Defined the loss, now we’ll have puro compute its gradient respect to the output neurons of the CNN mediante order onesto 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 sopra the nominator.

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

Sopra this Facebook sistema they claim that, despite being counter-intuitive, Categorical Cross-Entropy loss, or Softmax loss worked better than Binary Ciclocross-Entropy loss in 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 a multi-label scenario, the gradients get a 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 \(s_p\) in \(M\) is the CNN score for each positive class. As per Facebook paper, I introduce verso scaling factor \(1/M\) sicuro make the loss invariant to 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 durante the forward and backward passes of the layer:

## Forward pass: Loss computation

We first compute Softmax activations for each class and cloison them sopra probs. Then we compute the loss for each image mediante 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 onesto introduce class balancing. The batch loss will be the mean loss of the elements in the batch. We then chatib save the momento_loss puro schermo it and the probs puro use them in the backward pass.

## Backward pass: Gradients computation

Con the backward pass we need sicuro compute the gradients of each element of the batch respect to 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 esatto sbocco. For the positive classes in \(M\) we subtract 1 preciso the corresponding probs value and use scale_factor sicuro confronto the gradient expression. We compute the mean gradients of all the batch esatto run the backpropagation.

## Binary Cross-Entropy Loss

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