are provided, algorithm convergence is detailed, and estimator properties are outlined from the perspective of the literature on non-concave likelihood penalization. Related work on sentiment analysis from statistics, econometrics, and machine learning is surveyed and connected. Finally, the methods are applied in two detailed examples and we provide out-of-sample prediction studies to illustrate their effectiveness.Comment: Published in the Journal of the American Statistical Association 108, 2013, with discussion (rejoinder is here: http://arxiv.org/abs/1304.4200). Software is available in the textir package for

Sparsity with sign-coherent groups of variables via the cooperative-Lasso

We consider the problems of estimation and selection of parameters endowed with a known group structure, when the groups are assumed to be sign-coherent, that is, gathering either nonnegative, nonpositive or null parameters. To tackle this problem, we propose the cooperative-Lasso penalty. We derive the optimality conditions defining the cooperative-Lasso estimate for generalized linear models, and propose an efficient active set algorithm suited to high-dimensional problems. We study the asymptotic consistency of the estimator in the linear regression setup and derive its irrepresentable conditions, which are milder than the ones of the group-Lasso regarding the matching of groups with the sparsity pattern of the true parameters. We also address the problem of model selection in linear regression by deriving an approximation of the degrees of freedom of the cooperative-Lasso estimator. Simulations comparing the proposed estimator to the group and sparse group-Lasso comply with our theoretical results, showing consistent improvements in support recovery for sign-coherent groups. We finally propose two examples illustrating the wide applicability of the cooperative-Lasso: first to the processing of ordinal variables, where the penalty acts as a monotonicity prior; second to the processing of genomic data, where the set of differentially expressed probes is enriched by incorporating all the probes of the microarray that are related to the corresponding genes.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS520 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sparse functional regression models: minimax rates and contamination

In functional linear regression and functional generalized linear regression models, the effect of the predictor function is usually assumed to be spread across the index space. In this dissertation we consider the sparse functional linear model and the sparse functional generalized linear models (GLM), where the impact of the predictor process on the response is only via its value at one point in the index space, defined as the sensitive point. We are particularly interested in estimating the sensitive point. The minimax rate of convergence for estimating the parameters in sparse functional linear regression is derived. It is shown that the optimal rate for estimating the sensitive point depends on the roughness of the predictor function, which is quantified by a "generalized Hurst exponent". The least squares estimator (LSE) is shown to attain the optimal rate. Also, a lower bound is given on the minimax risk of estimating the parameters in sparse functional GLM, which also depends on the generalized Hurst exponent of the predictor process. The order of the minimax lower bound is the same as that of the weak convergence rate of the maximum likelihood estimator (MLE), given that the functional predictor behaves like a Brownian motion