Online covariance estimation for stochastic gradient descent under Markovian sampling

We investigate the online overlapping batch-means covariance estimator for Stochastic Gradient Descent (SGD) under Markovian sampling. Convergence rates of order $O\big(\sqrt{d}\,n^{-1/8}(\log n)^{1/4}\big)$ and $O\big(\sqrt{d}\,n^{-1/8}\big)$ are established under state-dependent and state-independent Markovian sampling, respectively, where $d$ is the dimensionality and $n$ denotes observations or SGD iterations. These rates match the best-known convergence rate for independent and identically distributed (i.i.d) data. Our analysis overcomes significant challenges that arise due to Markovian sampling, leading to the introduction of additional error terms and complex dependencies between the blocks of the batch-means covariance estimator. Moreover, we establish the convergence rate for the first four moments of the $\ell_2$ norm of the error of SGD dynamics under state-dependent Markovian data, which holds potential interest as an independent result. Numerical illustrations provide confidence intervals for SGD in linear and logistic regression models under Markovian sampling. Additionally, our method is applied to the strategic classification with logistic regression, where adversaries adaptively modify features during training to affect target class classification.