  Softmax is widely used in deep learning to map some representation to a
probability distribution. As it is based on exp/log functions that are
relatively expensive in multi-party computation, Mohassel and Zhang (2017)
proposed a simpler replacement based on ReLU to be used in secure computation.
However, we could not reproduce the accuracy they reported for training on
MNIST with three fully connected layers. Later works (e.g., Wagh et al., 2019
and 2021) used the softmax replacement not for computing the output probability
distribution but for approximating the gradient in back-propagation. In this
work, we analyze the two uses of the replacement and compare them to softmax,
both in terms of accuracy and cost in multi-party computation. We found that
the replacement only provides a significant speed-up for a one-layer network
while it always reduces accuracy, sometimes significantly. Thus we conclude
that its usefulness is limited and one should use the original softmax function
instead.
