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So we need sicuro compute the gradient of CE Loss respect each CNN class punteggio mediante \(s\)

Defined the loss, now we’ll have esatto compute its gradient respect to the output neurons of the CNN in 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 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 score of \(C_p\) (\(s_p\)) is per the nominator.

  • Caffe: SoftmaxWithLoss Layer. Is limited to multi-class classification.
  • Pytorch: CrossEntropyLoss. Is limited to multi-class classification.
  • TensorFlow: softmax_cross_entropy. Is limited preciso multi-class classification.

In 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 mediante Facebook or me using Softmax Loss for multi-label classification, which is not norma.

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

Where each \(s_p\) per \(M\) is the CNN risultato for each positive class. As in 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 supporto bgclive 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 mediante 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 scadenza_loss to display it and the probs sicuro use them durante 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 puro probs, we assign probs values puro delta. For the positive classes sopra \(M\) we subtract 1 puro the corresponding probs value and use scale_factor preciso confronto the gradient expression. We compute the mean gradients of all the batch to run the backpropagation.

Binary Ciclocross-Entropy Loss

Also called Sigmoid Ciclocampestre-Entropy loss. It is per Sigmoid activation plus verso 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 puro a excretion 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 in \(C\), as explained above. So when using this Loss, the formulation of Cross Entroypy Loss for binary problems is often used: