Defined the loss, now we’ll have onesto compute its gradient respect esatto the output neurons of the CNN durante order puro 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 durante the nominator.
- Caffe: SoftmaxWithLoss Layer. Is limited to multi-class classification.
- Pytorch: CrossEntropyLoss profilo caribbean cupid. Is limited puro multi-class classification.
- TensorFlow: softmax_cross_entropy. Is limited onesto multi-class classification.
Con this Facebook sistema they claim that, despite being counter-intuitive, Categorical Ciclocross-Entropy loss, or Softmax loss worked better than Binary Cross-Entropy loss durante their multi-label classification problem.
> Skip this part if you are not interested mediante Facebook or me using Softmax Loss for multi-label classification, which is not standard.
When Softmax loss is used is per multi-label ambiente, the gradients get verso 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 \(s_p\) sopra \(M\) is the CNN risultato for each positive class. As mediante Facebook paper, I introduce verso scaling factor \(1/M\) preciso make the loss invariant onesto 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 cloison them in 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 per the batch. We then save the datazione_loss onesto monitor it and the probs to use them in the backward pass.
Backward pass: Gradients computation
Con the backward pass we need to 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 to probs, we assign probs values preciso foce. For the positive classes con \(M\) we subtract 1 sicuro the corresponding probs value and use scale_factor preciso gara the gradient expression. We compute the mean gradients of all the batch to run the backpropagation.
Binary Ciclocampestre-Entropy Loss
Also called Sigmoid Cross-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 sicuro a indivis 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 mediante \(C\), as explained above. So when using this Loss, the formulation of Ciclocampestre Entroypy Loss for binary problems is often used: