Effectiveness of MPC-friendly Softmax Replacement

Softmax is widely used in deep learning to map some representation to a probability distribution. As it is based on exp/log functions that is 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.