Seminar on Math, Stat, and AI: Hong Hu

The Seminar on Math, Stat, and AI is an interdisciplinary seminar series focusing on problems at the intersection of statistics, probability, artificial intelligence and related fields.The upcoming seminar takes place on Friday, April 3rd at 10:30am in Maxwell-Dworkin G125. This week's speaker will be Hong Hu, assistant professor at the Courant Institute of Mathematical Sciences of NYU.

Title: Statistical Inference with De-biased Estimator in the Inconsistency Regime: Non-Gaussian Covariate and Unknown Covariance

Abstract: Statistical inference of a low-dimensional target in the high-dimensional regression is of great fundamental importance and practical interests. The problem can become challenging when either the unknown parameter $\beta$ or the covariance matrix $\Sigma$ of the covariates cannot be estimated consistently.

In this paper, we study the statistical inference of a linear projection $a^\top \beta$ of the unknown parameter in the high-dimensional linear model based on the idea of de-biasing a biased regularized estimator. Our focus is on the linear asymptotic regime: $n/p \asymp 1$ and we do not impose sparsity assumptions on $\beta$ or $\Sigma^{-1}$, which places us in the aforementioned inconsistency regime. Prior analyses of debiased estimator in this regime have relied on two strong assumptions: (i) Gaussian designs and (ii) known covariance, to construct de-biased estimators and establish asymptotic normality guarantee. In contrast, we establish the asymptotic normality of a de-biased ridge estimator under a general ensemble of design matrices beyond Gaussian. Moreover, in the semi-supervised setting with access to additional unlabeled covariate data, we demonstrate that valid statistical inference is attainable without requiring the exact knowledge of the covariance.