So we need preciso compute the gradient of CE Loss respect each CNN class score per \(s\)

Defined the loss, now we’ll have puro compute its gradient respect to the output neurons of the CNN in 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 niente. 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 puro multi-class classification.
• TensorFlow: softmax_cross_entropy. Is limited to multi-class classification.

Durante this Facebook rete informatica they claim that, despite being counter-intuitive, Categorical Ciclocross-Entropy loss, or Softmax loss worked better than Binary Ciclocross-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 campo, the gradients get per 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 \(s_p\) per \(M\) is the CNN conteggio for each positive class. As durante Facebook paper, I introduce per scaling factor \(1/M\) esatto 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 sopra the forward and backward passes of the layer:

Forward pass: Loss computation

We first compute Softmax activations for each class and cloison them per 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 to introduce class balancing. The batch loss will be the mean loss of the elements sopra the batch. We then save the momento_loss puro schermo it and https://datingranking.net/it/chappy-review/ the probs sicuro use them sopra the backward pass.

Backward pass: Gradients computation

Durante the backward pass we need sicuro compute the gradients of each element of the batch respect puro 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 preciso delta. For the positive classes in \(M\) we subtract 1 onesto the corresponding probs value and use scale_factor onesto competizione 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 verso 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 esatto per indivisible class should not influence the decision for another class. It’s called Binary Ciclocross-Entropy Loss because it sets up per binary classification problem between \(C’ = 2\) classes for every class sopra \(C\), as explained above. So when using this Loss, the formulation of Ciclocross Entroypy Loss for binary problems is often used: