I have read a similar question here: 1 neuron BCE loss VS 2 neurons CE loss that suggests there is no difference between softmax cross entropy loss and binary cross entropy loss, when choosing between two categories, since we can use complementary probability of a bernoulli distribution q = 1-p.
In practice, there is a difference because of different activation functions: BCE loss uses sigmoid activation, whereas CE loss uses softmax activation. CE(Softmax(X),Y)[0] ≠ BCE(Sigmoid(X[0]),Y[0]) $X,Y\in\mathbb{R}^{1\times2}$ for predictions and labels respectively. The other nuance is that the number of neurons in the final layer.
Although we could discard the information from the second neuron after applying softmax during inference, since it's just complementary of first... If the number of neurons in the hidden layer is 40 followed by 2 in the final layer we get 80 weight and 2 bias parameters, whereas the second diagram yield only 40 + 1.
If you were to design a neural network and choose between the two, what would you advise? I suspect the results would be similar but not exactly equal. The author of a linked question might have implemented it incorrectly if that is not the case. Yet, I'm still not quite sure about the difference in practice.
