We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. While there exist several appealing approaches based on convex relaxations and nonconvex heuristics, we focus on optimal solutions for the $\ell_0$-regularized formulation, a problem that is relatively unexplored due to computational challenges. Our methodology covers both high-dimensional linear regression and nonparametric sparse additive modeling with smooth components. Our algorithmic framework consists of approximate and exact algorithms. The approximate algorithms are based on coordinate descent and local search, with runtimes comparable to popular sparse learning algorithms. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer programming (MIP) problem to certified optimality. By exploiting the problem structure, our custom BnB algorithm can solve to optimality problem instances with $5 \times 10^6$ features in minutes to hours -- over $1000$ times larger than what is currently possible using state-of-the-art commercial MIP solvers. We also explore statistical properties of the $\ell_0$-based estimators. We demonstrate, theoretically and empirically, that our proposed estimators have an edge over popular group-sparse estimators in terms of statistical performance in various regimes.

Semi-supervised (SS) inference has received much attention in recent years. Apart from a moderate-sized labeled data, L, the SS setting is characterized by an additional, much larger sized, unlabeled data, U. The setting of |U| >> |L|, makes SS inference unique and different from the standard missing data problems, owing to natural violation of the so-called 'positivity' or 'overlap' assumption. However, most of the SS literature implicitly assumes L and U to be equally distributed, i.e., no selection bias in the labeling. Inferential challenges in missing at random (MAR) type labeling allowing for selection bias, are inevitably exacerbated by the decaying nature of the propensity score (PS). We address this gap for a prototype problem, the estimation of the response's mean. We propose a double robust SS (DRSS) mean estimator and give a complete characterization of its asymptotic properties. The proposed estimator is consistent as long as either the outcome or the PS model is correctly specified. When both models are correctly specified, we provide inference results with a non-standard consistency rate that depends on the smaller size |L|. The results are also extended to causal inference with imbalanced treatment groups. Further, we provide several novel choices of models and estimators of the decaying PS, including a novel offset logistic model and a stratified labeling model. We present their properties under both high and low dimensional settings. These may be of independent interest. Lastly, we present extensive simulations and also a real data application.

Conformal prediction constructs a confidence region for an unobserved response of a feature vector based on previous identically distributed and exchangeable observations of responses and features. It has a coverage guarantee at any nominal level without additional assumptions on their distribution. However, it requires a refitting procedure for all replacement candidates of the target response. In regression settings, this corresponds to an infinite number of model fit. Apart from relatively simple estimators that can be written as pieces of linear function of the response, efficiently computing such sets is difficult and is still considered as an open problem. We exploit the fact that, \emph{often}, conformal prediction sets are intervals whose boundaries can be efficiently approximated by classical root-finding software. We investigate how this approach can overcome many limitations of formerly used strategies and achieves calculations that have been unattainable so far. We discuss its complexity as well as its drawbacks and evaluate its efficiency through numerical experiments.

Rapid development of advanced modelling techniques gives an opportunity to develop tools that are more and more accurate. However as usually, everything comes with a price and in this case, the price to pay is to loose interpretability of a model while gaining on its accuracy and precision. For managers to control and effectively manage credit risk and for regulators to be convinced with model quality the price to pay is too high. In this paper, we show how to take credit scoring analytics in to the next level, namely we present comparison of various predictive models (logistic regression, logistic regression with weight of evidence transformations and modern artificial intelligence algorithms) and show that advanced tree based models give best results in prediction of client default. What is even more important and valuable we also show how to boost advanced models using techniques which allow to interpret them and made them more accessible for credit risk practitioners, resolving the crucial obstacle in widespread deployment of more complex, 'black box' models like random forests, gradient boosted or extreme gradient boosted trees. All this will be shown on the large dataset obtained from the Polish Credit Bureau to which all the banks and most of the lending companies in the country do report the credit files. In this paper the data from lending companies were used. The paper then compares state of the art best practices in credit risk modelling with new advanced modern statistical tools boosted by the latest developments in the field of interpretability and explainability of artificial intelligence algorithms. We believe that this is a valuable contribution when it comes to presentation of different modelling tools but what is even more important it is showing which methods might be used to get insight and understanding of AI methods in credit risk context.

This paper presents a Gaussian process (GP) model for estimating piecewise continuous regression functions. In scientific and engineering applications of regression analysis, the underlying regression functions are piecewise continuous in that data follow different continuous regression models for different regions of the data with possible discontinuities between the regions. However, many conventional GP regression approaches are not designed for piecewise regression analysis. We propose a new GP modeling approach for estimating an unknown piecewise continuous regression function. The new GP model seeks for a local GP estimate of an unknown regression function at each test location, using local data neighboring to the test location. To accommodate the possibilities of the local data from different regions, the local data is partitioned into two sides by a local linear boundary, and only the local data belonging to the same side as the test location is used for the regression estimate. This local split works very well when the input regions are bounded by smooth boundaries, so the local linear approximation of the smooth boundaries works well. We estimate the local linear boundary jointly with the other hyperparameters of the GP model, using the maximum likelihood approach. Its computation time is as low as the local GP's time. The superior numerical performance of the proposed approach over the conventional GP modeling approaches is shown using various simulated piecewise regression functions.

