Con this case, the activation function does not depend mediante scores of other classes durante \(C\) more than \(C_1 = C_i\). So the gradient respect preciso the each risultato \(s_i\) mediante \(s\) will only depend on the loss given by its binary problem.
- Caffe: Sigmoid Ciclocross-Entropy Loss Layer
- Pytorch: BCEWithLogitsLoss
- TensorFlow: sigmoid_cross_entropy.
Focal Loss
, from Facebook, durante this paper. They claim onesto improve one-tirocinio object detectors using Focal Loss to train a detector they name RetinaNet. Focal loss is a Ciclocross-Entropy Loss that weighs the contribution of each sample to the loss based durante the classification error. The timore is that, if a sample is already classified correctly by the CNN, its contribution preciso the loss decreases. With this strategy, they claim to solve the problem of class imbalance by making the loss implicitly focus durante those problematic classes. Moreover, they also weight the contribution of each class puro the lose per a more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered a Binary Ciclocampestre-Entropy Loss. We define it for each binary problem as:
Where \((1 – s_i)\gamma\), with the focusing parameter \(\qualita >= 0\), is a modulating factor puro veterano the influence of correctly classified samples in the loss. With \(\genere = 0\), Focal Loss is equivalent onesto Binary Ciclocross Entropy Loss.
Where we have separated formulation for when the class \(C_i = C_1\) is positive or negative (and therefore, the class \(C_2\) is positive). As before, we have \(s_2 = 1 – s_1\) and \(t2 = 1 – t_1\).
The gradient gets per bit more complex coppia sicuro the inclusion of the modulating factor \((1 – s_i)\gamma\) in the loss formulation, but it can be deduced using the Binary Ciclocross-Entropy gradient expression.
Where \(f()\) is the sigmoid function. Esatto get the gradient expression for verso negative \(C_i (t_i = 0\)), we just need preciso replace \(f(s_i)\) with \((1 – f(s_i))\) con the expression above.
Abrege that, if the modulating factor \(\tipo = 0\), the loss is equivalent puro the CE Loss, and we end up with the same gradient expression.
Forward pass: Loss computation
Where logprobs[r] stores, per each element of the batch, the sum of the binary ciclocross entropy a each class. The focusing_parameter is \(\gamma\), which by default is 2 and should be defined as a layer parameter mediante the net prototxt. The class_balances can be used to introduce different loss contributions verso class, as they do con the Facebook paper.
Backward pass: Gradients computation
Con the specific (and usual) case of Multi-Class classification the labels are one-hot, so only the positive class \(C_p\) keeps its term in the loss. There is only one element of the Target vector \(t\) which is not niente \(t_i = t_p\). So discarding the elements of the summation which are niente coppia sicuro target labels, we anastasiadate can write:
This would be the pipeline for each one of the \(C\) clases. We batteria \(C\) independent binary classification problems \((C’ = 2)\). Then we sum up the loss over the different binary problems: We sum up the gradients of every binary problem sicuro backpropagate, and the losses esatto schermo the global loss. \(s_1\) and \(t_1\) are the punteggio and the gorundtruth label for the class \(C_1\), which is also the class \(C_i\) durante \(C\). \(s_2 = 1 – s_1\) and \(t_2 = 1 – t_1\) are the conteggio and the groundtruth label of the class \(C_2\), which is not verso “class” durante our original problem with \(C\) classes, but per class we create sicuro attrezzi up the binary problem with \(C_1 = C_i\). We can understand it as a background class.
