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

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

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

Mediante this Facebook work 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 per Facebook or me using Softmax Loss for multi-label classification, which is not norma.

When Softmax loss is used is per multi-label sfondo, the gradients get a 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 \(s_p\) con \(M\) is the CNN score for each positive class. As mediante Facebook paper, I introduce verso scaling factor \(1/M\) to 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 store them per probs. Then we compute the loss for each image per 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 mediante the batch. We then save the momento_loss esatto monitor it and the probs esatto use them per the backward pass.

Backward pass: Gradients computation

Con the backward pass we need to compute the gradients of each element of the batch respect to each one of the classes scores \(s\). As the gradient for all the classes \(C\) except positive classes \(M\) is equal preciso probs, we assign probs values sicuro delta. For the positive classes durante \(M\) we subtract 1 to the corresponding probs value and use scale_factor puro gara the gradient expression. We compute the mean gradients of all the batch preciso run the backpropagation.

Binary Cross-Entropy Loss

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