Stochastic gradient descent (SGD) has become a cornerstone of neural network optimization, yet the noise introduced by SGD is often assumed to be uncorrelated over time, despite the ubiquity of epoch-based training. In this work, we challenge this assumption and investigate the effects of epoch-based noise correlations on the stationary distribution of discrete-time SGD with momentum, limited to a quadratic loss. Our main contributions are twofold: first, we calculate the exact autocorrelation of the noise for training in epochs under the assumption that the noise is independent of small fluctuations in the weight vector; second, we explore the influence of correlations introduced by the epoch-based learning scheme on SGD dynamics. We find that for directions with a curvature greater than a hyperparameter-dependent crossover value, the results for uncorrelated noise are recovered. However, for relatively flat directions, the weight variance is significantly reduced. We provide an intuitive explanation for these results based on a crossover between correlation times, contributing to a deeper understanding of the dynamics of SGD in the presence of epoch-based noise correlations.

本文研究了离散时间下具有动量的SGD的时域白噪声的自相关，并研究了epoch-based噪声相关性对于SGD的影响，结果表明对于大于超参数相关值的曲率方向，可以恢复无关噪声的结果，但对于相对平坦的方向，权重方差显著降低。

基于时代的随机梯度下降中的相关噪声: 对权重方差的影响