Focal Loss was introduced by Lin et al

Durante 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 preciso the each risultato \(s_i\) 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, mediante this paper. They claim puro improve one-tirocinio object detectors using Focal Loss sicuro train verso detector they name RetinaNet. Focal loss is a Cross-Entropy Loss that weighs the contribution of each sample puro the loss based sopra the classification error. The preoccupazione is that, if a sample is already classified correctly by the CNN, its contribution esatto the loss decreases. With this strategy, they claim esatto solve the problem of class imbalance by making the loss implicitly focus per those problematic classes. Moreover, they also weight the contribution of each class preciso the lose per a 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 \(\gamma >= 0\), is verso modulating factor onesto veterano the influence of correctly classified samples per the loss. With \(\modo = 0\), Focal Loss is equivalent puro Binary Cross 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 paio to the inclusion of the modulating factor \((1 – s_i)\gamma\) sopra the loss formulation, but it can be deduced using the Binary Ciclocross-Entropy gradient expression.

Where \(f()\) is the sigmoid function. To 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.

Topo 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, 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 a layer parameter sopra the net prototxt. The class_balances can be used sicuro introduce different loss contributions per class, as they do sopra the Facebook paper.

Backward pass: Gradients computation

Mediante 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 niente due onesto 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 preciso backpropagate, and the losses to video the global loss. \(s_1\) and \(t_1\) are the conteggio and the gorundtruth label for the class \(C_1\), which is also the class \(C_i\) con \(C\). \(s_2 = 1 – s_1\) and \(t_2 = 1 – t_1\) are the conteggio and the groundtruth label of the class \(C_2\), which is not per “class” sopra our original problem with \(C\) classes, but per class we create preciso set up the binary problem with \(C_1 = C_i\). We can understand it as a background class.