Author(s): Zhang, Yuqian; Bradic, Jelena | Abstract: We provide a high-dimensional semi-supervised inference framework focused on the mean and variance of the response. Our data are comprised of an extensive set of observations regarding the covariate vectors and a much smaller set of labeled observations where we observe both the response as well as the covariates. We allow the size of the covariates to be much larger than the sample size and impose weak conditions on a statistical form of the data. We provide new estimators of the mean and variance of the response that extend some of the recent results presented in low-dimensional models. In particular, at times we will not necessitate consistent estimation of the functional form of the data. Together with estimation of the population mean and variance, we provide their asymptotic distribution and confidence intervals where we showcase gains in efficiency compared to the sample mean and variance. Our procedure, with minor modifications, is then presented to make important contributions regarding inference about average treatment effects. We also investigate the robustness of estimation and coverage and showcase widespread applicability and generality of the proposed method.
The main contribution of our work is both construction of new estimates as well as an asymptotic normality results that accommodate semi-supervised as well as a high-dimensionality phenomenon. Recent results of @cite_15 @cite_3 @cite_33 consider the class of graph-oriented semi-supervised learning algorithms and proceed to establish estimation or prediction properties of semi-supervised estimators when the number of observed responses was much higher than the number of features in the data. New work of @cite_11 @cite_10 @cite_21 develops semi-supervised approaches for estimation of the conditional mean of the responses. They utilize the additional information to obtain information about the marginal distribution of the covariates in order to then reduce the bias of the local linear estimator. Meanwhile, blue @cite_7 proposed a semi-supervised estimator of the explained variance; apart from discussing a different question, their task requires exact model specification. Semi-supervised inference in the context of classification has had a long tradition; see blue @cite_8 @cite_20 @cite_9 @cite_1 . A major limitation of this line of work is that it has lacked formal statistical inference results.
@cite_33 @cite_7 @cite_8 @cite_9 @cite_21 @cite_1 @cite_3 @cite_15 @cite_10 @cite_20 @cite_11
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We propose a general semi-supervised inference framework focused on the estimation of the population mean. As usual in semi-supervised settings, there exists an unlabeled sample of covariate vectors and a labeled sample consisting of covariate vectors along with real-valued responses ("labels"). Otherwise, the formulation is "assumption-lean" in that no major conditions are imposed on the statistical or functional form of the data. We consider both the ideal semi-supervised setting where infinitely many unlabeled samples are available, as well as the ordinary semi-supervised setting in which only a finite number of unlabeled samples is available. Estimators are proposed along with corresponding confidence intervals for the population mean. Theoretical analysis on both the asymptotic distribution and @math -risk for the proposed procedures are given. Surprisingly, the proposed estimators, based on a simple form of the least squares method, outperform the ordinary sample mean. The simple, transparent form of the estimator lends confidence to the perception that its asymptotic improvement over the ordinary sample mean also nearly holds even for moderate size samples. The method is further extended to a nonparametric setting, in which the oracle rate can be achieved asymptotically. The proposed estimators are further illustrated by simulation studies and a real data example involving estimation of the homeless population. We consider the linear regression problem under semi-supervised settings wherein the available data typically consists of: (i) a small or moderate sized 'labeled' data, and (ii) a much larger sized 'unlabeled' data. Such data arises naturally from settings where the outcome, unlike the covariates, is expensive to obtain, a frequent scenario in modern studies involving large databases like electronic medical records (EMR). Supervised estimators like the ordinary least squares (OLS) estimator utilize only the labeled data. It is often of interest to investigate if and when the unlabeled data can be exploited to improve estimation of the regression parameter in the adopted linear model. In this paper, we propose a class of 'Efficient and Adaptive Semi-Supervised Estimators' (EASE) to improve estimation efficiency. The EASE are two-step estimators adaptive to model mis-specification, leading to improved (optimal in some cases) efficiency under model mis-specification, and equal (optimal) efficiency under a linear model. This adaptive property, often unaddressed in the existing literature, is crucial for advocating 'safe' use of the unlabeled data. The construction of EASE primarily involves a flexible 'semi-non-parametric' imputation, including a smoothing step that works well even when the number of covariates is not small; and a follow up 'refitting' step along with a cross-validation (CV) strategy both of which have useful practical as well as theoretical implications towards addressing two important issues: under-smoothing and over-fitting. We establish asymptotic results including consistency, asymptotic normality and the adaptive properties of EASE. We also provide influence function expansions and a 'double' CV strategy for inference. The results are further validated through extensive simulations, followed by application to an EMR study on auto-immunity. In many modern machine learning applications, the outcome is expensive or time consuming to collect whereas the predictor information is easy to obtain. Semiâ€ supervised (SS) learning aims at utilizing large amounts of â€˜unlabelledâ€™ data along with small amounts of â€˜labelledâ€™ data to improve the efficiency of a classical supervised approach. Though numerous SS learning classification and prediction procedures have been proposed in recent years, no methods currently exist to evaluate the prediction performance of a working regression model. In the context of developing phenotyping algorithms derived from electronic medical records, we present an efficient twoâ€ step estimation procedure for evaluating a binary classifier based on various prediction performance measures in the SS setting. In step I, the labelled data are used to obtain a nonâ€ parametrically calibrated estimate of the conditional risk function. In step II, SS estimates of the prediction accuracy parameters are constructed based on the estimated conditional risk function and the unlabelled data. We demonstrate that, under mild regularity conditions, the estimators proposed are consistent and asymptotically normal. Importantly, the asymptotic variance of the SS estimators is always smaller than that of the supervised counterparts under correct model specification. We also correct for potential overfitting bias in the SS estimators in finite samples with crossâ€ validation and we develop a perturbation resampling procedure to approximate their distributions. Our proposals are evaluated through extensive simulation studies and illustrated with two real electronic medical record studies aiming to develop phenotyping algorithms for rheumatoid arthritis and multiple sclerosis. We investigate a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of @math sample points, provided a small training subset of @math labeled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider random geometric graphs, with connection radius @math , to represent the geometry of the data set. Functionals which model the task reward the regularity of the estimator function and impose or reward the agreement with the training data. Here we consider the discrete @math -Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases, while the number of training points remains fixed. We uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. We rigorously obtain almost optimal ranges on the scaling of @math for the asymptotic consistency to hold. We prove that the minimizers of the discrete functionals in random setting converge uniformly to the desired continuum limit. Furthermore we discover that for the standard model used there is a restrictive upper bound on how quickly @math must converge to zero as @math . We introduce a new model which is as simple as the original model, but overcomes this restriction.