I found much to like about this paper. It was very clear and well-written, and the proposed method is elegant, intuitive, novel (to the best of my knowledge), seemingly well motivated, and widely applicable. In addition, the main ideas in the paper are well-supported by appropriate numerical and real data examples. I would like to see the authors compare their method to two other methods that I would see as major competitors: first, the relaxed LASSO (Meinshausen 2007) wherein we fit the lasso and then do an unpenalized fit using only the variables that the lasso has selected. As I understand the field, this is the most popular method and the one that Hastie, Tibshirani, and Wainwright (2015) recommend for users who want to avoid the bias of the lasso (but note that the shrinkage "bias" is sometimes desirable to counteract the selection bias or "winner's curse" that the selected coefficients may suffer from). Second, best-subsets itself is now feasible using modern mixed-integer optimization methods (Bertsimas, Kind, and Mazumder, 2016) (if one of these competitors outperforms your method, it would not make me reconsider my recommendation to accept, since the method you propose applies to a much more general class of problems).

We consider the problem of learning a sparse graph underlying an undirected Gaussian graphical model, a key problem in statistical machine learning. Given $n$ samples from a multivariate Gaussian distribution with $p$ variables, the goal is to estimate the $p \times p$ inverse covariance matrix (aka precision matrix), assuming it is sparse (i.e., has a few nonzero entries). We propose GraphL0BnB, a new estimator based on an $\ell_0$-penalized version of the pseudolikelihood function, while most earlier approaches are based on the $\ell_1$-relaxation. Our estimator can be formulated as a convex mixed integer program (MIP) which can be difficult to compute at scale using off-the-shelf commercial solvers. To solve the MIP, we propose a custom nonlinear branch-and-bound (BnB) framework that solves node relaxations with tailored first-order methods. As a by-product of our BnB framework, we propose large-scale solvers for obtaining good primal solutions that are of independent interest. We derive novel statistical guarantees (estimation and variable selection) for our estimator and discuss how our approach improves upon existing estimators. Our numerical experiments on real/synthetic datasets suggest that our method can solve, to near-optimality, problem instances with $p = 10^4$ -- corresponding to a symmetric matrix of size $p \times p$ with $p^2/2$ binary variables. We demonstrate the usefulness of GraphL0BnB versus various state-of-the-art approaches on a range of datasets.

We extend best-subset selection to linear Multi-Task Learning (MTL), where a set of linear models are jointly trained on a collection of datasets (``tasks''). Allowing the regression coefficients of tasks to have different sparsity patterns (i.e., different supports), we propose a modeling framework for MTL that encourages models to share information across tasks, for a given covariate, through separately 1) shrinking the coefficient supports together, and/or 2) shrinking the coefficient values together. This allows models to borrow strength during variable selection even when the coefficient values differ markedly between tasks. We express our modeling framework as a Mixed-Integer Program, and propose efficient and scalable algorithms based on block coordinate descent and combinatorial local search. We show our estimator achieves statistically optimal prediction rates. Importantly, our theory characterizes how our estimator leverages the shared support information across tasks to achieve better variable selection performance. We evaluate the performance of our method in simulations and two biology applications. Our proposed approaches outperform other sparse MTL methods in variable selection and prediction accuracy. Interestingly, penalties that shrink the supports together often outperform penalties that shrink the coefficient values together. We will release an R package implementing our methods.

We introduce L0Learn: an open-source package for sparse regression and classification using L0 regularization. L0Learn implements scalable, approximate algorithms, based on coordinate descent and local combinatorial optimization. The package is built using C++ and has a user-friendly R interface. Our experiments indicate that L0Learn can scale to problems with millions of features, achieving competitive run times with state-of-the-art sparse learning packages. L0Learn is available on both CRAN and GitHub.

Unlike many valuable resources, real-time data is both abundant and growing rapidly. But it also needs to be handled with great care. That was one of the key takeaways from an online workshop produced by Science Business' Data Rules group, which explored what the rapid growth in real-time data means for artificial intelligence (AI). Real-time data is increasingly feeding machine learning systems that then adjust the algorithms they use to make decisions, such as which news item to display on your screen or which product to recommend. "With AI, especially, you want to make sure that the data that you have is consistent, replicable and also valid," noted Chris Atherton, senior research engagement officer at GÉANT, who described how his organisation transmits data captured by the European Space Agency's satellites to researchers across the world.

Voice-enabled technologies like Siri have gone from a novelty to a routine way to interact with technology in the past decade. In the coming years, our devices will only get chattier as the market for voice-enabled apps, technologies and services continues to expand. But the growth of voice-enabled technology is not universal. For much of the world, technology remains frustratingly silent. "Speech is a natural way for people to interact with devices, but we haven't realized the full potential of that yet because so much of the world is shut out from these technologies," said Mark Mazumder, a Ph.D. student at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) and the Graduate School of Arts and Sciences.

In the existing methods for solving matrix completion, such as singular value thresholding (SVT), soft-impute and fixed point continuation (FPCA) algorithms, it is typically required to repeatedly implement singular value decompositions (SVD) of matrices.When the size of the matrix in question is large, the computational complexity of finding a solution is costly. To reduce this expensive computational complexity, we apply Kronecker products to handle the matrix completion problem. In particular, we propose using Kronecker factorization, which approximates a matrix by the Kronecker product of several matrices of smaller sizes. Weintroduce Kronecker factorization into the soft-impute framework and devise an effective matrix completion algorithm.Especially when the factorized matrices have about the samesizes, the computational complexity of our algorithm is improved substantially.