What is a good objective function for allowing close to 0 predictions?
Please be aware that neural networks are normally badly calibrated. Essentially this means that for a binary classification, neural networks are good at keeping the prediction score of a sample in the right area (above 50% or below 50%, depending on the class), but the actual values do not have to represent plausible probabilites of the real world.
There is a lot of research going on about how to calibrate neural networks, e.g. this approach.