This paper presents a comprehensive exploration of the theoretical properties inherent in the Adaptive Lasso and the Transfer Lasso. The Adaptive Lasso, a well-established method, employs regularization divided by initial estimators and is characterized by asymptotic normality and variable selection consistency. In contrast, the recently proposed Transfer Lasso employs regularization subtracted by initial estimators with the demonstrated capacity to curtail non-asymptotic estimation errors. A pivotal question thus emerges: Given the distinct ways the Adaptive Lasso and the Transfer Lasso employ initial estimators, what benefits or drawbacks does this disparity confer upon each method? This paper conducts a theoretical examination of the asymptotic properties of the Transfer Lasso, thereby elucidating its differentiation from the Adaptive Lasso. Informed by the findings of this analysis, we introduce a novel method, one that amalgamates the strengths and compensates for the weaknesses of both methods. The paper concludes with validations of our theory and comparisons of the methods via simulation experiments.
We describe SharpSAT-TD, our submission to the unweighted and weighted tracks of the Model Counting Competition in 2021-2023, which has won in total $6$ first places in different tracks of the competition. SharpSAT-TD is based on SharpSAT [Thurley, SAT 2006], with the primary novel modification being the use of tree decompositions in the variable selection heuristic as introduced by the authors in [CP 2021]. Unlike the version of SharpSAT-TD evaluated in [CP 2021], the current version that is available in https://github.com/Laakeri/sharpsat-td features also other significant modifications compared to the original SharpSAT, for example, a new preprocessor.
In this paper we propose a novel variable selection method for two-view settings, or for vector-valued supervised learning problems. Our framework is able to handle extremely large scale selection tasks, where number of data samples could be even millions. In a nutshell, our method performs variable selection by iteratively selecting variables that are highly correlated with the output variables, but which are not correlated with the previously chosen variables. To measure the correlation, our method uses the concept of projection operators and their algebra. With the projection operators the relationship, correlation, between sets of input and output variables can also be expressed by kernel functions, thus nonlinear correlation models can be exploited as well. We experimentally validate our approach, showing on both synthetic and real data its scalability and the relevance of the selected features. Keywords: Supervised variable selection, vector-valued learning, projection-valued measure, reproducing kernel Hilbert space
Cox models with time-dependent coefficients and covariates are widely used in survival analysis. In high-dimensional settings, sparse regularization techniques are employed for variable selection, but existing methods for time-dependent Cox models lack flexibility in enforcing specific sparsity patterns (i.e., covariate structures). We propose a flexible framework for variable selection in time-dependent Cox models, accommodating complex selection rules. Our method can adapt to arbitrary grouping structures, including interaction selection, temporal, spatial, tree, and directed acyclic graph structures. It achieves accurate estimation with low false alarm rates. We develop the sox package, implementing a network flow algorithm for efficiently solving models with complex covariate structures. sox offers a user-friendly interface for specifying grouping structures and delivers fast computation. Through examples, including a case study on identifying predictors of time to all-cause death in atrial fibrillation patients, we demonstrate the practical application of our method with specific selection rules.
Combinatorial optimization problems require an exhaustive search to find the optimal solution. A convenient approach to solving combinatorial optimization tasks in the form of Mixed Integer Linear Programs is Branch-and-Bound. Branch-and-Bound solver splits a task into two parts dividing the domain of an integer variable, then it solves them recursively, producing a tree of nested sub-tasks. The efficiency of the solver depends on the branchning heuristic used to select a variable for splitting. In the present work, we propose a reinforcement learning method that can efficiently learn the branching heuristic. We view the variable selection task as a tree Markov Decision Process, prove that the Bellman operator adapted for the tree Markov Decision Process is contracting in mean, and propose a modified learning objective for the reinforcement learning agent. Our agent requires less training data and produces smaller trees compared to previous reinforcement learning methods.
Artificial neural networks (ANNs) are powerful machine learning methods used in many modern applications such as facial recognition, machine translation, and cancer diagnostics. A common issue with ANNs is that they usually have millions or billions of trainable parameters, and therefore tend to overfit to the training data. This is especially problematic in applications where it is important to have reliable uncertainty estimates. Bayesian neural networks (BNN) can improve on this, since they incorporate parameter uncertainty. In addition, latent binary Bayesian neural networks (LBBNN) also take into account structural uncertainty by allowing the weights to be turned on or off, enabling inference in the joint space of weights and structures. In this paper, we will consider two extensions to the LBBNN method: Firstly, by using the local reparametrization trick (LRT) to sample the hidden units directly, we get a more computationally efficient algorithm. More importantly, by using normalizing flows on the variational posterior distribution of the LBBNN parameters, the network learns a more flexible variational posterior distribution than the mean field Gaussian. Experimental results show that this improves predictive power compared to the LBBNN method, while also obtaining more sparse networks. We perform two simulation studies. In the first study, we consider variable selection in a logistic regression setting, where the more flexible variational distribution leads to improved results. In the second study, we compare predictive uncertainty based on data generated from two-dimensional Gaussian distributions. Here, we argue that our Bayesian methods lead to more realistic estimates of predictive uncertainty.
Identifying differential equation governing dynamical system is an important problem with wide applications. Copula Entropy (CE) is a mathematical concept for measuring statistical independence in information theory. In this paper we propose a method for identifying differential equation of dynamical systems with CE. The problem is considered as a variable selection problem and solved with the previously proposed CE-based method for variable selection. The proposed method composed of two components: the difference operator and the CE estimator. Since both components can be done non-parametrically, the proposed method is therefore model-free and hyperparameter-free. The simulation experiment with the 3D Lorenz system verified the effectiveness of the proposed method.
Synthetic electronic health records (EHRs) that are both realistic and preserve privacy can serve as an alternative to real EHRs for machine learning (ML) modeling and statistical analysis. However, generating high-fidelity and granular electronic health record (EHR) data in its original, highly-dimensional form poses challenges for existing methods due to the complexities inherent in high-dimensional data. In this paper, we propose Hierarchical Autoregressive Language mOdel (HALO) for generating longitudinal high-dimensional EHR, which preserve the statistical properties of real EHR and can be used to train accurate ML models without privacy concerns. Our HALO method, designed as a hierarchical autoregressive model, generates a probability density function of medical codes, clinical visits, and patient records, allowing for the generation of realistic EHR data in its original, unaggregated form without the need for variable selection or aggregation. Additionally, our model also produces high-quality continuous variables in a longitudinal and probabilistic manner. We conducted extensive experiments and demonstrate that HALO can generate high-fidelity EHR data with high-dimensional disease code probabilities (d > 10,000), disease co-occurrence probabilities within visits (d > 1,000,000), and conditional probabilities across consecutive visits (d > 5,000,000) and achieve above 0.9 R2 correlation in comparison to real EHR data. This performance then enables downstream ML models trained on its synthetic data to achieve comparable accuracy to models trained on real data (0.938 AUROC with HALO data vs. 0.943 with real data). Finally, using a combination of real and synthetic data enhances the accuracy of ML models beyond that achieved by using only real EHR data.
This paper considers a multi-environment linear regression model in which data from multiple experimental settings are collected. The joint distribution of the response variable and covariates may vary across different environments, yet the conditional expectations of $y$ given the unknown set of important variables are invariant. Such a statistical model is related to the problem of endogeneity, causal inference, and transfer learning. The motivation behind it is illustrated by how the goals of prediction and attribution are inherent in estimating the true parameter and the important variable set. We construct a novel environment invariant linear least squares (EILLS) objective function, a multi-environment version of linear least-squares regression that leverages the above conditional expectation invariance structure and heterogeneity among different environments to determine the true parameter. Our proposed method is applicable without any additional structural knowledge and can identify the true parameter under a near-minimal identification condition. We establish non-asymptotic $\ell_2$ error bounds on the estimation error for the EILLS estimator in the presence of spurious variables. Moreover, we further show that the $\ell_0$ penalized EILLS estimator can achieve variable selection consistency in high-dimensional regimes. These non-asymptotic results demonstrate the sample efficiency of the EILLS estimator and its capability to circumvent the curse of endogeneity in an algorithmic manner without any prior structural knowledge. To the best of our knowledge, this paper is the first to realize statistically efficient invariance learning in the general linear model.
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.
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}.
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 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.
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.
In high dimensional variable selection problems, statisticians often seek to design multiple testing procedures that control the False Discovery Rate (FDR), while concurrently identifying a greater number of relevant variables. Model-X methods, such as Knockoffs and conditional randomization tests, achieve the primary goal of finite-sample FDR control, assuming a known distribution of covariates. However, whether these methods can also achieve the secondary goal of maximizing discoveries remains uncertain. In fact, designing procedures to discover more relevant variables with finite-sample FDR control is a largely open question, even within the arguably simplest linear models. In this paper, we develop near-optimal multiple testing procedures for high dimensional Bayesian linear models with isotropic covariates. We introduce Model-X procedures that provably control the frequentist FDR from finite samples, even when the model is misspecified, and conjecturally achieve near-optimal power when the data follow the Bayesian linear model. Our proposed procedure, PoEdCe, incorporates three key ingredients: Posterior Expectation, distilled Conditional randomization test (dCRT), and the Benjamini-Hochberg procedure with e-values (eBH). The optimality conjecture of PoEdCe is based on a heuristic calculation of its asymptotic true positive proportion (TPP) and false discovery proportion (FDP), which is supported by methods from statistical physics as well as extensive numerical simulations. Our result establishes the Bayesian linear model as a benchmark for comparing the power of various multiple testing procedures.
Bayesian optimization (BO) is a class of popular methods for expensive black-box optimization, and has been widely applied to many scenarios. However, BO suffers from the curse of dimensionality, and scaling it to high-dimensional problems is still a challenge. In this paper, we propose a variable selection method MCTS-VS based on Monte Carlo tree search (MCTS), to iteratively select and optimize a subset of variables. That is, MCTS-VS constructs a low-dimensional subspace via MCTS and optimizes in the subspace with any BO algorithm. We give a theoretical analysis of the general variable selection method to reveal how it can work. Experiments on high-dimensional synthetic functions and real-world problems (i.e., NAS-bench problems and MuJoCo locomotion tasks) show that MCTS-VS equipped with a proper BO optimizer can achieve state-of-the-art performance.
Variable selection is an important problem in statistics and machine learning. Copula Entropy (CE) is a mathematical concept for measuring statistical independence and has been applied to variable selection recently. In this paper we propose to apply the CE-based method for variable selection to survival analysis. The idea is to measure the correlation between variables and time-to-event with CE and then select variables according to their CE value. Experiments on simulated data and two real cancer data were conducted to compare the proposed method with two related methods: random survival forest and Lasso-Cox. Experimental results showed that the proposed method can select the 'right' variables out that are more interpretable and lead to better prediction performance.
Bayesian variable selection is a powerful tool for data analysis, as it offers a principled method for variable selection that accounts for prior information and uncertainty. However, wider adoption of Bayesian variable selection has been hampered by computational challenges, especially in difficult regimes with a large number of covariates P or non-conjugate likelihoods. To scale to the large P regime we introduce an efficient MCMC scheme whose cost per iteration is sublinear in P. In addition we show how this scheme can be extended to generalized linear models for count data, which are prevalent in biology, ecology, economics, and beyond. In particular we design efficient algorithms for variable selection in binomial and negative binomial regression, which includes logistic regression as a special case. In experiments we demonstrate the effectiveness of our methods, including on cancer and maize genomic data.
Survival analysis is a fundamental area of focus in biomedical research, particularly in the context of personalized medicine. This prominence is due to the increasing prevalence of large and high-dimensional datasets, such as omics and medical image data. However, the literature on non-linear regression algorithms and variable selection techniques for interval-censoring is either limited or non-existent, particularly in the context of neural networks. Our objective is to introduce a novel predictive framework tailored for interval-censored regression tasks, rooted in Accelerated Failure Time (AFT) models. Our strategy comprises two key components: i) a variable selection phase leveraging recent advances on sparse neural network architectures, ii) a regression model targeting prediction of the interval-censored response. To assess the performance of our novel algorithm, we conducted a comprehensive evaluation through both numerical experiments and real-world applications that encompass scenarios related to diabetes and physical activity. Our results outperform traditional AFT algorithms, particularly in scenarios featuring non-linear relationships.
In the presence of grouped covariates, we propose a framework for boosting that allows to enforce sparsity within and between groups. By using component-wise and group-wise gradient boosting at the same time with adjusted degrees of freedom, a model with similar properties as the sparse group lasso can be fitted through boosting. We show that within-group and between-group sparsity can be controlled by a mixing parameter and discuss similarities and differences to the mixing parameter in the sparse group lasso. With simulations, gene data as well as agricultural data we show the effectiveness and predictive competitiveness of this estimator. The data and simulations suggest, that in the presence of grouped variables the use of sparse group boosting is associated with less biased variable selection and higher predictability compared to component-wise boosting. Additionally, we propose a way of reducing bias in component-wise boosting through the degrees of freedom.