How is the error calculated with multiple output neurons in the neural network? Machine Learning books generally explains that the error calculated for a given sample $i$ is:
$e_i = y_i - \hat{y_i}$
Where $\hat{y}$ is the target output and $y$ is the actual output given by the network. So, a loss function $L$ is calculated:
$L = \frac{1}{2N}\sum^{N}_{i=1}(e_i)^2$
The above scenario is explained for a binary classification/regression problem. Now, let's assume a MLP network with $m$ neurons in the output layer for a multiclass classification problem (generally one neuron per class).
What changes in the equations above? Since we now have multiple outputs, both $e_i$ and $y_i$ should be a vector?
Assuming you're using softmax on the last layer for classification, it sounds like a simple application of cross entropy loss from here on out: https://datascience.stackexchange.com/questions/20296/cross-entropy-loss-explanation
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