In regression problems involving vector-valued outputs (or equivalently, multiple responses), it is well known that the maximum likelihood estimator (MLE), which takes noise covariance structure into account, can be significantly more accurate than the ordinary least squares (OLS) estimator. However, existing literature compares OLS and MLE in terms of their asymptotic, not finite sample, guarantees. More crucially, computing the MLE in general requires solving a non-convex optimization problem and is not known to be efficiently solvable. We provide finite sample upper and lower bounds on the estimation error of OLS and MLE, in two popular models: a) Pooled model, b) Seemingly Unrelated Regression (SUR) model. We provide precise instances where the MLE is significantly more accurate than OLS. Furthermore, for both models, we show that the output of a computationally efficient alternating minimization procedure enjoys the same performance guarantee as MLE, up to universal constants. Finally, we show that for high-dimensional settings as well, the alternating minimization procedure leads to significantly more accurate solutions than the corresponding OLS solutions but with error bound that depends only logarithmically on the data dimensionality.

Linear regression studies the problem of estimating a model parameter $\beta^* \in \R^p$, from $n$ observations $\{(y_i,x_i)\}_{i=1}^n$ from linear model $y_i = \langle \x_i,\beta^* \rangle + \epsilon_i$. We consider a significant generalization in which the relationship between $\langle x_i,\beta^* \rangle$ and $y_i$ is noisy, quantized to a single bit, potentially nonlinear, noninvertible, as well as unknown. This model is known as the single-index model in statistics, and, among other things, it represents a significant generalization of one-bit compressed sensing. We propose a novel spectral-based estimation procedure and show that we can recover $\beta^*$ in settings (i.e., classes of link function $f$) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on the (unknown) functional relationship between $y_i$ and $\langle x_i,\beta^* \rangle$. We also consider the high dimensional setting where $\beta^*$ is sparse, and introduce a two-stage nonconvex framework that addresses estimation challenges in high dimensional regimes where $p \gg n$. For a broad class of link functions between $\langle x_i,\beta^* \rangle$ and $y_i$, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.

We extend the theory of boosting for regression problems to the online learning setting. Generalizing from the batch setting for boosting, the notion of a weak learning algorithm is modeled as an online learning algorithm with linear loss functions that competes with a base class of regression functions, while a strong learning algorithm is an online learning algorithm with smooth convex loss functions that competes with a larger class of regression functions. Our main result is an online gradient boosting algorithm which converts a weak online learning algorithm into a strong one where the larger class of functions is the linear span of the base class. We also give a simpler boosting algorithm that converts a weak online learning algorithm into a strong one where the larger class of functions is the convex hull of the base class, and prove its optimality.

We propose a class of closed-form estimators for GLMs under high-dimensional sampling regimes. Our class of estimators is based on deriving closed-form variants of the vanilla unregularized MLE but which are (a) well-defined even under high-dimensional settings, and (b) available in closed-form. We then perform thresholding operations on this MLE variant to obtain our class of estimators. We derive a unified statistical analysis of our class of estimators, and show that it enjoys strong statistical guarantees in both parameter error as well as variable selection, that surprisingly match those of the more complex regularized GLM MLEs, even while our closed-form estimators are computationally much simpler. We derive instantiations of our class of closed-form estimators, as well as corollaries of our general theorem, for the special cases of logistic, exponential and Poisson regression models. We corroborate the surprising statistical and computational performance of our class of estimators via extensive simulations.

Imaging neuroscience links human behavior to aspects of brain biology in ever-increasing datasets. Existing neuroimaging methods typically perform either discovery of unknown neural structure or testing of neural structure associated with mental tasks. However, testing hypotheses on the neural correlates underlying larger sets of mental tasks necessitates adequate representations for the observations. We therefore propose to blend representation modelling and task classification into a unified statistical learning problem. A multinomial logistic regression is introduced that is constrained by factored coefficients and coupled with an autoencoder. We show that this approach yields more accurate and interpretable neural models of psychological tasks in a reference dataset, as well as better generalization to other datasets.

High dimensional regression benefits from sparsity promoting regularizations. Screening rules leverage the known sparsity of the solution by ignoring some variables in the optimization, hence speeding up solvers. When the procedure is proven not to discard features wrongly the rules are said to be safe. In this paper we derive new safe rules for generalized linear models regularized with L1 and L1/L2 norms. The rules are based on duality gap computations and spherical safe regions whose diameters converge to zero. This allows to discard safely more variables, in particular for low regularization parameters. The GAP Safe rule can cope with any iterative solver and we illustrate its performance on coordinate descent for multi-task Lasso, binary and multinomial logistic regression, demonstrating significant speed ups on all tested datasets with respect to previous safe rules.

Learning to predict multi-label outputs is challenging, but in many problems there is a natural metric on the outputs that can be used to improve predictions. In this paper we develop a loss function for multi-label learning, based on the Wasserstein distance. The Wasserstein distance provides a natural notion of dissimilarity for probability measures. Although optimizing with respect to the exact Wasserstein distance is costly, recent work has described a regularized approximation that is efficiently computed. We describe an efficient learning algorithm based on this regularization, as well as a novel extension of the Wasserstein distance from probability measures to unnormalized measures. We also describe a statistical learning bound for the loss. The Wasserstein loss can encourage smoothness of the predictions with respect to a chosen metric on the output space. We demonstrate this property on a real-data tag prediction problem, using the Yahoo Flickr Creative Commons dataset, outperforming a baseline that doesn't use the metric.

We develop an approach for feature elimination in statistical learning with kernel machines, based on recursive elimination of features.We present theoretical properties of this method and show that it is uniformly consistent in finding the correct feature space under certain generalized assumptions.We present four case studies to show that the assumptions are met in most practical situations and present simulation results to demonstrate performance of the proposed approach.

We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. Called a Sparse Superposition or Sparse Regression codebook, this structure is motivated by an analogous construction proposed recently by Barron and Joseph for communication over an AWGN channel. For i.i.d Gaussian sources and minimum-distance encoding, we show that such a code can attain the Shannon rate-distortion function with the optimal error exponent, for all distortions below a specified value. It is also shown that sparse regression codes are robust in the following sense: a codebook designed to compress an i.i.d Gaussian source of variance $\sigma^2$ with (squared-error) distortion $D$ can compress any ergodic source of variance less than $\sigma^2$ to within distortion $D$. Thus the sparse regression ensemble retains many of the good covering properties of the i.i.d random Gaussian ensemble, while having having a compact representation in terms of a matrix whose size is a low-order polynomial in the block-length.

Accurate real-time monitoring systems of influenza outbreaks help public health officials make informed decisions that may help save lives. We show that information extracted from cloud-based electronic health records databases, in combination with machine learning techniques and historical epidemiological information, have the potential to accurately and reliably provide near real-time regional predictions of flu outbreaks in the United States.