  Online statistical inference facilitates real-time analysis of sequentially
collected data, making it different from traditional methods that rely on
static datasets. This paper introduces a novel approach to online inference in
high-dimensional generalized linear models, where we update regression
coefficient estimates and their standard errors upon each new data arrival. In
contrast to existing methods that either require full dataset access or
large-dimensional summary statistics storage, our method operates in a
single-pass mode, significantly reducing both time and space complexity. The
core of our methodological innovation lies in an adaptive stochastic gradient
descent algorithm tailored for dynamic objective functions, coupled with a
novel online debiasing procedure. This allows us to maintain low-dimensional
summary statistics while effectively controlling optimization errors introduced
by the dynamically changing loss functions. We demonstrate that our method,
termed the Approximated Debiased Lasso (ADL), not only mitigates the need for
the bounded individual probability condition but also significantly improves
numerical performance. Numerical experiments demonstrate that the proposed ADL
method consistently exhibits robust performance across various covariance
matrix structures.
