Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regression

Stochastic gradient descent (SGD) has emerged as the quintessential method in a data scientist's toolbox. Using SGD for high-stakes applications requires, however, careful quantification of the associated uncertainty. Towards that end, in this work, we establish a high-dimensional Central Limit Theorem (CLT) for linear functionals of online SGD iterates for overparametrized least-squares regression with non-isotropic Gaussian inputs. Our result shows that a CLT holds even when the dimensionality is of order exponential in the number of iterations of the online SGD, which, to the best of our knowledge, is the first such result. In order to use the developed result in practice, we further develop an online approach for estimating the expectation and the variance terms appearing in the CLT, and establish high-probability bounds for the developed online estimator. Furthermore, we propose a two-step fully online bias-correction methodology which together with the CLT result and the variance estimation result, provides a fully online and data-driven way to numerically construct confidence intervals, thereby enabling practical high-dimensional algorithmic inference with SGD. We also extend our results to a class of single-index models, based on the Gaussian Stein's identity. We also provide numerical simulations to verify our theoretical findings in practice.