This is the NN 101 explanation: mean-square loss is for regression, cross-entropy is for classification. NN 201 explanation: mean-square is about finite but co

This is the NN 101 explanation: mean-square loss is for regression, cross-entropy is for classification. NN 201 explanation: mean-square is about finite but continuous errors (residuals) of predicted value vs true value of the output, cross-entropy is for distributions over discrete sets of categories. NN 501 explanation: the task of the NN and the form of its outputs should be defined in terms of the "shape" or nature of the residuals of its predictions. Mean-square corresponds to predicting means with Gaussian residuals, cross-entropy corresponds to predicting over discrete outputs with multinomial (mutually exclusive) structure. Indeed you can derive any loss function you want by first defining the expected form of the residuals and then deriving the negative log-likelihood of the associated distribution.