Implicit Regularization of SGD via Thermophoresis

A central ingredient in the impressive predictive performance of deep neural networks is optimization via stochastic gradient descent (SGD). While some theoretical progress has been made, the effect of SGD in neural networks is still unclear, especially during the early phase of training. Here we generalize the theory of thermophoresis from statistical mechanics and show that there exists an effective entropic force from SGD that pushes to reduce the gradient variance. We study this effect in detail in a simple two-layer model, where the thermophoretic force functions to decreases the weight norm and activation rate of the units. The strength of this effect is proportional to squared learning rate and inverse batch size, and is more effective during the early phase of training when the model's predictions are poor. Lastly we test our quantitative predictions with experiments on various models and datasets.