Three Factors Influencing Minima in SGD

We study the properties of the endpoint of stochastic gradient descent (SGD). By approximating SGD as a stochastic differential equation (SDE) we consider the Boltzmann-Gibbs equilibrium distribution of that SDE under the assumption of isotropic variance in loss gradients. Through this analysis, we find that three factors – learning rate, batch size and the variance of the loss gradients – control the trade-off between the depth and width of the minima found by SGD, with wider minima favoured by a higher ratio of learning rate to batch size. We have direct control over the learning rate and batch size, while the variance is determined by the choice of model architecture, model parameterization and dataset. In the equilibrium distribution only the ratio of learning rate to batch size appears, implying that the equilibrium distribution is invariant under a simultaneous rescaling of learning rate and batch size by the same amount. We then explore experimentally how learning rate and batch size affect SGD from two perspectives: the endpoint of SGD and the dynamics that lead up to it. For the endpoint, the experiments suggest the endpoint of SGD is invariant under simultaneous rescaling of batch size and learning rate, and also that a higher ratio leads to flatter minima, both findings are consistent with our theoretical analysis. We note experimentally that the dynamics also seem to be invariant under the same rescaling of learning rate and batch size, which we explore showing that one can exchange batch size and learning rate for cyclical learning rate schedule. Next, we illustrate how noise affects memorization, showing that high noise levels lead to better generalization. Finally, we find experimentally that the invariance under simultaneous rescaling of learning rate and batch size breaks down if the learning rate gets too large or the batch size gets too small.