# So we need to compute the gradient of CE Loss respect each CNN class score sopra \(s\)

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 score of \(C_p\) (\(s_p\)) is con the nominator.

- Caffe: SoftmaxWithLoss Layer. Is limited to multi-class classification.
- Pytorch: CrossEntropyLoss. Is limited esatto multi-class classification.
- TensorFlow: softmax_cross_entropy. Is limited esatto multi-class classification.

Per this Facebook work they claim that, despite being counter-intuitive, Categorical Ciclocampestre-Entropy loss, or Softmax loss worked better than Binary Ciclocampestre-Entropy loss durante their multi-label classification problem.

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

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

Where each \(s_p\) con \(M\) is the CNN conteggio for each positive class. As sopra Facebook paper, I introduce verso scaling factor \(1/M\) sicuro make the loss invariant preciso 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 per the forward and backward passes of the layer:

## Forward pass: Loss computation

We first compute Softmax activations for each class and paravent them in probs. Then we compute the loss for each image sopra 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 sicuro introduce class balancing. The batch loss will be the mean loss of the elements mediante the batch https://datingranking.net/it/angelreturn-review/. We then save the giorno_loss to monitor it and the probs to use them sopra the backward pass.

## Backward pass: Gradients computation

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

## Binary Ciclocross-Entropy Loss

Also called Sigmoid Ciclocross-Entropy loss. It is a Sigmoid activation plus a 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 onesto verso excretion class should not influence the decision for another class. It’s called Binary Ciclocampestre-Entropy Loss because it sets up a binary classification problem between \(C’ = 2\) classes for every class per \(C\), as explained above. So when using this Loss, the formulation of Cross Entroypy Loss for binary problems is often used: