We establish sample complexity guarantees for estimating the covariance matrix of a strongly log-concave smooth distribution using the unadjusted Langevin algorithm (ULA). We quantitatively compare our complexity estimates on single-chain ULA with embarrassingly parallel ULA and derive that the sample complexity of the single-chain approach is smaller than that of embarrassingly parallel ULA by a logarithmic factor in the dimension and the reciprocal of the prescribed precision, with the difference arising from effective bias reduction through burn-in. The key technical contribution is a concentration bound for the sample covariance matrix around its expectation, derived via a log-Sobolev inequality for the joint distribution of ULA iterates.

翻译：本文针对使用非调整朗之万算法估计强对数凹光滑分布协方差矩阵的问题，建立了样本复杂度的理论保证。我们定量比较了单链ULA与完全并行ULA的复杂度估计，并推导出在维度与预设精度倒数的对数因子意义上，单链方法的样本复杂度低于完全并行ULA，这一差异源于预热阶段带来的有效偏差缩减。核心理论贡献是通过ULA迭代值联合分布的对数索伯列夫不等式，推导出样本协方差矩阵围绕其期望值的集中界。