A Bayesian Perspective on Generalization and Stochastic Gradient Descent

TL;DR

This paper investigates why stochastic gradient descent finds minima that generalize well, revealing that the Bayesian evidence favors flat minima and that an optimal batch size exists proportional to learning rate and dataset size.

Contribution

It introduces a Bayesian explanation for generalization in neural networks and derives a formula for the optimal batch size based on noise scale, validated empirically.

Findings

Bayesian evidence favors flat minima for better generalization.

An optimal batch size proportional to learning rate and dataset size maximizes test accuracy.

Empirical validation confirms the theoretical predictions.

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 Zhang et al. (2016), 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.