A Bayesian Perspective on Generalization and Stochastic Gradient Descent

This paper tackles two related questions at the heart of machine learning; how can we predict if a minimum will generalize to the test set, and why does stochastic gradient descent find minima that generalize well? Our work is inspired by Zhang et al. (2017), who showed deep networks can easily memorize randomly labeled training data, despite generalizing well when shown real labels of the same inputs. We show here that the same phenomenon occurs in small linear models. These observations are explained by evaluating the Bayesian evidence, which penalizes sharp minima but is invariant to model parameterization. We also explore the "generalization gap" between small and large batch training, identifying an optimum batch size which maximizes the test set accuracy. Interpreting stochastic gradient descent as a stochastic differential equation, we identify a "noise scale" $g = \epsilon (\frac{N}{B} - 1) \approx \epsilon N/B$, where $\epsilon$ is the learning rate, $N$ training set size and $B$ batch size. Consequently the optimum batch size is proportional to the learning rate and the training set size, $B_{opt} \propto \epsilon N$. We verify these predictions empirically.