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

Defined the loss, now we’ll have to compute its gradient respect preciso the output neurons of the CNN con 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 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 sopra the nominator.

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

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

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

When Softmax loss is used is a multi-label campo, the gradients get a 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 \(s_p\) per \(M\) is the CNN score for each positive class. As per Facebook paper, I introduce verso scaling factor \(1/M\) puro 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 mediante the forward and backward passes of the layer:

Forward pass: Loss computation

We first compute Softmax activations for each class and cloison 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 onesto introduce class balancing. The batch loss will be the mean loss of the elements in the batch. We then save the datazione_loss to display it and the probs puro use them durante the backward pass.

Backward pass: Gradients computation

Con the backward pass we need onesto compute the gradients of each element of the batch respect esatto 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 esatto sbocco. For the positive classes durante \(M\) we subtract 1 to the corresponding probs value and use scale_factor esatto incontro the gradient expression. We compute the mean gradients of all the batch sicuro run the backpropagation.

Binary Ciclocampestre-Entropy Loss

Also called Sigmoid Ciclocross-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 esatto verso certain 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 per \(C\), as explained above. So when using this Loss, the formulation of Ciclocampestre Entroypy Loss for binary problems is often used: