Statistical Inference for Linear Functionals of Online Least-squares SGD when $t \gtrsim d^{1+\delta}$

Abstract

Stochastic Gradient Descent (SGD) has become a cornerstone method in modern data science. However, deploying SGD in high-stakes applications necessitates rigorous quantification of its inherent uncertainty. In this work, we establish \emph{non-asymptotic Berry--Esseen bounds} for linear functionals of online least-squares SGD, thereby providing a Gaussian Central Limit Theorem (CLT) in a \emph{growing-dimensional regime}. Existing approaches to high-dimensional inference for projection parameters, such as~\cite{chang2023inference}, rely on inverting empirical covariance matrices and require at least $t \gtrsim d^{3/2}$ iterations to achieve finite-sample Berry--Esseen guarantees, rendering them computationally expensive and restrictive in the allowable dimensional scaling. In contrast, we show that a CLT holds for SGD iterates when the number of iterations grows as $t \gtrsim d^{1+\delta}$ for any $\delta > 0$, significantly extending the dimensional regime permitted by prior works while improving computational efficiency. The proposed online SGD-based procedure operates in $\mathcal{O}(td)$ time and requires only $\mathcal{O}(d)$ memory, in contrast to the $\mathcal{O}(td^2 + d^3)$ runtime of covariance-inversion methods. To render the theory practically applicable, we further develop an \emph{online variance estimator} for the asymptotic variance appearing in the CLT and establish \emph{high-probability deviation bounds} for this estimator. Collectively, these results yield the first fully online and data-driven framework for constructing confidence intervals for SGD iterates in the near-optimal scaling regime $t \gtrsim d^{1+\delta}$.