Using BERT for co-reference resolving, what's the loss function?
NER is approached as a sequence-labeling problem and the end of the network, there is a categorical distribution estimated by softmax that is trained using cross-entropy loss.
The paper you are specifically asking about claims to be based on the End-to-end Neural Coreference Resolution paper that does something more tricky. They explicitly consider all spans to the entities spans and compute the probability for that. Nevertheless, when they get the probabilities they still do in principle the cross-entropy loss (cf. their code).