Defined the loss, now we’ll have puro compute its gradient respect to the output neurons of the CNN sopra 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 nulla. 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 punteggio of \(C_p\) (\(s_p\)) is durante 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 preciso multi-class classification.

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

> Skip this part if you are not interested in Facebook or me using https://datingranking.net/it/chatstep-review/ Softmax Loss for multi-label classification, which is not canone.

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

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

## Forward pass: Loss computation

We first compute Softmax activations for each class and abri 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 esatto introduce class balancing. The batch loss will be the mean loss of the elements per the batch. We then save the momento_loss puro display it and the probs sicuro use them con the backward pass.

## Backward pass: Gradients computation

Sopra the backward pass we need preciso 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 sicuro probs, we assign probs values onesto delta. For the positive classes con \(M\) we subtract 1 to the corresponding probs value and use scale_factor sicuro incontro the gradient expression. We compute the mean gradients of all the batch preciso run the backpropagation.

## Binary Ciclocampestre-Entropy Loss

Also called Sigmoid Ciclocampestre-Entropy loss. It is a 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 to a insecable 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 sopra \(C\), as explained above. So when using this Loss, the formulation of Cross Entroypy Loss for binary problems is often used: