  Generalization is one of the most important problems in deep learning (DL).
In the overparameterized regime in neural networks, there exist many low-loss
solutions that fit the training data equally well. The key question is which
solution is more generalizable. Empirical studies showed a strong correlation
between flatness of the loss landscape at a solution and its generalizability,
and stochastic gradient descent (SGD) is crucial in finding the flat solutions.
To understand how SGD drives the learning system to flat solutions, we
construct a simple model whose loss landscape has a continuous set of
degenerate (or near degenerate) minima. By solving the Fokker-Planck equation
of the underlying stochastic learning dynamics, we show that due to its strong
anisotropy the SGD noise introduces an additional effective loss term that
decreases with flatness and has an overall strength that increases with the
learning rate and batch-to-batch variation. We find that the additional
landscape-dependent SGD-loss breaks the degeneracy and serves as an effective
regularization for finding flat solutions. Furthermore, a stronger SGD noise
shortens the convergence time to the flat solutions. However, we identify an
upper bound for the SGD noise beyond which the system fails to converge. Our
results not only elucidate the role of SGD for generalization they may also
have important implications for hyperparameter selection for learning
efficiently without divergence.
