Is SGD a Bayesian sampler? Well, almost

Overparameterised deep neural networks (DNNs) are highly expressive and so can, in principle, generate almost any function that fits a training dataset with zero error. The vast majority of these functions will perform poorly on unseen data, and yet in practice DNNs often generalise remarkably well. This success suggests that a trained DNN must have a strong inductive bias towards functions with low generalisation error. Here we empirically investigate this inductive bias by calculating, for a range of architectures and datasets, the probability PSGD(fmid S) that an overparameterised DNN, trained with stochastic gradient descent (SGD) or one of its variants, converges on a function f consistent with a training set S. We also use Gaussian processes to estimate the Bayesian posterior probability PB(fmid S) that the DNN expresses f upon random sampling of its parameters, conditioned on S. Our main findings are that PSGD(fmid S) correlates remarkably well with PB(fmid S) and that PB(fmid S) is strongly biased towards low-error and low complexity functions. These results imply that strong inductive bias in the parameter-function map (which determines PB(fmid S)), rather than a special property of SGD, is the primary explanation for why DNNs generalise so well in the overparameterised regime. While our results suggest that the Bayesian posterior PB(fmid S) is the first order determinant of PSGD(fmid S), there remain second order differences that are sensitive to hyperparameter tuning. A function probability picture, based on PSGD(fmid S) and/or PB(fmid S), can shed new light on the way that variations in architecture or hyperparameter settings such as batch size, learning rate, and optimiser choice, affect DNN performance.