Linear regression models depend directly on the design matrix and its properties. Techniques that efficiently estimate model coefficients by partitioning rows of the design matrix are increasingly popular for large-scale problems because they fit well with modern parallel computing architectures. We propose a simple measure of {\em concordance} between a design matrix and a subset of its rows that estimates how well a subset captures the variance-covariance structure of a larger data set. We illustrate the use of this measure in a heuristic method for selecting row partition sizes that balance statistical and computational efficiency goals in real-world problems.

In the last twenty-five years (1990-2014), algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 200 billion factor speedup in solving Mixed Integer Optimization (MIO) problems. We present a MIO approach for solving the classical best subset selection problem of choosing $k$ out of $p$ features in linear regression given $n$ observations. We develop a discrete extension of modern first order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with $n$ in the 1000s and $p$ in the 100s in minutes to provable optimality, and finds near optimal solutions for $n$ in the 100s and $p$ in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than {\texttt {Lasso}} and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.

We consider the problem of learning the inhomogeneous intensity of a counting process, under a sparse segmentation assumption. We introduce a weighted total-variation penalization, using data-driven weights that correctly scale the penalization along the observation interval. We prove that this leads to a sharp tuning of the convex relaxation of the segmentation prior, by stating oracle inequalities with fast rates of convergence, and consistency for change-points detection. This provides first theoretical guarantees for segmentation with a convex proxy beyond the standard i.i.d signal + white noise setting. We introduce a fast algorithm to solve this convex problem. Numerical experiments illustrate our approach on simulated and on a high-frequency genomics dataset.

This thesis studies two problems in modern statistics. First, we study selective inference, or inference for hypothesis that are chosen after looking at the data. The motiving application is inference for regression coefficients selected by the lasso. We present the Condition-on-Selection method that allows for valid selective inference, and study its application to the lasso, and several other selection algorithms. In the second part, we consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to structure learning. In previous work, authors have considered structure learning of Gaussian graphical models and structure learning of discrete models. Our approach is a natural generalization of these two lines of work to the mixed case. The penalization scheme involves a novel symmetric use of the group-lasso norm and follows naturally from a particular parametrization of the model. We provide conditions under which our estimator is model selection consistent in the high-dimensional regime.

In this paper we implement a Local Linear Regression Ensemble Committee (LOLREC) to predict 1-day-ahead returns of 453 assets form the S&P500. The estimates and the historical returns of the committees are used to compute the weights of the portfolio from the 453 stock. The proposed method outperforms benchmark portfolio selection strategies that optimize the growth rate of the capital. We investigate the effect of algorithm parameter m: the number of selected stocks on achieved average annual yields. Results suggest the algorithm's practical usefulness in everyday trading.

Mixture of Experts (MoE) is a popular framework for modeling heterogeneity in data for regression, classification and clustering. For continuous data which we consider here in the context of regression and cluster analysis, MoE usually use normal experts, that is, expert components following the Gaussian distribution. However, for a set of data containing a group or groups of observations with asymmetric behavior, heavy tails or atypical observations, the use of normal experts may be unsuitable and can unduly affect the fit of the MoE model. In this paper, we introduce new non-normal mixture of experts (NNMoE) which can deal with these issues regarding possibly skewed, heavy-tailed data and with outliers. The proposed models are the skew-normal MoE and the robust $t$ MoE and skew $t$ MoE, respectively named SNMoE, TMoE and STMoE. We develop dedicated expectation-maximization (EM) and expectation conditional maximization (ECM) algorithms to estimate the parameters of the proposed models by monotonically maximizing the observed data log-likelihood. We describe how the presented models can be used in prediction and in model-based clustering of regression data. Numerical experiments carried out on simulated data show the effectiveness and the robustness of the proposed models in terms modeling non-linear regression functions as well as in model-based clustering. Then, to show their usefulness for practical applications, the proposed models are applied to the real-world data of tone perception for musical data analysis, and the one of temperature anomalies for the analysis of climate change data.

