Sopra this case, the activation function does not depend durante scores of other classes mediante \(C\) more than \(C_1 = C_i\). So the gradient respect to the each punteggio \(s_i\) in \(s\) 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, con this paper. They claim onesto improve one-stage object detectors using Focal Loss preciso train per detector they name RetinaNet. Focal loss is a Ciclocampestre-Entropy Loss that weighs the contribution of each sample sicuro the loss based con the classification error. The timore is that, if verso sample is already classified correctly by the CNN, its contribution sicuro the loss decreases. With this strategy, they claim to solve the problem of class imbalance by making the loss implicitly focus in those problematic classes. Moreover, they also weight the contribution of each class onesto the lose con per more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered a Binary Cross-Entropy Loss. We define it for each binary problem as:
Where \((1 – s_i)\gamma\), with the focusing parameter \(\genere >= 0\), is per modulating factor esatto reduce the influence of correctly classified samples in the loss. With \(\genere = 0\), Focal Loss is equivalent sicuro 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 preciso the inclusion of the modulating factor \((1 – s_i)\gamma\) sopra the loss formulation, but it can be deduced using the Binary Cross-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 to replace \(f(s_i)\) with \((1 – f(s_i))\) per the expression above.
Sorcio that, if the modulating factor \(\gamma = 0\), the loss is equivalent preciso 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 cross entropy a each class. The focusing_parameter is \(\gamma\), which by default is 2 and should be defined as per layer parameter per the net https://datingranking.net/it/beetalk-review prototxt. The class_balances can be used preciso introduce different loss contributions verso 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 con the loss. There is only one element of the Target vector \(t\) which is not zero \(t_i = t_p\). So discarding the elements of the summation which are niente due sicuro target labels, we 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 puro backpropagate, and the losses preciso filmato 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 \(C_i\) per \(C\). \(s_2 = 1 – s_1\) and \(t_2 = 1 – t_1\) are the risultato and the groundtruth label of the class \(C_2\), which is not per “class” per our original problem with \(C\) classes, but verso class we create esatto set up the binary problem with \(C_1 = C_i\). We can understand it as verso retroterra class.