We establish sample complexity guarantees for estimating the covariance matrix of strongly log-concave smooth distributions 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：本文针对使用未调整朗之万算法（ULA）估计强对数凹平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与易并行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易并行ULA，这一差异源于通过老化过程实现的有效偏差削减。核心理论贡献在于：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。