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Register a new userTransfer Learning for High-dimensional Linear Regression: Prediction, Estimation, and Minimax Optimality
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Sai Li
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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This paper considers the estimation and prediction of a high-dimensional linear regression in the setting of transfer learning, using samples from the target model as well as auxiliary samples from different but possibly related regression models. When the set of informative auxiliary samples is known, an estimator and a predictor are proposed and their optimality is established. The optimal rates of convergence for prediction and estimation are faster than the corresponding rates without using the auxiliary samples. This implies that knowledge from the informative auxiliary samples can be transferred to improve the learning performance of the target problem. In the case that the set of informative auxiliary samples is unknown, we propose a data-driven procedure for transfer learning, called Trans-Lasso, and reveal its robustness to non-informative auxiliary samples and its efficiency in knowledge transfer. The proposed procedures are demonstrated in numerical studies and are applied to a dataset concerning the associations among gene expressions. It is shown that Trans-Lasso leads to improved performance in gene expression prediction in a target tissue by incorporating the data from multiple different tissues as auxiliary samples.
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There are many scenarios such as the electronic health records where the outcome is much more difficult to collect than the covariates. In this paper, we consider the linear regression problem with such a data structure under the high dimensionality. Our goal is to investigate when and how the unlabeled data can be exploited to improve the estimation and inference of the regression parameters in linear models, especially in light of the fact that such linear models may be misspecified in data analysis. In particular, we address the following two important questions. (1) Can we use the labeled data as well as the unlabeled data to construct a semi-supervised estimator such that its convergence rate is faster than the supervised estimators? (2) Can we construct confidence intervals or hypothesis tests that are guaranteed to be more efficient or powerful than the supervised estimators? To address the first question, we establish the minimax lower bound for parameter estimation in the semi-supervised setting. We show that the upper bound from the supervised estimators that only use the labeled data cannot attain this lower bound. We close this gap by proposing a new semi-supervised estimator which attains the lower bound. To address the second question, based on our proposed semi-supervised estimator, we propose two additional estimators for semi-supervised inference, the efficient estimator and the safe estimator. The former is fully efficient if the unknown conditional mean function is estimated consistently, but may not be more efficient than the supervised approach otherwise. The latter usually does not aim to provide fully efficient inference, but is guaranteed to be no worse than the supervised approach, no matter whether the linear model is correctly specified or the conditional mean function is consistently estimated.
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With the availability of high dimensional genetic biomarkers, it is of interest to identify heterogeneous effects of these predictors on patients survival, along with proper statistical inference. Censored quantile regression has emerged as a powerful tool for detecting heterogeneous effects of covariates on survival outcomes. To our knowledge, there is little work available to draw inference on the effects of high dimensional predictors for censored quantile regression. This paper proposes a novel procedure to draw inference on all predictors within the framework of global censored quantile regression, which investigates covariate-response associations over an interval of quantile levels, instead of a few discrete values. The proposed estimator combines a sequence of low dimensional model estimates that are based on multi-sample splittings and variable selection. We show that, under some regularity conditions, the estimator is consistent and asymptotically follows a Gaussian process indexed by the quantile level. Simulation studies indicate that our procedure can properly quantify the uncertainty of the estimates in high dimensional settings. We apply our method to analyze the heterogeneous effects of SNPs residing in lung cancer pathways on patients survival, using the Boston Lung Cancer Survival Cohort, a cancer epidemiology study on the molecular mechanism of lung cancer.
Neural networks are usually not the tool of choice for nonparametric high-dimensional problems where the number of input features is much larger than the number of observations. Though neural networks can approximate complex multivariate functions, they generally require a large number of training observations to obtain reasonable fits, unless one can learn the appropriate network structure. In this manuscript, we show that neural networks can be applied successfully to high-dimensional settings if the true function falls in a low dimensional subspace, and proper regularization is used. We propose fitting a neural network with a sparse group lasso penalty on the first-layer input weights. This results in a neural net that only uses a small subset of the original features. In addition, we characterize the statistical convergence of the penalized empirical risk minimizer to the optimal neural network: we show that the excess risk of this penalized estimator only grows with the logarithm of the number of input features; and we show that the weights of irrelevant features converge to zero. Via simulation studies and data analyses, we show that these sparse-input neural networks outperform existing nonparametric high-dimensional estimation methods when the data has complex higher-order interactions.
We study a mean-field spike and slab variational Bayes (VB) approximation to Bayesian model selection priors in sparse high-dimensional linear regression. Under compatibility conditions on the design matrix, oracle inequalities are derived for the mean-field VB approximation, implying that it converges to the sparse truth at the optimal rate and gives optimal prediction of the response vector. The empirical performance of our algorithm is studied, showing that it works comparably well as other state-of-the-art Bayesian variable selection methods. We also numerically demonstrate that the widely used coordinate-ascent variational inference (CAVI) algorithm can be highly sensitive to the parameter updating order, leading to potentially poor performance. To mitigate this, we propose a novel prioritized updating scheme that uses a data-driven updating order and performs better in simulations. The variational algorithm is implemented in the R package sparsevb.
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