  We propose sparsemax, a new activation function similar to the traditional
softmax, but able to output sparse probabilities. After deriving its
properties, we show how its Jacobian can be efficiently computed, enabling its
use in a network trained with backpropagation. Then, we propose a new smooth
and convex loss function which is the sparsemax analogue of the logistic loss.
We reveal an unexpected connection between this new loss and the Huber
classification loss. We obtain promising empirical results in multi-label
classification problems and in attention-based neural networks for natural
language inference. For the latter, we achieve a similar performance as the
traditional softmax, but with a selective, more compact, attention focus.
