Early Stopping as Nonparametric Variational Inference

We show that unconverged stochastic gradient descent can be interpreted as sampling from a nonparametric approximate posterior distribution. This distribution is implicitly defined by the transformation of an initial distribution by a sequence of optimization steps. By tracking the change in entropy of this distribution during optimization, we give a scalable, unbiased estimate of a variational lower bound on the log marginal likelihood. This bound can be used to optimize hyperparameters instead of cross-validation. This Bayesian interpretation of SGD also suggests new overfitting-resistant optimization procedures, and gives a theoretical foundation for early stopping and ensembling. We investigate the properties of this marginal likelihood estimator on neural network models.