Online covariance estimation for stochastic gradient descent under Markovian sampling

We study the online overlapping batch-means covariance estimator for Stochastic Gradient Descent (SGD) under Markovian sampling. We show that the convergence rates of the covariance estimator are $O\big(\sqrt{d}\,n^{-1/8}(\log n)^{1/4}\big)$ and $O\big(\sqrt{d}\,n^{-1/8}\big)$ under state-dependent and state-independent Markovian sampling, respectively, with $d$ representing dimensionality and $n$ denoting the number of observations or SGD iterations. Remarkably, these rates match the best-known convergence rate previously established for the independent and identically distributed ($\iid$) case by \cite{zhu2021online}, up to logarithmic factors. 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. To validate our theoretical findings, we provide numerical illustrations to derive confidence intervals for SGD when training linear and logistic regression models under Markovian sampling. Additionally, we apply our approach to tackle the intriguing problem of strategic classification with logistic regression, where adversaries can adaptively modify features during the training process to increase their chances of being classified in a specific target class.