So we need onesto compute the gradient of CE Loss respect each CNN class punteggio durante \(s\)
Defined the loss, now we’ll have puro compute its gradient respect sicuro the output neurons of the CNN in order sicuro backpropagate it through the net and optimize the defined loss function tuning the net parameters. The loss terms coming from the negative classes are zero. However, the loss gradient respect those negative classes is not cancelled, since the Softmax of the positive class also depends on the negative classes scores.

The gradient expression will be the same for all \(C\) except for the ground truth class \(C_p\), because the conteggio of \(C_p\) (\(s_p\)) is in the nominator.

- Caffe: SoftmaxWithLoss Layer. Is limited sicuro multi-class classification.
- Pytorch: CrossEntropyLoss. Is limited onesto multi-class classification.
- TensorFlow: softmax_cross_entropy. Is limited preciso multi-class classification.

In this Facebook rete di emittenti they claim that, despite being counter-intuitive, Categorical Ciclocampestre-Entropy loss, or Softmax loss worked better than Binary Ciclocross-Entropy loss con their multi-label classification problem.

> Skip this part if you are not interested sopra Facebook or me using Softmax Loss for multi-label classification, which is not standard.

When Softmax loss is used is per multi-label cornice, the gradients get per bit more complex, since the loss contains an element for each positive class. Consider \(M\) are the positive classes of a sample. The CE Loss with Softmax activations would be:

Where each \(s_p\) per \(M\) is the CNN risultato for each positive class. As mediante Facebook paper, I introduce verso scaling factor \(1/M\) esatto make the loss invariant puro the number of positive classes, which ple.

As Caffe Softmax with Loss layer nor Multinomial Logistic Loss Layer accept multi-label targets, I implemented my own PyCaffe Softmax loss layer, following the specifications of the Facebook paper. Caffe python layers let’s us easily customize the operations done mediante the forward and backward passes of the layer:

## Forward pass: Loss computation

We first compute Softmax activations for each class and cloison them in probs. Then we compute the loss for each image per the batch considering there might be more than one positive label. We use an scale_factor (\(M\)) and we also multiply losses by the labels, which can be binary or real numbers, so they can be used for instance esatto introduce class balancing. The batch loss will be the mean loss of the elements durante the batch. We then save the momento_loss to monitor it and the probs preciso use them per the backward pass.

## Backward pass: Gradients computation

Per the backward pass we need puro compute the gradients of each element of the batch respect onesto each one of the classes scores \(s\). As the gradient for all the classes \(C\) except positive classes \(M\) is equal onesto probs, we assign probs values puro sbocco. For the positive classes con \(M\) we subtract 1 puro the corresponding probs value and use scale_factor sicuro confronto the gradient expression. We compute the mean gradients of all the batch preciso run the backpropagation.

## Binary Ciclocross-Entropy Loss

Also called Sigmoid Ciclocross-Entropy loss. It is verso Sigmoid activation plus verso Ciclocampestre-Entropy loss. Unlike Softmax loss it is independent for each vector component (class), meaning that the loss computed for every CNN output vector component is not affected by other component values. That’s why it is used for multi-label classification, were the insight of an element belonging puro a un class should not influence the decision for another class. It’s called Binary Ciclocross-Entropy Loss because it sets up a binary classification problem between \(C’ = 2\) classes for every class in \(C\), as explained above. So when using this Loss, the formulation of Cross Entroypy Loss for binary problems is often used: