We consider the variable selection problem for two-sample tests, aiming to select the most informative variables to determine whether two collections of samples follow the same distribution. To address this, we propose a novel framework based on the kernel maximum mean discrepancy (MMD). Our approach seeks a subset of variables with a pre-specified size that maximizes the variance-regularized kernel MMD statistic. We focus on three commonly used types of kernels: linear, quadratic, and Gaussian. From a computational perspective, we derive mixed-integer programming formulations and propose exact and approximation algorithms with performance guarantees to solve these formulations. From a statistical viewpoint, we derive the rate of testing power of our framework under appropriate conditions. These results show that the sample size requirements for the three kernels depend crucially on the number of selected variables, rather than the data dimension. Experimental results on synthetic and real datasets demonstrate the superior performance of our method, compared to other variable selection frameworks, particularly in high-dimensional settings.
Sparse model identification enables nonlinear dynamical system discovery from data. However, the control of false discoveries for sparse model identification is challenging, especially in the low-data and high-noise limit. In this paper, we perform a theoretical study on ensemble sparse model discovery, which shows empirical success in terms of accuracy and robustness to noise. In particular, we analyse the bootstrapping-based sequential thresholding least-squares estimator. We show that this bootstrapping-based ensembling technique can perform a provably correct variable selection procedure with an exponential convergence rate of the error rate. In addition, we show that the ensemble sparse model discovery method can perform computationally efficient uncertainty estimation, compared to expensive Bayesian uncertainty quantification methods via MCMC. We demonstrate the convergence properties and connection to uncertainty quantification in various numerical studies on synthetic sparse linear regression and sparse model discovery. The experiments on sparse linear regression support that the bootstrapping-based sequential thresholding least-squares method has better performance for sparse variable selection compared to LASSO, thresholding least-squares, and bootstrapping-based LASSO. In the sparse model discovery experiment, we show that the bootstrapping-based sequential thresholding least-squares method can provide valid uncertainty quantification, converging to a delta measure centered around the true value with increased sample sizes. Finally, we highlight the improved robustness to hyperparameter selection under shifting noise and sparsity levels of the bootstrapping-based sequential thresholding least-squares method compared to other sparse regression methods.
Dynamic treatment regimes or policies are a sequence of decision functions over multiple stages that are tailored to individual features. One important class of treatment policies in practice, namely multi-stage stationary treatment policies, prescribes treatment assignment probabilities using the same decision function across stages, where the decision is based on the same set of features consisting of time-evolving variables (e.g., routinely collected disease biomarkers). Although there has been extensive literature on constructing valid inference for the value function associated with dynamic treatment policies, little work has focused on the policies themselves, especially in the presence of high-dimensional feature variables. We aim to fill the gap in this work. Specifically, we first estimate the multi-stage stationary treatment policy using an augmented inverse probability weighted estimator for the value function to increase asymptotic efficiency, and further apply a penalty to select important feature variables. We then construct one-step improvements of the policy parameter estimators for valid inference. Theoretically, we show that the improved estimators are asymptotically normal, even if nuisance parameters are estimated at a slow convergence rate and the dimension of the feature variables increases with the sample size. Our numerical studies demonstrate that the proposed method estimates a sparse policy with a near-optimal value function and conducts valid inference for the policy parameters.
Federated learning has become a popular tool in the big data era nowadays. It trains a centralized model based on data from different clients while keeping data decentralized. In this paper, we propose a federated sparse sliced inverse regression algorithm for the first time. Our method can simultaneously estimate the central dimension reduction subspace and perform variable selection in a federated setting. We transform this federated high-dimensional sparse sliced inverse regression problem into a convex optimization problem by constructing the covariance matrix safely and losslessly. We then use a linearized alternating direction method of multipliers algorithm to estimate the central subspace. We also give approaches of Bayesian information criterion and hold-out validation to ascertain the dimension of the central subspace and the hyper-parameter of the algorithm. We establish an upper bound of the statistical error rate of our estimator under the heterogeneous setting. We demonstrate the effectiveness of our method through simulations and real world applications.
The brown marmorated stink bug (BMSB), Halyomorpha halys, is an invasive insect pest of global importance that damages several crops, compromising agri-food production. Field monitoring procedures are fundamental to perform risk assessment operations, in order to promptly face crop infestations and avoid economical losses. To improve pest management, spectral cameras mounted on Unmanned Aerial Vehicles (UAVs) and other Internet of Things (IoT) devices, such as smart traps or unmanned ground vehicles, could be used as an innovative technology allowing fast, efficient and real-time monitoring of insect infestations. The present study consists in a preliminary evaluation at the laboratory level of Near Infrared Hyperspectral Imaging (NIR-HSI) as a possible technology to detect BMSB specimens on different vegetal backgrounds, overcoming the problem of BMSB mimicry. Hyperspectral images of BMSB were acquired in the 980-1660 nm range, considering different vegetal backgrounds selected to mimic a real field application scene. Classification models were obtained following two different chemometric approaches. The first approach was focused on modelling spectral information and selecting relevant spectral regions for discrimination by means of sparse-based variable selection coupled with Soft Partial Least Squares Discriminant Analysis (s-Soft PLS-DA) classification algorithm. The second approach was based on modelling spatial and spectral features contained in the hyperspectral images using Convolutional Neural Networks (CNN). Finally, to further improve BMSB detection ability, the two strategies were merged, considering only the spectral regions selected by s-Soft PLS-DA for CNN modelling.
Relating a set of variables X to a response y is crucial in chemometrics. A quantitative prediction objective can be enriched by qualitative data interpretation, for instance by locating the most influential features. When high-dimensional problems arise, dimension reduction techniques can be used. Most notable are projections (e.g. Partial Least Squares or PLS ) or variable selections (e.g. lasso). Sparse partial least squares combine both strategies, by blending variable selection into PLS. The variant presented in this paper, Dual-sPLS, generalizes the classical PLS1 algorithm. It provides balance between accurate prediction and efficient interpretation. It is based on penalizations inspired by classical regression methods (lasso, group lasso, least squares, ridge) and uses the dual norm notion. The resulting sparsity is enforced by an intuitive shrinking ratio parameter. Dual-sPLS favorably compares to similar regression methods, on simulated and real chemical data. Code is provided as an open-source package in R: \url{https://CRAN.R-project.org/package=dual.spls}.
We consider the problem of best subset selection (BSS) under high-dimensional sparse linear regression model. Recently, Guo et al. (2020) showed that the model selection performance of BSS depends on a certain identifiability margin, a measure that captures the model discriminative power of BSS under a general correlation structure that is robust to the design dependence, unlike its computational surrogates such as LASSO, SCAD, MCP, etc. Expanding on this, we further broaden the theoretical understanding of best subset selection in this paper and show that the complexities of the residualized signals, the portion of the signals orthogonal to the true active features, and spurious projections, describing the projection operators associated with the irrelevant features, also play fundamental roles in characterizing the margin condition for model consistency of BSS. In particular, we establish both necessary and sufficient margin conditions depending only on the identifiability margin and the two complexity measures. We also partially extend our sufficiency result to the case of high-dimensional sparse generalized linear models (GLMs).
The problem of jointly analysing functional connectomics and behavioral data is extremely challenging owing to the complex interactions between the two domains. In addition, clinical rs-fMRI studies often have to contend with limited samples, especially in the case of rare disorders. This data-starved regimen can severely restrict the reliability of classical machine learning or deep learning designed to predict behavior from connectivity data. In this work, we approach this problem from the lens of representation learning and bayesian modeling. To model the distributional characteristics of the domains, we first examine the ability of approaches such as Bayesian Linear Regression, Stochastic Search Variable Selection after performing a classical covariance decomposition. Finally, we present a fully bayesian formulation for joint representation learning and prediction. We present preliminary results on a subset of a publicly available clinical rs-fMRI study on patients with Autism Spectrum Disorder.
Constraint programming is known for being an efficient approach for solving combinatorial problems. Important design choices in a solver are the branching heuristics, which are designed to lead the search to the best solutions in a minimum amount of time. However, developing these heuristics is a time-consuming process that requires problem-specific expertise. This observation has motivated many efforts to use machine learning to automatically learn efficient heuristics without expert intervention. To the best of our knowledge, it is still an open research question. Although several generic variable-selection heuristics are available in the literature, the options for a generic value-selection heuristic are more scarce. In this paper, we propose to tackle this issue by introducing a generic learning procedure that can be used to obtain a value-selection heuristic inside a constraint programming solver. This has been achieved thanks to the combination of a deep Q-learning algorithm, a tailored reward signal, and a heterogeneous graph neural network architecture. Experiments on graph coloring, maximum independent set, and maximum cut problems show that our framework is able to find better solutions close to optimality without requiring a large amounts of backtracks while being generic.
We study estimation and testing in the Poisson regression model with noisy high dimensional covariates, which has wide applications in analyzing noisy big data. Correcting for the estimation bias due to the covariate noise leads to a non-convex target function to minimize. Treating the high dimensional issue further leads us to augment an amenable penalty term to the target function. We propose to estimate the regression parameter through minimizing the penalized target function. We derive the L1 and L2 convergence rates of the estimator and prove the variable selection consistency. We further establish the asymptotic normality of any subset of the parameters, where the subset can have infinitely many components as long as its cardinality grows sufficiently slow. We develop Wald and score tests based on the asymptotic normality of the estimator, which permits testing of linear functions of the members if the subset. We examine the finite sample performance of the proposed tests by extensive simulation. Finally, the proposed method is successfully applied to the Alzheimer's Disease Neuroimaging Initiative study, which motivated this work initially.
Rapid advancements in collection and dissemination of multi-platform molecular and genomics data has resulted in enormous opportunities to aggregate such data in order to understand, prevent, and treat human diseases. While significant improvements have been made in multi-omic data integration methods to discover biological markers and mechanisms underlying both prognosis and treatment, the precise cellular functions governing these complex mechanisms still need detailed and data-driven de-novo evaluations. We propose a framework called Functional Integrative Bayesian Analysis of High-dimensional Multiplatform Genomic Data (fiBAG), that allows simultaneous identification of upstream functional evidence of proteogenomic biomarkers and the incorporation of such knowledge in Bayesian variable selection models to improve signal detection. fiBAG employs a conflation of Gaussian process models to quantify (possibly non-linear) functional evidence via Bayes factors, which are then mapped to a novel calibrated spike-and-slab prior, thus guiding selection and providing functional relevance to the associations with patient outcomes. Using simulations, we illustrate how integrative methods with functional calibration have higher power to detect disease related markers than non-integrative approaches. We demonstrate the profitability of fiBAG via a pan-cancer analysis of 14 cancer types to identify and assess the cellular mechanisms of proteogenomic markers associated with cancer stemness and patient survival.
This paper presents a methodology for integrating machine learning techniques into metaheuristics for solving combinatorial optimization problems. Namely, we propose a general machine learning framework for neighbor generation in metaheuristic search. We first define an efficient neighborhood structure constructed by applying a transformation to a selected subset of variables from the current solution. Then, the key of the proposed methodology is to generate promising neighbors by selecting a proper subset of variables that contains a descent of the objective in the solution space. To learn a good variable selection strategy, we formulate the problem as a classification task that exploits structural information from the characteristics of the problem and from high-quality solutions. We validate our methodology on two metaheuristic applications: a Tabu Search scheme for solving a Wireless Network Optimization problem and a Large Neighborhood Search heuristic for solving Mixed-Integer Programs. The experimental results show that our approach is able to achieve a satisfactory trade-off between the exploration of a larger solution space and the exploitation of high-quality solution regions on both applications.
We consider a problem in Multi-Task Learning (MTL) where multiple linear models are jointly trained on a collection of datasets ("tasks"). A key novelty of our framework is that it allows the sparsity pattern of regression coefficients and the values of non-zero coefficients to differ across tasks while still leveraging partially shared structure. Our methods encourage models to share information across tasks through separately encouraging 1) coefficient supports, and/or 2) nonzero coefficient values to be similar. This allows models to borrow strength during variable selection even when non-zero coefficient values differ across tasks. We propose a novel mixed-integer programming formulation for our estimator. We develop custom scalable algorithms based on block coordinate descent and combinatorial local search to obtain high-quality (approximate) solutions for our estimator. Additionally, we propose a novel exact optimization algorithm to obtain globally optimal solutions. We investigate the theoretical properties of our estimators. We formally show how our estimators leverage the shared support information across tasks to achieve better variable selection performance. We evaluate the performance of our methods in simulations and two biomedical applications. Our proposed approaches appear to outperform other sparse MTL methods in variable selection and prediction accuracy. We provide the sMTL package on CRAN.
This paper evaluates Machine Learning (ML) in establishing ratemaking for new insurance schemes. To make the evaluation feasible, we established expected indemnities as premiums. Then, we use ML to forecast indemnities using a minimum set of variables. The analysis simulates the introduction of an income insurance scheme, the so-called Income Stabilization Tool (IST), in Italy as a case study using farm-level data from the FADN from 2008-2018. We predicted the expected IST indemnities using three ML tools, LASSO, Elastic Net, and Boosting, that perform variable selection, comparing with the Generalized Linear Model (baseline) usually adopted in insurance investigations. Furthermore, Tweedie distribution is implemented to consider the peculiarity shape of the indemnities function, characterized by zero-inflated, no-negative value, and asymmetric fat-tail. The robustness of the results was evaluated by comparing the econometric and economic performance of the models. Specifically, ML has obtained the best goodness-of-fit than baseline, using a small and stable selection of regressors and significantly reducing the gathering cost of information. However, Boosting enabled it to obtain the best economic performance, balancing the most and most minor risky subjects optimally and achieving good economic sustainability. These findings suggest how machine learning can be successfully applied in agricultural insurance.This study represents one of the first to use ML and Tweedie distribution in agricultural insurance, demonstrating its potential to overcome multiple issues.
Identifying factors that affect participation is key to a successful insurance scheme. This study's challenges involve using many factors that could affect insurance participation to make a better forecast.Huge numbers of factors affect participation, making evaluation difficult. These interrelated factors can mask the influence on adhesion predictions, making them misleading.This study evaluated how 66 common characteristics affect insurance participation choices. We relied on individual farm data from FADN from 2016 to 2019 with type 1 (Fieldcrops) farming with 10,926 observations.We use three Machine Learning (ML) approaches (LASSO, Boosting, Random Forest) compare them to the GLM model used in insurance modelling. ML methodologies can use a large set of information efficiently by performing the variable selection. A highly accurate parsimonious model helps us understand the factors affecting insurance participation and design better products.ML predicts fairly well despite the complexity of insurance participation problem. Our results suggest Boosting performs better than the other two ML tools using a smaller set of regressors. The proposed ML tools identify which variables explain participation choice. This information includes the number of cases in which single variables are selected and their relative importance in affecting participation.Focusing on the subset of information that best explains insurance participation could reduce the cost of designing insurance schemes.
The problem of covariate-shift generalization has attracted intensive research attention. Previous stable learning algorithms employ sample reweighting schemes to decorrelate the covariates when there is no explicit domain information about training data. However, with finite samples, it is difficult to achieve the desirable weights that ensure perfect independence to get rid of the unstable variables. Besides, decorrelating within stable variables may bring about high variance of learned models because of the over-reduced effective sample size. A tremendous sample size is required for these algorithms to work. In this paper, with theoretical justification, we propose SVI (Sparse Variable Independence) for the covariate-shift generalization problem. We introduce sparsity constraint to compensate for the imperfectness of sample reweighting under the finite-sample setting in previous methods. Furthermore, we organically combine independence-based sample reweighting and sparsity-based variable selection in an iterative way to avoid decorrelating within stable variables, increasing the effective sample size to alleviate variance inflation. Experiments on both synthetic and real-world datasets demonstrate the improvement of covariate-shift generalization performance brought by SVI.
Tensor Gaussian graphical models (GGMs), interpreting conditional independence structures within tensor data, have important applications in numerous areas. Yet, the available tensor data in one single study is often limited due to high acquisition costs. Although relevant studies can provide additional data, it remains an open question how to pool such heterogeneous data. In this paper, we propose a transfer learning framework for tensor GGMs, which takes full advantage of informative auxiliary domains even when non-informative auxiliary domains are present, benefiting from the carefully designed data-adaptive weights. Our theoretical analysis shows substantial improvement of estimation errors and variable selection consistency on the target domain under much relaxed conditions, by leveraging information from auxiliary domains. Extensive numerical experiments are conducted on both synthetic tensor graphs and a brain functional connectivity network data, which demonstrates the satisfactory performance of the proposed method.
Variable selection problem for the nonlinear Cox regression model is considered. In survival analysis, one main objective is to identify the covariates that are associated with the risk of experiencing the event of interest. The Cox proportional hazard model is being used extensively in survival analysis in studying the relationship between survival times and covariates, where the model assumes that the covariate has a log-linear effect on the hazard function. However, this linearity assumption may not be satisfied in practice. In order to extract a representative subset of features, various variable selection approaches have been proposed for survival data under the linear Cox model. However, there exists little literature on variable selection for the nonlinear Cox model. To break this gap, we extend the recently developed deep learning-based variable selection model LassoNet to survival data. Simulations are provided to demonstrate the validity and effectiveness of the proposed method. Finally, we apply the proposed methodology to analyze a real data set on diffuse large B-cell lymphoma.
This paper proposes a new algorithm for an automatic variable selection procedure in High Dimensional Graphical Models. The algorithm selects the relevant variables for the node of interest on the basis of mutual information. Several contributions in literature have investigated the use of mutual information in selecting the appropriate number of relevant features in a large data-set, but most of them have focused on binary outcomes or required high computational effort. The algorithm here proposed overcomes these drawbacks as it is an extension of Chow and Liu's algorithm. Once, the probabilistic structure of a High Dimensional Graphical Model is determined via the said algorithm, the best path-step, including variables with the most explanatory/predictive power for a variable of interest, is determined via the computation of the entropy coefficient of determination. The latter, being based on the notion of (symmetric) Kullback-Leibler divergence, turns out to be closely connected to the mutual information of the involved variables. The application of the algorithm to a wide range of real-word and publicly data-sets has highlighted its potential and greater effectiveness compared to alternative extant methods.
Kernel methods provide a flexible and theoretically grounded approach to nonlinear and nonparametric learning. While memory and run-time requirements hinder their applicability to large datasets, many low-rank kernel approximations, such as random Fourier features, were recently developed to scale up such kernel methods. However, these scalable approaches are based on approximations of isotropic kernels, which cannot remove the influence of irrelevant features. In this work, we design random Fourier features for a family of automatic relevance determination (ARD) kernels, and introduce RFFNet, a new large-scale kernel method that learns the kernel relevances' on the fly via first-order stochastic optimization. We present an effective initialization scheme for the method's non-convex objective function, evaluate if hard-thresholding RFFNet's learned relevances yield a sensible rule for variable selection, and perform an extensive ablation study of RFFNet's components. Numerical validation on simulated and real-world data shows that our approach has a small memory footprint and run-time, achieves low prediction error, and effectively identifies relevant features, thus leading to more interpretable solutions. We supply users with an efficient, PyTorch-based library, that adheres to the scikit-learn standard API and code for fully reproducing our results.