Sopra this case, the activation function does not depend mediante scores of other classes sopra \(C\) more than \(C_1 = C_i\). So the gradient respect onesto the each punteggio \(s_i\) sopra \(s\) will only depend on the loss given by its binary problem.
- Caffe: Sigmoid Cross-Entropy Loss Layer
- Pytorch: BCEWithLogitsLoss
- TensorFlow: sigmoid_cross_entropy.
Focal Loss
, from Facebook, con this paper. They claim esatto improve one-stage object detectors using Focal Loss esatto train per detector they name RetinaNet. Focal loss is per Ciclocampestre-Entropy Loss that weighs the contribution of each sample sicuro the loss based per the classification error. The ispirazione is that, if verso sample is already classified correctly by the CNN, its contribution sicuro the loss decreases. With this strategy, they claim esatto solve the problem of class imbalance by making the loss implicitly focus mediante those problematic classes. Moreover, they also weight the contribution of each class onesto the lose sopra a more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered verso Binary Ciclocampestre-Entropy Loss. We define it for each binary problem as:
Where \((1 – s_i)\gamma\), with the focusing parameter \(\qualita >= 0\), is verso modulating factor onesto scampato the influence of correctly classified samples mediante the loss. With \(\tipo = 0\), Focal Loss is equivalent onesto Binary Ciclocampestre 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 due preciso the inclusion of the modulating factor \((1 – s_i)\gamma\) sopra the loss formulation, but it can be deduced using the Binary Ciclocampestre-Entropy gradient expression.
Where \(f()\) is the sigmoid function. Sicuro get the gradient expression for a negative \(C_i (t_i = 0\)), we just need esatto replace \(f(s_i)\) with \((1 – f(s_i))\) sopra the expression above.
Abrege that, if the modulating factor \(\gamma = 0\), the loss is equivalent to the CE Loss, and we end up with the same gradient expression.
Forward pass: Loss computation
Where logprobs[r] stores, a each element of the batch, the sum of the binary ciclocampestre entropy per each class. The focusing_parameter is \(\gamma\), which by default is 2 and should be defined as a layer parameter sopra the net prototxt. The class_balances can be used sicuro introduce different loss contributions per 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 con 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 zero paio preciso target labels, we can write:
This would be the pipeline for each one of the \(C\) clases. We servizio \(C\) 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 sicuro filmato the global loss. \(s_1\) and \(t_1\) are funziona chatki the conteggio and the gorundtruth label for the class \(C_1\), which is also the class \(C_i\) sopra \(C\). \(s_2 = 1 – s_1\) and \(t_2 = 1 – t_1\) are the risultato and the groundtruth label of the class \(C_2\), which is not a “class” in our original problem with \(C\) classes, but verso class we create onesto serie up the binary problem with \(C_1 = C_i\). We can understand it as per sostrato class.