Defined the loss, now we’ll have puro compute its gradient respect puro the output neurons of the CNN mediante order preciso 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 conteggio of \(C_p\) (\(s_p\)) is mediante the nominator.
- Caffe: SoftmaxWithLoss Layer. Is limited to multi-class classification.
- Pytorch: CrossEntropyLoss. Is limited sicuro multi-class classification.
- TensorFlow: softmax_cross_entropy. Is limited to multi-class classification.
Mediante this Facebook work they claim that, despite being counter-intuitive, Categorical Cross-Entropy loss, or Softmax loss worked better than Binary Ciclocampestre-Entropy loss mediante their multi-label classification problem.
> Skip this part if you are not interested durante Facebook or me using Softmax Loss for multi-label classification, which is not standard.
When Softmax loss is used is a multi-label cornice, the gradients get verso bit more complex, since the loss contains an element for each https://datingranking.net/it/charmdate-review positive class. Consider \(M\) are the positive classes of a sample. The CE Loss with Softmax activations would be:
Where each \(s_p\) sopra \(M\) is the CNN risultato for each positive class. As con Facebook paper, I introduce per scaling factor \(1/M\) onesto 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 per the forward and backward passes of the layer:
Forward pass: Loss computation
We first compute Softmax activations for each class and abri them durante 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 puro introduce class balancing. The batch loss will be the mean loss of the elements in the batch. We then save the momento_loss sicuro display it and the probs sicuro use them in the backward pass.
Backward pass: Gradients computation
Con the backward pass we need esatto compute the gradients of each element of the batch respect puro each one of the classes scores \(s\). As the gradient for all the classes \(C\) except positive classes \(M\) is equal esatto probs, we assign probs values to delta. For the positive classes sopra \(M\) we subtract 1 preciso the corresponding probs value and use scale_factor preciso scontro the gradient expression. We compute the mean gradients of all the batch to run the backpropagation.
Binary Ciclocross-Entropy Loss
Also called Sigmoid Ciclocross-Entropy loss. It is per Sigmoid activation plus verso Cross-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 a un class should not influence the decision for another class. It’s called Binary Ciclocross-Entropy Loss because it sets up verso binary classification problem between \(C’ = 2\) classes for every class durante \(C\), as explained above. So when using this Loss, the formulation of Ciclocampestre Entroypy Loss for binary problems is often used: