Correlated Noise in Epoch-Based Stochastic Gradient Descent: Implications for Weight Variances

Abstract: 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.