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1. A note on centering in subsample selection for linear regression

   https://doi.org/10.1002/sta4.525  

   Wang, HaiYing (December 2022, Stat)

   Centring is a commonly used technique in linear regression analysis. With centred data on both the responses and covariates, the ordinary least squares estimator of the slope parameter can be calculated from a model without the intercept. If a subsample is selected from a centred full data, the subsample is typically uncentred. In this case, is it still appropriate to fit a model without the intercept? The answer is yes, and we show that the least squares estimator on the slope parameter obtained from a model without the intercept is unbiased and it has a smaller variance covariance matrix in the Loewner order than that obtained from a model with the intercept. We further show that for noninformative weighted subsampling when a weighted least squares estimator is used, using the full data weighted means to relocate the subsample improves the estimation efficiency. 

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2. Augmented balancing weights as linear regression

   https://doi.org/10.1093/jrsssb/qkaf019  

   Bruns-Smith, David; Dukes, Oliver; Feller, Avi; Ogburn, Elizabeth L (April 2025, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract We provide a novel characterization of augmented balancing weights, also known as automatic debiased machine learning. These popular doubly robust estimators combine outcome modelling with balancing weights—weights that achieve covariate balance directly instead of estimating and inverting the propensity score. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine those of the original outcome model with those from unpenalized ordinary least-squares (OLS). Under certain choices of regularization parameters, the augmented estimator in fact collapses to the OLS estimator alone. We then extend these results to specific outcome and weighting models. We first show that the augmented estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression—implying a novel analysis of undersmoothing. When the weighting model is instead lasso-penalized, we demonstrate a familiar ‘double selection’ property. Our framework opens the black box on this increasingly popular class of estimators, bridges the gap between existing results on the semiparametric efficiency of undersmoothed and doubly robust estimators, and provides new insights into the performance of augmented balancing weights. 

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3. Sequential Domain Adaptation by Synthesizing Distributionally Robust Experts

   Taskesen, Bahar; Yue, Man-Chung; Blanchet, Jose; Kuhn, Daniel; Nguyen, Viet Anh (January 2021, Proceedings of the 38th International Conference on Machine Learning)

   Meila, Marina and (Ed.)

   Least squares estimators, when trained on a few target domain samples, may predict poorly. Supervised domain adaptation aims to improve the predictive accuracy by exploiting additional labeled training samples from a source distribution that is close to the target distribution. Given available data, we investigate novel strategies to synthesize a family of least squares estimator experts that are robust with regard to moment conditions. When these moment conditions are specified using Kullback-Leibler or Wasserstein-type divergences, we can find the robust estimators efficiently using convex optimization. We use the Bernstein online aggregation algorithm on the proposed family of robust experts to generate predictions for the sequential stream of target test samples. Numerical experiments on real data show that the robust strategies systematically outperform non-robust interpolations of the empirical least squares estimators. 

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4. Drawing inferences for High‐dimensional Linear Models: A Selection‐assisted Partial Regression and Smoothing Approach

   Zhei, Fei Zhu (January 2018, Biometrics)

   Inference for high-dimensional models is challenging as regular asymptotic the- ories are not applicable. This paper proposes a new framework of simultaneous estimation and inference for high-dimensional linear models. By smoothing over par- tial regression estimates based on a given variable selection scheme, we reduce the problem to a low-dimensional least squares estimation. The procedure, termed as Selection-assisted Partial Regression and Smoothing (SPARES), utilizes data split- ting along with variable selection and partial regression. We show that the SPARES estimator is asymptotically unbiased and normal, and derive its variance via a non- parametric delta method. The utility of the procedure is evaluated under various simulation scenarios and via comparisons with the de-biased LASSO estimators, a major competitor. We apply the method to analyze two genomic datasets and obtain biologically meaningful results. 

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5. Distributed Least Squares in Small Space via Sketching and Bias Reduction

   Garg, Sachin; Tan, Kevin; Dereziński, Michał (December 2024, Advances in Neural Information Processing Systems (NeurIPS 2024))

   Matrix sketching is a powerful tool for reducing the size of large data matrices. Yet there are fundamental limitations to this size reduction when we want to recover an accurate estimator for a task such as least square regression. We show that these limitations can be circumvented in the distributed setting by designing sketching methods that minimize the bias of the estimator, rather than its error. In particular, we give a sparse sketching method running in optimal space and current matrix multiplication time, which recovers a nearly-unbiased least squares estimator using two passes over the data. This leads to new communication-efficient distributed averaging algorithms for least squares and related tasks, which directly improve on several prior approaches. Our key novelty is a new bias analysis for sketched least squares, giving a sharp characterization of its dependence on the sketch sparsity. The techniques include new higher moment restricted Bai-Silverstein inequalities, which are of independent interest to the non-asymptotic analysis of deterministic equivalents for random matrices that arise from sketching. 

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