  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, and find that SGD noise is anti-correlated in time.
Second, we explore the influence of these anti-correlations 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, and our variance prediction leads to a considerable
reduction in loss fluctuations as compared to the constant weight variance
assumption.
