Con 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 risultato \(s_i\) in \(s\) will only depend on the loss given by its binary problem.

• Caffe: Sigmoid Ciclocampestre-Entropy Loss Layer
• Pytorch: BCEWithLogitsLoss
• TensorFlow: sigmoid_cross_entropy.

Focal Loss

, from Facebook, in this paper. They claim to improve one-stage object detectors using Focal Loss preciso train verso detector they name RetinaNet. Focal loss is a Cross-Entropy Loss that weighs the contribution of each sample onesto the loss based sopra the classification error. The idea is that, if beautifulpeople a sample is already classified correctly by the CNN, its contribution esatto the loss decreases. With this strategy, they claim preciso 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 puro the lose con per more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered a Binary Ciclocross-Entropy Loss. We define it for each binary problem as:

Where \((1 – s_i)\gamma\), with the focusing parameter \(\modo >= 0\), is per modulating factor to scampato the influence of correctly classified samples per the loss. With \(\modo = 0\), 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 verso bit more complex coppia sicuro the inclusion of the modulating factor \((1 – s_i)\gamma\) con 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 verso negative \(C_i (t_i = 0\)), we just need onesto replace \(f(s_i)\) with \((1 – f(s_i))\) con the expression above.

Sorcio that, if the modulating factor \(\varieta = 0\), the loss is equivalent onesto 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 ciclocampestre entropy a 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 puro 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 durante 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 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 to 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 \(C_i\) in \(C\). \(s_2 = 1 – s_1\) and \(t_2 = 1 – t_1\) are the score and the groundtruth label of the class \(C_2\), which is not per “class” in our original problem with \(C\) classes, but a class we create preciso serie up the binary problem with \(C_1 = C_i\). We can understand it as verso preparazione class.