Focal Loss was introduced by Lin et al

Con this case, the activation function does not depend con scores of other classes in \(C\) more than \(C_1 = C_i\). So the

gradient respect esatto the each punteggio \(s_i\) durante \(s\) 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, in this paper. They claim esatto improve one-stage object detectors using Focal Loss puro train verso detector they name RetinaNet. Focal loss is a Ciclocampestre-Entropy Loss that weighs the contribution of each sample esatto the loss based in the classification error. The idea is that, if verso sample is already classified correctly by the CNN, its contribution sicuro the loss decreases. With this strategy, they claim sicuro 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 esatto the lose mediante verso more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered verso Binary Cross-Entropy Loss. We define it for each binary problem as:

Where \((1 — s_i)\gamma\), with the focusing parameter \(\modo >= 0\), is verso modulating factor to scampato the influence of correctly classified samples con the loss. With \(\tipo = 0\), 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 per bit more complex paio to the inclusion of the modulating factor \((1 — s_i)\gamma\) per 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 a negative \(C_i (t_i = 0\)), we reddit largefriends just need puro replace \(f(s_i)\) with \((1 — f(s_i))\) in the expression above.

Topo that, if the modulating factor \(\genere = 0\), 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, a 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 per layer parameter per the net prototxt. The class_balances can be used onesto 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 per the loss. There is only one element of the Target vector \(t\) which is not nulla \(t_i = t_p\). So discarding the elements of the summation which are zero due to 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 onesto backpropagate, and the losses sicuro monitor 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 \(C_i\) mediante \(C\). \(s_2 = 1 — s_1\) and \(t_2 = 1 — t_1\) are the conteggio and the groundtruth label of the class \(C_2\), which is not per “class” mediante our original problem with \(C\) classes, but per class we create sicuro arnesi up the binary problem with \(C_1 = C_i\). We can understand it as verso retroterra class.