We study the dynamics of a continuous-time model of the Stochastic Gradient Descent (SGD) for the least-square problem. Indeed, pursuing the work of Li et al. (2019), we analyze Stochastic Differential Equations (SDEs) that model SGD either in the case of the training loss (finite samples) or the population one (online setting). A key qualitative feature of the dynamics is the existence of a perfect interpolator of the data, irrespective of the sample size. In both scenarios, we provide precise, non-asymptotic rates of convergence to the (possibly degenerate) stationary distribution. Additionally, we describe this asymptotic distribution, offering estimates of its mean, deviations from it, and a proof of the emergence of heavy-tails related to the step-size magnitude. Numerical simulations supporting our findings are also presented.

我们研究了最小二乘问题的连续时间随机梯度下降（SGD）模型的动力学。我们通过分析随机微分方程(SDE)，在训练损失（有限样本）或总体损失（在线设置）的情况下建模SGD来追求Li等人(2019)的研究成果。该动力学的一个关键特征是无论样本大小如何，都存在与数据完美插值器。在这两种情况下，我们提供了收敛到（可能退化的）稳态分布的精确非渐近速率。此外，我们描述了渐近分布，给出了其均值、与之偏差的估计，并证明了与步长大小有关的重尾现象的出现。我们还呈现了支持我们发现的数值模拟结果。

最小二乘随机梯度下降的随机微分方程模型