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.

翻译：本文研究了在马尔可夫采样下，随机梯度下降（SGD）的在线重叠批均值协方差估计量。在状态依赖和状态独立马尔可夫采样下，分别建立了 $O\big(\sqrt{d}\,n^{-1/8}(\log n)^{1/4}\big)$ 和 $O\big(\sqrt{d}\,n^{-1/8}\big)$ 阶的收敛速率，其中 $d$ 为维度，$n$ 表示观测值或SGD迭代次数。这些速率与独立同分布（i.i.d）数据的最佳已知收敛速率相匹配。我们的分析克服了马尔可夫采样带来的重大挑战，由此引入了额外的误差项以及批均值协方差估计量块之间的复杂依赖关系。此外，我们在状态依赖马尔可夫数据下建立了SGD动力学误差 $\ell_2$ 范数前四阶矩的收敛速率，该结果本身具有独立的研究价值。数值实验为线性回归和逻辑回归模型在马尔可夫采样下的SGD提供了置信区间。同时，我们的方法被应用于逻辑回归的对抗性分类场景，其中对抗方在训练过程中自适应地修改特征以影响目标类别的分类结果。