Defined the loss, now we’ll have puro compute its gradient respect onesto the output neurons of the CNN durante 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 niente. 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 per the nominator.

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

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

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

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 per sample. The CE Loss with Softmax activations would be:

Where each \(s_p\) con \(M\) is the CNN punteggio for each positive class. As durante Facebook paper, I introduce per scaling factor \(1/M\) preciso make the loss invariant onesto the number of positive classes, which ple.

As Caffe Softmax with Loss https://datingranking.net/it/japan-cupid-review/ 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 paravent them con 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 sicuro introduce class balancing. The batch loss will be the mean loss of the elements sopra the batch. We then save the momento_loss preciso schermo it and the probs puro use them mediante the backward pass.

## Backward pass: Gradients computation

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

## Binary Ciclocampestre-Entropy Loss

Also called Sigmoid Ciclocampestre-Entropy loss. It is verso Sigmoid activation plus a 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 onesto verso indivisible 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 Ciclocross Entroypy Loss for binary problems is often used: