Identifying the most suitable variables to represent the state is a fundamental challenge in Reinforcement Learning (RL). These variables must efficiently capture the information necessary for making optimal decisions. In order to address this problem, in this paper, we introduce the Transfer Entropy Redundancy Criterion (TERC), an information-theoretic criterion, which determines if there is \textit{entropy transferred} from state variables to actions during training. We define an algorithm based on TERC that provably excludes variables from the state that have no effect on the final performance of the agent, resulting in more sample efficient learning. Experimental results show that this speed-up is present across three different algorithm classes (represented by tabular Q-learning, Actor-Critic, and Proximal Policy Optimization (PPO)) in a variety of environments. Furthermore, to highlight the differences between the proposed methodology and the current state-of-the-art feature selection approaches, we present a series of controlled experiments on synthetic data, before generalizing to real-world decision-making tasks. We also introduce a representation of the problem that compactly captures the transfer of information from state variables to actions as Bayesian networks.
Gaussian graphical models emerge in a wide range of fields. They model the statistical relationships between variables as a graph, where an edge between two variables indicates conditional dependence. Unfortunately, well-established estimators, such as the graphical lasso or neighborhood selection, are known to be susceptible to a high prevalence of false edge detections. False detections may encourage inaccurate or even incorrect scientific interpretations, with major implications in applications, such as biomedicine or healthcare. In this paper, we introduce a nodewise variable selection approach to graph learning and provably control the false discovery rate of the selected edge set at a self-estimated level. A novel fusion method of the individual neighborhoods outputs an undirected graph estimate. The proposed method is parameter-free and does not require tuning by the user. Benchmarks against competing false discovery rate controlling methods in numerical experiments considering different graph topologies show a significant gain in performance.
Sparse principal component analysis (PCA) aims at mapping large dimensional data to a linear subspace of lower dimension. By imposing loading vectors to be sparse, it performs the double duty of dimension reduction and variable selection. Sparse PCA algorithms are usually expressed as a trade-off between explained variance and sparsity of the loading vectors (i.e., number of selected variables). As a high explained variance is not necessarily synonymous with relevant information, these methods are prone to select irrelevant variables. To overcome this issue, we propose an alternative formulation of sparse PCA driven by the false discovery rate (FDR). We then leverage the Terminating-Random Experiments (T-Rex) selector to automatically determine an FDR-controlled support of the loading vectors. A major advantage of the resulting T-Rex PCA is that no sparsity parameter tuning is required. Numerical experiments and a stock market data example demonstrate a significant performance improvement.
We propose SDORE, a Semi-supervised Deep Sobolev Regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep ReQU neural networks to minimize the empirical risk with gradient norm regularization, allowing the approximation of the regularization term by unlabeled data. Our study includes a thorough analysis of the convergence rates of SDORE in $L^{2}$-norm, achieving the minimax optimality. Further, we establish a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift. These theoretical findings offer valuable insights for selecting regularization parameters and determining the size of the neural network, while showcasing the provable advantage of leveraging unlabeled data in semi-supervised learning. To the best of our knowledge, SDORE is the first provable neural network-based approach that simultaneously estimates the regression function and its gradient, with diverse applications such as nonparametric variable selection. The effectiveness of SDORE is validated through an extensive range of numerical simulations.
We propose a deep neural network (DNN) based least distance (LD) estimator (DNN-LD) for a multivariate regression problem, addressing the limitations of the conventional methods. Due to the flexibility of a DNN structure, both linear and nonlinear conditional mean functions can be easily modeled, and a multivariate regression model can be realized by simply adding extra nodes at the output layer. The proposed method is more efficient in capturing the dependency structure among responses than the least squares loss, and robust to outliers. In addition, we consider $L_1$-type penalization for variable selection, crucial in analyzing high-dimensional data. Namely, we propose what we call (A)GDNN-LD estimator that enjoys variable selection and model estimation simultaneously, by applying the (adaptive) group Lasso penalty to weight parameters in the DNN structure. For the computation, we propose a quadratic smoothing approximation method to facilitate optimizing the non-smooth objective function based on the least distance loss. The simulation studies and a real data analysis demonstrate the promising performance of the proposed method.
