  The softmax function combined with a cross-entropy loss is a principled
approach to modeling probability distributions that has become ubiquitous in
deep learning. The softmax function is defined by a lone hyperparameter, the
temperature, that is commonly set to one or regarded as a way to tune model
confidence after training; however, less is known about how the temperature
impacts training dynamics or generalization performance. In this work we
develop a theory of early learning for models trained with
softmax-cross-entropy loss and show that the learning dynamics depend crucially
on the inverse-temperature $\beta$ as well as the magnitude of the logits at
initialization, $||\beta{\bf z}||_{2}$. We follow up these analytic results
with a large-scale empirical study of a variety of model architectures trained
on CIFAR10, ImageNet, and IMDB sentiment analysis. We find that generalization
performance depends strongly on the temperature, but only weakly on the initial
logit magnitude. We provide evidence that the dependence of generalization on
$\beta$ is not due to changes in model confidence, but is a dynamical
phenomenon. It follows that the addition of $\beta$ as a tunable hyperparameter
is key to maximizing model performance. Although we find the optimal $\beta$ to
be sensitive to the architecture, our results suggest that tuning $\beta$ over
the range $10^{-2}$ to $10^1$ improves performance over all architectures
studied. We find that smaller $\beta$ may lead to better peak performance at
the cost of learning stability.
