In high dimensional settings, sparse structures are crucial for efficiency, either in term of memory, computation or performance. In some contexts, it is natural to handle more refined structures than pure sparsity, such as for instance group sparsity. Sparse-Group Lasso has recently been introduced in the context of linear regression to enforce sparsity both at the feature level and at the group level. We adapt to the case of Sparse-Group Lasso recent safe screening rules that discard early in the solver irrelevant features/groups. Such rules have led to important speed-ups for a wide range of iterative methods. Thanks to dual gap computations, we provide new safe screening rules for Sparse-Group Lasso and show significant gains in term of computing time for a coordinate descent implementation.

We compute approximate solutions to L0 regularized linear regression using L1 regularization, also known as the Lasso, as an initialization step. Our algorithm, the Lass-0 ("Lass-zero"), uses a computationally efficient stepwise search to determine a locally optimal L0 solution given any L1 regularization solution. We present theoretical results of consistency under orthogonality and appropriate handling of redundant features. Empirically, we use synthetic data to demonstrate that Lass-0 solutions are closer to the true sparse support than L1 regularization models. Additionally, in real-world data Lass-0 finds more parsimonious solutions than L1 regularization while maintaining similar predictive accuracy.

Privacy concern has been increasingly important in many machine learning (ML) problems. We study empirical risk minimization (ERM) problems under secure multi-party computation (MPC) frameworks. Main technical tools for MPC have been developed based on cryptography. One of limitations in current cryptographically private ML is that it is computationally intractable to evaluate non-linear functions such as logarithmic functions or exponential functions. Therefore, for a class of ERM problems such as logistic regression in which non-linear function evaluations are required, one can only obtain approximate solutions. In this paper, we introduce a novel cryptographically private tool called secure approximation guarantee (SAG) method. The key property of SAG method is that, given an arbitrary approximate solution, it can provide a non-probabilistic assumption-free bound on the approximation quality under cryptographically secure computation framework. We demonstrate the benefit of the SAG method by applying it to several problems including a practical privacy-preserving data analysis task on genomic and clinical information.

Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are nonparametric probabilistic models and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. This paper develops a new approximate Bayesian learning scheme that enables DGPs to be applied to a range of medium to large scale regression problems for the first time. The new method uses an approximate Expectation Propagation procedure and a novel and efficient extension of the probabilistic backpropagation algorithm for learning. We evaluate the new method for non-linear regression on eleven real-world datasets, showing that it always outperforms GP regression and is almost always better than state-of-the-art deterministic and sampling-based approximate inference methods for Bayesian neural networks. As a by-product, this work provides a comprehensive analysis of six approximate Bayesian methods for training neural networks.

Due to the economic and social impacts of tourism, both private and public sectors are interested in precisely forecasting the tourism demand volume in a timely manner. With recent advances in social networks, more people use online resources to plan their future trips. In this paper we explore the application of Wikipedia usage trends (WUTs) in tourism analysis. We propose a framework that deploys WUTs for forecasting the tourism demand of Hawaii. We also propose a data-driven approach, using WUTs, to estimate the behavior of tourists when they plan their trips.

In this paper, we consider a statistical problem of learning a linear model from noisy samples. Existing work has focused on approximating the least squares solution by using leverage-based scores as an importance sampling distribution. However, no finite sample statistical guarantees and no computationally efficient optimal sampling strategies have been proposed. To evaluate the statistical properties of different sampling strategies, we propose a simple yet effective estimator, which is easy for theoretical analysis and is useful in multitask linear regression. We derive the exact mean square error of the proposed estimator for any given sampling scores. Based on minimizing the mean square error, we propose the optimal sampling scores for both estimator and predictor, and show that they are influenced by the noise-to-signal ratio. Numerical simulations match the theoretical analysis well.

Good predictors of ICU Mortality have the potential to identify high-risk patients earlier, improve ICU resource allocation, or create more accurate population-level risk models. Machine learning practitioners typically make choices about how to represent features in a particular model, but these choices are seldom evaluated quantitatively. This study compares the performance of different representations of clinical event data from MIMIC II in a logistic regression model to predict 36-hour ICU mortality. The most common representations are linear (normalized counts) and binary (yes/no). These, along with a new representation termed "hill", are compared using both L1 and L2 regularization. Results indicate that the introduced "hill" representation outperforms both the binary and linear representations, the hill representation thus has the potential to improve existing models of ICU mortality.

This article proposes Multinomial Probit Bayesian Additive Regression Trees (MPBART) as a multinomial probit extension of BART - Bayesian Additive Regression Trees (Chipman et al (2010)). MPBART is flexible to allow inclusion of predictors that describe the observed units as well as the available choice alternatives. Through two simulation studies and four real data examples, we show that MPBART exhibits very good predictive performance in comparison to other discrete choice and multiclass classification methods. To implement MPBART, we have developed an R package mpbart available freely from CRAN repositories.

In high-dimensional estimation, analysts are faced with more parameters $p$ than available observations $n$, and asymptotic analysis of performance allows the ratio $p/n\rightarrow \infty$. This situation makes regularization both necessary and desirable in order for estimators to possess theoretical guarantees. However, the amount of regularization, often determined by one or more tuning parameters, is integral to achieving good performance. In practice, choosing the tuning parameter is done through resampling methods (e.g. cross-validation), generalized information criteria, or reformulating the optimization problem (e.g. square-root lasso or scaled sparse regression). Each of these techniques comes with varying levels of theoretical guarantee for the low- or high-dimensional regimes. However, there are some notable deficiencies in the literature. The theory, and sometimes practice, of many methods relies on either the knowledge or estimation of the variance parameter, which is difficult to estimate in high dimensions. In this paper, we provide theoretical intuition suggesting that some previously proposed approaches based on information criteria work poorly in high dimensions. We introduce a suite of new risk estimators leveraging the burgeoning literature on high-dimensional variance estimation. Finally, we compare our proposal to many existing methods for choosing the tuning parameters for lasso regression by providing an extensive simulation to examine their finite sample performance. We find that our new estimators perform quite well, often better than the existing approaches across a wide range of simulation conditions and evaluation criteria.

In high-dimensional data, structured noise caused by observed and unobserved factors affecting multiple target variables simultaneously, imposes a serious challenge for modeling, by masking the often weak signal. Therefore, (1) explaining away the structured noise in multiple-output regression is of paramount importance. Additionally, (2) assumptions about the correlation structure of the regression weights are needed. We note that both can be formulated in a natural way in a latent variable model, in which both the interesting signal and the noise are mediated through the same latent factors. Under this assumption, the signal model then borrows strength from the noise model by encouraging similar effects on correlated targets. We introduce a hyperparameter for the \emph{latent signal-to-noise ratio} which turns out to be important for modelling weak signals, and an ordered infinite-dimensional shrinkage prior that resolves the rotational unidentifiability in reduced-rank regression models. Simulations and prediction experiments with metabolite, gene expression, FMRI measurement, and macroeconomic time series data show that our model equals or exceeds the state-of-the-art performance and, in particular, outperforms the standard approach of assuming independent noise and signal models.