Abrupt environmental changes can lead to evolutionary shifts in not only the optimal trait value, but also the rate of adaptation and the diffusion variance in trait evolution. While several methods exist for detecting shifts in optimal values, few explicitly model shifts in both evolutionary variance and adaptation rates. We use a multi-optima and multi-variance Ornstein-Uhlenbeck (OU) process model to describe trait evolution with shifts in both optimal value and diffusion variance and analyze how covariance between species is affected when shifts in variance occur along the phylogeny. We propose a new method that simultaneously detects shifts in both variance and optimal values by formulating the problem as a variable selection task using an L1-penalized loss function. Our method is implemented in the R package ShiVa (Detection of evolutionary Shifts in Variance). Through simulations, we compare ShiVa with methods that only consider shifts in optimal values (l1ou; PhylogeneticEM), and PCMFit. Our method demonstrates improved predictive ability and significantly reduces false positives in detecting optimal value shifts when variance shifts are present. When only shifts in optimal value occur, our method performs comparably to existing approaches. Applying ShiVa to empirical data from cordylid lizards , we find that it outperforms l1ou and PhylogeneticEM, achieving the highest log-likelihood and lowest BIC.
This paper considers the problem of variable selection allowing for parameter instability. It distinguishes between signal and pseudo-signal variables that are correlated with the target variable, and noise variables that are not, and investigate the asymptotic properties of the One Covariate at a Time Multiple Testing (OCMT) method proposed by Chudik et al. (2018) under parameter insatiability. It is established that OCMT continues to asymptotically select an approximating model that includes all the signals and none of the noise variables. Properties of post selection regressions are also investigated, and in-sample fit of the selected regression is shown to have the oracle property. The theoretical results support the use of unweighted observations at the selection stage of OCMT, whilst applying down-weighting of observations only at the forecasting stage. Monte Carlo and empirical applications show that OCMT without down-weighting at the selection stage yields smaller mean squared forecast errors compared to Lasso, Adaptive Lasso, and boosting.
Classification and probability estimation have broad applications in modern machine learning and data science applications, including biology, medicine, engineering, and computer science. The recent development of a class of weighted Support Vector Machines (wSVMs) has shown great values in robustly predicting the class probability and classification for various problems with high accuracy. The current framework is based on the $\ell^2$-norm regularized binary wSVMs optimization problem, which only works with dense features and has poor performance at sparse features with redundant noise in most real applications. The sparse learning process requires a prescreen of the important variables for each binary wSVMs for accurately estimating pairwise conditional probability. In this paper, we proposed novel wSVMs frameworks that incorporate automatic variable selection with accurate probability estimation for sparse learning problems. We developed efficient algorithms for effective variable selection for solving either the $\ell^1$-norm or elastic net regularized binary wSVMs optimization problems. The binary class probability is then estimated either by the $\ell^2$-norm regularized wSVMs framework with selected variables or by elastic net regularized wSVMs directly. The two-step approach of $\ell^1$-norm followed by $\ell^2$-norm wSVMs show a great advantage in both automatic variable selection and reliable probability estimators with the most efficient time. The elastic net regularized wSVMs offer the best performance in terms of variable selection and probability estimation with the additional advantage of variable grouping in the compensation of more computation time for high dimensional problems. The proposed wSVMs-based sparse learning methods have wide applications and can be further extended to $K$-class problems through ensemble learning.
Random Forest is a machine learning method that offers many advantages, including the ability to easily measure variable importance. Class balancing technique is a well-known solution to deal with class imbalance problem. However, it has not been actively studied on RF variable importance. In this paper, we study the effect of class balancing on RF variable importance. Our simulation results show that over-sampling is effective in correctly measuring variable importance in class imbalanced situations with small sample size, while under-sampling fails to differentiate important and non-informative variables. We then propose a variable selection algorithm that utilizes RF variable importance and its confidence interval. Through an experimental study using many real and artificial datasets, we demonstrate that our proposed algorithm efficiently selects an optimal feature set, leading to improved prediction performance in class imbalance problem.
This paper addresses a key question in economic forecasting: does pure noise truly lack predictive power? Economists typically conduct variable selection to eliminate noises from predictors. Yet, we prove a compelling result that in most economic forecasts, the inclusion of noises in predictions yields greater benefits than its exclusion. Furthermore, if the total number of predictors is not sufficiently large, intentionally adding more noises yields superior forecast performance, outperforming benchmark predictors relying on dimension reduction. The intuition lies in economic predictive signals being densely distributed among regression coefficients, maintaining modest forecast bias while diversifying away overall variance, even when a significant proportion of predictors constitute pure noises. One of our empirical demonstrations shows that intentionally adding 300~6,000 pure noises to the Welch and Goyal (2008) dataset achieves a noteworthy 10% out-of-sample R square accuracy in forecasting the annual U.S. equity premium. The performance surpasses the majority of sophisticated machine learning models.
Biomedical research now commonly integrates diverse data types or views from the same individuals to better understand the pathobiology of complex diseases, but the challenge lies in meaningfully integrating these diverse views. Existing methods often require the same type of data from all views (cross-sectional data only or longitudinal data only) or do not consider any class outcome in the integration method, presenting limitations. To overcome these limitations, we have developed a pipeline that harnesses the power of statistical and deep learning methods to integrate cross-sectional and longitudinal data from multiple sources. Additionally, it identifies key variables contributing to the association between views and the separation among classes, providing deeper biological insights. This pipeline includes variable selection/ranking using linear and nonlinear methods, feature extraction using functional principal component analysis and Euler characteristics, and joint integration and classification using dense feed-forward networks and recurrent neural networks. We applied this pipeline to cross-sectional and longitudinal multi-omics data (metagenomics, transcriptomics, and metabolomics) from an inflammatory bowel disease (IBD) study and we identified microbial pathways, metabolites, and genes that discriminate by IBD status, providing information on the etiology of IBD. We conducted simulations to compare the two feature extraction methods. The proposed pipeline is available from the following GitHub repository: https://github.com/lasandrall/DeepIDA-GRU.
Combinatorial optimization finds an optimal solution within a discrete set of variables and constraints. The field has seen tremendous progress both in research and industry. With the success of deep learning in the past decade, a recent trend in combinatorial optimization has been to improve state-of-the-art combinatorial optimization solvers by replacing key heuristic components with machine learning (ML) models. In this paper, we investigate two essential aspects of machine learning algorithms for combinatorial optimization: temporal characteristics and attention. We argue that for the task of variable selection in the branch-and-bound (B&B) algorithm, incorporating the temporal information as well as the bipartite graph attention improves the solver's performance. We support our claims with intuitions and numerical results over several standard datasets used in the literature and competitions. Code is available at: https://developer.huaweicloud.com/develop/aigallery/notebook/detail?id=047c6cf2-8463-40d7-b92f-7b2ca998e935
Hurricanes cause significant economic and human costs, requiring individuals to make critical evacuation decisions under uncertainty and stress. To enhance the understanding of this decision-making process, we propose using Bayesian Networks (BNs) to model evacuation decisions during hurricanes. We collected questionnaire data from two significant hurricane events: Hurricane Harvey and Hurricane Irma. We employed a data-driven approach by first conducting variable selection using mutual information, followed by BN structure learning with two constraint-based algorithms. The robustness of the learned structures was enhanced by model averaging based on bootstrap resampling. We examined and compared the learned structures of both hurricanes, revealing potential causal relationships among key predictors of evacuation, including risk perception, information received from media, suggestions from family and friends, and neighbors evacuating. Our findings highlight the significant role of social influence, providing valuable insights into the process of evacuation decision-making. Our results demonstrate the applicability and effectiveness of data-driven BN modeling in evacuation decision making.
Medical prediction applications often need to deal with small sample sizes compared to the number of covariates. Such data pose problems for prediction and variable selection, especially when the covariate-response relationship is complicated. To address these challenges, we propose to incorporate co-data, i.e. external information on the covariates, into Bayesian additive regression trees (BART), a sum-of-trees prediction model that utilizes priors on the tree parameters to prevent overfitting. To incorporate co-data, an empirical Bayes (EB) framework is developed that estimates, assisted by a co-data model, prior covariate weights in the BART model. The proposed method can handle multiple types of co-data simultaneously. Furthermore, the proposed EB framework enables the estimation of the other hyperparameters of BART as well, rendering an appealing alternative to cross-validation. We show that the method finds relevant covariates and that it improves prediction compared to default BART in simulations. If the covariate-response relationship is nonlinear, the method benefits from the flexibility of BART to outperform regression-based co-data learners. Finally, the use of co-data enhances prediction in an application to diffuse large B-cell lymphoma prognosis based on clinical covariates, gene mutations, DNA translocations, and DNA copy number data. Keywords: Bayesian additive regression trees; Empirical Bayes; Co-data; High-dimensional data; Omics; Prediction
This paper proposes a new method for estimating high-dimensional binary choice models. The model we consider is semiparametric, placing no distributional assumptions on the error term, allowing for heteroskedastic errors, and permitting endogenous regressors. Our proposed approaches extend the special regressor estimator originally proposed by Lewbel (2000). This estimator becomes impractical in high-dimensional settings due to the curse of dimensionality associated with high-dimensional conditional density estimation. To overcome this challenge, we introduce an innovative data-driven dimension reduction method for nonparametric kernel estimators, which constitutes the main innovation of this work. The method combines distance covariance-based screening with cross-validation (CV) procedures, rendering the special regressor estimation feasible in high dimensions. Using the new feasible conditional density estimator, we address the variable and moment (instrumental variable) selection problems for these models. We apply penalized least squares (LS) and Generalized Method of Moments (GMM) estimators with a smoothly clipped absolute deviation (SCAD) penalty. A comprehensive analysis of the oracle and asymptotic properties of these estimators is provided. Monte Carlo simulations are employed to demonstrate the effectiveness of our proposed procedures in finite sample scenarios.