In this article a surprising result is demonstrated using the neural tangent kernel. This kernel is defined as the inner product of the vector of the gradient of an underlying model evaluated at training points. This kernel is used to perform kernel regression. The surprising thing is that the accuracy of that regression is independent of the accuracy of the underlying network.

We study how masking and predicting tokens in an unsupervised fashion can give rise to linguistic structures and downstream performance gains. Recent theories have suggested that pretrained language models acquire useful inductive biases through masks that implicitly act as cloze reductions for downstream tasks. While appealing, we show that the success of the random masking strategy used in practice cannot be explained by such cloze-like masks alone. We construct cloze-like masks using task-specific lexicons for three different classification datasets and show that the majority of pretrained performance gains come from generic masks that are not associated with the lexicon. To explain the empirical success of these generic masks, we demonstrate a correspondence between the Masked Language Model (MLM) objective and existing methods for learning statistical dependencies in graphical models. Using this, we derive a method for extracting these learned statistical dependencies in MLMs and show that these dependencies encode useful inductive biases in the form of syntactic structures. In an unsupervised parsing evaluation, simply forming a minimum spanning tree on the implied statistical dependence structure outperforms a classic method for unsupervised parsing (58.74 vs. 55.91 UUAS).

Multi-task learning is increasingly used to investigate the association structure between multiple responses and a single set of predictor variables in many applications. In the era of big data, the coexistence of incomplete outcomes, large number of responses, and high dimensionality in predictors poses unprecedented challenges in estimation, prediction, and computation. In this paper, we propose a scalable and computationally efficient procedure, called PEER, for large-scale multi-response regression with incomplete outcomes, where both the numbers of responses and predictors can be high-dimensional. Motivated by sparse factor regression, we convert the multi-response regression into a set of univariate-response regressions, which can be efficiently implemented in parallel. Under some mild regularity conditions, we show that PEER enjoys nice sampling properties including consistency in estimation, prediction, and variable selection. Extensive simulation studies show that our proposal compares favorably with several existing methods in estimation accuracy, variable selection, and computation efficiency.

We study in this paper lower bounds for the generalization error of models derived from multi-layer neural networks, in the regime where the size of the layers is commensurate with the number of samples in the training data. We show that unbiased estimators have unacceptable performance for such nonlinear networks in this regime. We derive explicit generalization lower bounds for general biased estimators, in the cases of linear regression and of two-layered networks. In the linear case the bound is asymptotically tight. In the nonlinear case, we provide a comparison of our bounds with an empirical study of the stochastic gradient descent algorithm. The analysis uses elements from the theory of large random matrices.

In a business context where an unconstrained real-world application were The potential risk of machine learning algorithms to unintentionally to be replaced with a fairer one, such extreme discrepancies embed and reproduce bias and therefore discriminating would not be viable because individuals who were substantially various sub populations in high-stakes decisionmaking negatively impacted would probably not accept applications has given rise to the new research the change and switch to a competitor. Based on our findings, field of fair machine learning (Kamiran and Calders 2009; we therefore propose algorithmic post-processing procedures Corbett-Davies et al. 2018; Barocas, Hardt, and Narayanan to adjust for unwanted, extreme discrepancies between 2019). Plenty of quantitative measures of fairness have been unconstrained and fair methods in order to enable a proposed (Dwork et al. 2011; Hardt, Price, and Srebro 2016; smooth transition from an "unfair" to a fairer model. Chouldechova 2017; Berk et al. 2018) which opened up The main contributions of this paper are: the way for three types of algorithms that seek to satisfy them: First, the pre-processing approach which modifies - We empirically examine the evolution of fair regression the data representation prior to using classical algorithms outputs compared to unconstrained predictors and demonstrate (Kamiran and Calders 2012; Zemel et al. 2013). Second, that some variations on the individual level may be the in-processing approach which intervenes during the unacceptable in practice. To the best of our knowledge we learning phase by adding a fairness constraint to the optimization offer the first investigation of this kind; objective (Kamishima et al. 2012; Zafar et al. - We propose a range of post-processing algorithms to mitigate 2017; Zhang, Lemoine, and Mitchell 2018). Third, the postprocessing this effect and therefore provide mechanisms to approach which adjusts the outputs of classical implement fair regression in practice.