A standard cross-entropy loss function can be applied to soft labels. Training (examples https://oflameron.imgbb.com/ ) by the maximum likelihood method with a softmax classifier and hard labels may never converge - softmax cannot predict a probability exactly equal to 0 or 1, so it will endlessly continue training larger and larger weights, making more and more extreme predictions. http://shmeleff.com Extensive experience in creating data sets for final tuning of neural networks.