Quantile regression is a powerful tool for inferring how covariates affect specific percentiles of the response distribution. Existing methods either estimate conditional quantiles separately for each quantile of interest or estimate the entire conditional distribution using semi- or non-parametric models. The former often produce inadequate models for real data and do not share information across quantiles, while the latter are characterized by complex and constrained models that can be difficult to interpret and computationally inefficient. Neither approach is well-suited for quantile-specific subset selection. Instead, we pose the fundamental problems of linear quantile estimation, uncertainty quantification, and subset selection from a Bayesian decision analysis perspective. For any Bayesian regression model -- including, but not limited to existing Bayesian quantile regression models -- we derive optimal point estimates, interpretable uncertainty quantification, and scalable subset selection techniques for all model-based conditional quantiles. Our approach introduces a quantile-focused squared error loss that enables efficient, closed-form computing and maintains a close relationship with Wasserstein-based density estimation. In an extensive simulation study, our methods demonstrate substantial gains in quantile estimation accuracy, inference, and variable selection over frequentist and Bayesian competitors. We use these tools to identify and quantify the heterogeneous impacts of multiple social stressors and environmental exposures on educational outcomes across the full spectrum of low-, medium-, and high-achieving students in North Carolina.
Two-sample testing decides whether two datasets are generated from the same distribution. This paper studies variable selection for two-sample testing, the task being to identify the variables (or dimensions) responsible for the discrepancies between the two distributions. This task is relevant to many problems of pattern analysis and machine learning, such as dataset shift adaptation, causal inference and model validation. Our approach is based on a two-sample test based on the Maximum Mean Discrepancy (MMD). We optimise the Automatic Relevance Detection (ARD) weights defined for individual variables to maximise the power of the MMD-based test. For this optimisation, we introduce sparse regularisation and propose two methods for dealing with the issue of selecting an appropriate regularisation parameter. One method determines the regularisation parameter in a data-driven way, and the other aggregates the results of different regularisation parameters. We confirm the validity of the proposed methods by systematic comparisons with baseline methods, and demonstrate their usefulness in exploratory analysis of high-dimensional traffic simulation data. Preliminary theoretical analyses are also provided, including a rigorous definition of variable selection for two-sample testing.
High-dimensional longitudinal data is increasingly used in a wide range of scientific studies. To properly account for dependence between longitudinal observations, statistical methods for high-dimensional linear mixed models (LMMs) have been developed. However, few packages implementing these high-dimensional LMMs are available in the statistical software R. Additionally, some packages suffer from scalability issues. This work presents an efficient and accurate Bayesian framework for high-dimensional LMMs. We use empirical Bayes estimators of hyperparameters for increased flexibility and an Expectation-Conditional-Minimization (ECM) algorithm for computationally efficient maximum a posteriori probability (MAP) estimation of parameters. The novelty of the approach lies in its partitioning and parameter expansion as well as its fast and scalable computation. We illustrate Linear Mixed Modeling with PaRtitiOned empirical Bayes ECM (LMM-PROBE) in simulation studies evaluating fixed and random effects estimation along with computation time. A real-world example is provided using data from a study of lupus in children, where we identify genes and clinical factors associated with a new lupus biomarker and predict the biomarker over time. Supplementary materials are available online.
Conditional independence testing (CIT) is a common task in machine learning, e.g., for variable selection, and a main component of constraint-based causal discovery. While most current CIT approaches assume that all variables are numerical or all variables are categorical, many real-world applications involve mixed-type datasets that include numerical and categorical variables. Non-parametric CIT can be conducted using conditional mutual information (CMI) estimators combined with a local permutation scheme. Recently, two novel CMI estimators for mixed-type datasets based on k-nearest-neighbors (k-NN) have been proposed. As with any k-NN method, these estimators rely on the definition of a distance metric. One approach computes distances by a one-hot encoding of the categorical variables, essentially treating categorical variables as discrete-numerical, while the other expresses CMI by entropy terms where the categorical variables appear as conditions only. In this work, we study these estimators and propose a variation of the former approach that does not treat categorical variables as numeric. Our numerical experiments show that our variant detects dependencies more robustly across different data distributions and preprocessing types.
Dictionary learning is a popular method for obtaining sparse linear representations for high dimensional data, with many applications in image classification, signal processing and machine learning. In this paper, we introduce a novel dictionary learning method based on a recent variable selection algorithm called Feature Selection with Annealing (FSA). Because FSA uses an L0 constraint instead of the L1 penalty, it does not introduce any bias in the coefficients and obtains a more accurate sparse representation. Furthermore, the L0 constraint makes it easy to directly specify the desired sparsity level instead of indirectly through a L1 penalty. Finally, experimental validation on real gray-scale images shows that the proposed method obtains higher accuracy and efficiency in dictionary learning compared to classical methods based on the L1 penalty.