A Bayesian Perspective on Generalization and Stochastic Gradient Descent

Samuel L. Smith and Quoc V. Le

Feb 15, 2018 (modified: Feb 15, 2018) ICLR 2018 Conference Blind Submission readers: everyone Show Bibtex
  • Abstract: We consider two 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 responds to \citet{zhang2016understanding}, who showed deep neural networks can easily memorize randomly labeled training data, despite generalizing well on real labels of the same inputs. We show that the same phenomenon occurs in small linear models. These observations are explained by the Bayesian evidence, which penalizes sharp minima but is invariant to model parameterization. We also demonstrate that, when one holds the learning rate fixed, there is an optimum batch size which maximizes the test set accuracy. We propose that the noise introduced by small mini-batches drives the parameters towards minima whose evidence is large. Interpreting stochastic gradient descent as a stochastic differential equation, we identify the ``noise scale" $g = \epsilon (\frac{N}{B} - 1) \approx \epsilon N/B$, where $\epsilon$ is the learning rate, $N$ the training set size and $B$ the batch size. Consequently the optimum batch size is proportional to both the learning rate and the size of the training set, $B_{opt} \propto \epsilon N$. We verify these predictions empirically.
  • TL;DR: Generalization is strongly correlated with the Bayesian evidence, and gradient noise drives SGD towards minima whose evidence is large.
  • Keywords: generalization, stochastic gradient descent, stochastic differential equations, scaling rules, large batch training, bayes theorem, batch size