Defined the loss, now we’ll have sicuro compute its gradient respect onesto the output neurons of the CNN con order to 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 per the nominator.
- Caffe: SoftmaxWithLoss Layer. Is limited to multi-class classification.
- Pytorch: CrossEntropyLoss. Is limited onesto multi-class classification.
- TensorFlow: softmax_cross_entropy. Is limited preciso multi-class classification.
Per this Facebook rete di emittenti they claim that, despite being counter-intuitive, Categorical Cross-Entropy loss, or Softmax loss worked better than Binary Ciclocross-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 norma.
When Softmax loss is used is a multi-label campo, the gradients get verso 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 \(s_p\) con \(M\) is the CNN conteggio for each positive class. As con Facebook paper, I introduce verso scaling factor \(1/M\) onesto make the loss invariant puro 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 in the forward and backward passes of the layer:
Forward pass: Loss computation
We first compute Softmax activations for each class and panneau them con probs. Then we compute the loss for each image in 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 con the batch. We then save the giorno_loss sicuro video it and the probs to use them mediante the backward pass.
Backward pass: Gradients computation
Sopra the backward pass we need puro 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 esatto probs, we assign probs values sicuro delta. For the positive classes mediante \(M\) we subtract 1 onesto the corresponding probs value and use scale_factor onesto confronto the gradient expression. We compute the mean gradients of all the batch sicuro run the backpropagation.
Binary Ciclocross-Entropy Loss
Also called Sigmoid Ciclocampestre-Entropy loss. It is a Sigmoid activation plus per 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 sicuro a certain class should not influence the decision for another class. It’s called Binary Cross-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:
