Per this case, the activation function does not depend durante scores of other classes sopra \(C\) more than \(C_1 = C_i\). So the gradient respect sicuro the each risultato \(s_i\) per \(s\) 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, per this paper. They claim preciso improve one-stage object detectors using Focal Loss sicuro train per detector they name RetinaNet. Focal loss is verso Ciclocampestre-Entropy Loss that weighs the contribution of each sample to the loss based per the classification error. The idea is that, if verso sample is already classified correctly by the CNN, its contribution onesto the loss decreases. With this strategy, they claim puro solve the problem of class imbalance by making the loss implicitly focus con those problematic classes. Moreover, they also weight the contribution of each class to the lose in a more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered per Binary Cross-Entropy Loss. We define it for each binary problem as:
Where \((1 – s_i)\gamma\), with the focusing parameter \(\tipo >= 0\), is verso modulating factor sicuro scampato the influence of correctly classified samples mediante the loss. With \(\qualita = 0\), Focal Loss is equivalent preciso 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 a bit more complex due esatto the inclusion of the modulating factor \((1 – s_i)\gamma\) per the loss formulation, but it can be deduced using the Binary Cross-Entropy gradient expression.
Where \(f()\) is the sigmoid function. Sicuro get the gradient expression for per negative \(C_i (t_i = 0\)), we just need preciso replace \(f(s_i)\) with \((1 – f(s_i))\) sopra the expression above.
Ratto that, if the modulating factor \(\qualita = 0\), the loss is equivalent esatto the CE Loss, and we end up with the same gradient expression.
Forward pass: Loss computation
Where logprobs[r] stores, verso each element of the batch, the sum of the binary cross entropy verso each class. The focusing_parameter is \(\gamma\), which by default is 2 and should be defined as verso layer parameter mediante the net prototxt. The class_balances can be used sicuro introduce different loss contributions a class, as they do per the Facebook paper.
Backward pass: Gradients computation
Durante the specific (and usual) case of Multi-Class classification the labels are one-hot, so only the positive class \(C_p\) keeps its term per the loss. There is only one element of the Target vector https://datingranking.net/it/colombiancupid-review/ \(t\) which is not nulla \(t_i = t_p\). So discarding the elements of the summation which are zero due onesto target labels, we can write:
This would be the pipeline for each one of the \(C\) clases. We serie \(C\) 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 esatto backpropagate, and the losses preciso videoclip 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 \(C_i\) in \(C\). \(s_2 = 1 – s_1\) and \(t_2 = 1 – t_1\) are the conteggio and the groundtruth label of the class \(C_2\), which is not per “class” per our original problem with \(C\) classes, but verso class we create to servizio up the binary problem with \(C_1 = C_i\). We can understand it as verso retroterra class.
