We introduce S++, a simple, robust, and deployable framework for training a neural network (NN) using private data from multiple sources, using secret-shared secure function evaluation. In short, consider a virtual third party to whom every data-holder sends their inputs, and which computes the neural network: in our case, this virtual third party is actually a set of servers which individually learn nothing, even with a malicious (but non-colluding) adversary. Previous work in this area has been limited to just one specific activation function: ReLU, rendering the approach impractical for many use-cases. For the first time, we provide fast and verifiable protocols for all common activation functions and optimize them for running in a secret-shared manner. The ability to quickly, verifiably, and robustly compute exponentiation, softmax, sigmoid, etc., allows us to use previously written NNs without modification, vastly reducing developer effort and complexity of code. In recent times, ReLU has been found to converge much faster and be more computationally efficient as compared to non-linear functions like sigmoid or tanh. However, we argue that it would be remiss not to extend the mechanism to non-linear functions such as the logistic sigmoid, tanh, and softmax that are fundamental due to their ability to express outputs as probabilities and their universal approximation property. Their contribution in RNNs and a few recent advancements also makes them more relevant.

Among the best practices for training a Neural Network is to normalize your data to obtain a mean close to 0. Normalizing the data generally speeds up learning and leads to faster convergence. Also, the (logistic) sigmoid function is hardly ever used anymore as an activation function in hidden layers of Neural Networks, because the tanh function (among others) seems to be strictly superior. While this might not be immediately evident, there are very similar reasons for why this is the case. The tanh function is quite similar to the logistic sigmoid. The main difference, however, is that the tanh function outputs results between -1 and 1, while the sigmoid function outputs values that are between 0 and 1 -- therefore they are always positive.