  In a multi-class classification problem, it is standard to model the output
of a neural network as a categorical distribution conditioned on the inputs.
The output must therefore be positive and sum to one, which is traditionally
enforced by a softmax. This probabilistic mapping allows to use the maximum
likelihood principle, which leads to the well-known log-softmax loss. However
the choice of the softmax function seems somehow arbitrary as there are many
other possible normalizing functions. It is thus unclear why the log-softmax
loss would perform better than other loss alternatives. In particular Vincent
et al. (2015) recently introduced a class of loss functions, called the
spherical family, for which there exists an efficient algorithm to compute the
updates of the output weights irrespective of the output size. In this paper,
we explore several loss functions from this family as possible alternatives to
the traditional log-softmax. In particular, we focus our investigation on
spherical bounds of the log-softmax loss and on two spherical log-likelihood
losses, namely the log-Spherical Softmax suggested by Vincent et al. (2015) and
the log-Taylor Softmax that we introduce. Although these alternatives do not
yield as good results as the log-softmax loss on two language modeling tasks,
they surprisingly outperform it in our experiments on MNIST and CIFAR-10,
suggesting that they might be relevant in a broad range of applications.
