  We take a Bayesian perspective to illustrate a connection between training
speed and the marginal likelihood in linear models. This provides two major
insights: first, that a measure of a model's training speed can be used to
estimate its marginal likelihood. Second, that this measure, under certain
conditions, predicts the relative weighting of models in linear model
combinations trained to minimize a regression loss. We verify our results in
model selection tasks for linear models and for the infinite-width limit of
deep neural networks. We further provide encouraging empirical evidence that
the intuition developed in these settings also holds for deep neural networks
trained with stochastic gradient descent. Our results suggest a promising new
direction towards explaining why neural networks trained with stochastic
gradient descent are biased towards functions that generalize well.