Finding statistically significant high-order interaction features in predictive modeling is important but challenging task. The difficulty lies in the fact that, for a recent applications with high-dimensional covariates, the number of possible high-order interaction features would be extremely large. Identifying statistically significant features from such a huge pool of candidates would be highly challenging both in computational and statistical senses. To work with this problem, we consider a two stage algorithm where we first select a set of high-order interaction features by marginal screening, and then make statistical inferences on the regression model fitted only with the selected features. Such statistical inferences are called post-selection inference (PSI), and receiving an increasing attention in the literature. One of the seminal recent advancements in PSI literature is the works by Lee et al. where the authors presented an algorithmic framework for computing exact sampling distributions in PSI. A main challenge when applying their approach to our high-order interaction models is to cope with the fact that PSI in general depends not only on the selected features but also on the unselected features, making it hard to apply to our extremely high-dimensional high-order interaction models. The goal of this paper is to overcome this difficulty by introducing a novel efficient method for PSI. Our key idea is to exploit the underlying tree structure among high-order interaction features, and to develop a pruning method of the tree which enables us to quickly identify a group of unselected features that are guaranteed to have no influence on PSI. The experimental results indicate that the proposed method allows us to reliably identify statistically significant high-order interaction features with reasonable computational cost.

Statistical models with constrained probability distributions are abundant in machine learning. Some examples include regression models with norm constraints (e.g., Lasso), probit, many copula models, and latent Dirichlet allocation (LDA). Bayesian inference involving probability distributions confined to constrained domains could be quite challenging for commonly used sampling algorithms. In this paper, we propose a novel augmentation technique that handles a wide range of constraints by mapping the constrained domain to a sphere in the augmented space. By moving freely on the surface of this sphere, sampling algorithms handle constraints implicitly and generate proposals that remain within boundaries when mapped back to the original space. Our proposed method, called {Spherical Augmentation}, provides a mathematically natural and computationally efficient framework for sampling from constrained probability distributions. We show the advantages of our method over state-of-the-art sampling algorithms, such as exact Hamiltonian Monte Carlo, using several examples including truncated Gaussian distributions, Bayesian Lasso, Bayesian bridge regression, reconstruction of quantized stationary Gaussian process, and LDA for topic modeling.

We introduce GAMSEL (Generalized Additive Model Selection), a penalized likelihood approach for fitting sparse generalized additive models in high dimension. Our method interpolates between null, linear and additive models by allowing the effect of each variable to be estimated as being either zero, linear, or a low-complexity curve, as determined by the data. We present a blockwise coordinate descent procedure for efficiently optimizing the penalized likelihood objective over a dense grid of the tuning parameter, producing a regularization path of additive models. We demonstrate the performance of our method on both real and simulated data examples, and compare it with existing techniques for additive model selection.

Forward stagewise regression follows a very simple strategy for constructing a sequence of sparse regression estimates: it starts with all coefficients equal to zero, and iteratively updates the coefficient (by a small amount $\epsilon$) of the variable that achieves the maximal absolute inner product with the current residual. This procedure has an interesting connection to the lasso: under some conditions, it is known that the sequence of forward stagewise estimates exactly coincides with the lasso path, as the step size $\epsilon$ goes to zero. Furthermore, essentially the same equivalence holds outside of least squares regression, with the minimization of a differentiable convex loss function subject to an $\ell_1$ norm constraint (the stagewise algorithm now updates the coefficient corresponding to the maximal absolute component of the gradient). Even when they do not match their $\ell_1$-constrained analogues, stagewise estimates provide a useful approximation, and are computationally appealing. Their success in sparse modeling motivates the question: can a simple, effective strategy like forward stagewise be applied more broadly in other regularization settings, beyond the $\ell_1$ norm and sparsity? The current paper is an attempt to do just this. We present a general framework for stagewise estimation, which yields fast algorithms for problems such as group-structured learning, matrix completion, image denoising, and more.