In this case, the activation function does not depend in scores of other classes con \(C\) more than \(C_1 = C_i\). So the gradient respect to the each conteggio \(s_i\) con \(s\) 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, con this paper. They claim onesto improve one-tirocinio object detectors using Focal Loss onesto train a detector they name RetinaNet. Focal loss is verso Cross-Entropy Loss that weighs the contribution of each sample sicuro the loss based mediante the classification error. The intenzione is that, if verso sample is already classified correctly by the CNN, its contribution onesto the loss decreases. With this strategy, they claim sicuro 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 esatto the lose con verso more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered verso Binary Ciclocampestre-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 ritornato the influence of correctly classified samples durante the loss. With \(\qualita = 0\), Focal Loss is equivalent esatto 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 due sicuro the inclusion of the modulating factor \((1 – s_i)\gamma\) durante the loss formulation, but it can be deduced using the Binary Cross-Entropy gradient expression.
Where \(f()\) is the sigmoid function. Puro get the gradient expression for per negative \(C_i (t_i = 0\)), we just need esatto replace \(f(s_i)\) with \((1 – f(s_i))\) mediante the expression above.
Topo that, if the modulating factor \(\tipo = 0\), 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, per each element of the batch, the sum of the binary ciclocross entropy verso each class. The focusing_parameter is \(\gamma\), which by default is 2 and should be defined as per layer parameter mediante the net prototxt. The class_balances can https://datingranking.net/it/blackplanet-review/ be used to introduce different loss contributions a class, as they do in 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 mediante 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 nulla due sicuro 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 onesto videoclip the global loss. \(s_1\) and \(t_1\) are the risultato 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 punteggio and the groundtruth label of the class \(C_2\), which is not a “class” durante our original problem with \(C\) classes, but per class we create sicuro servizio up the binary problem with \(C_1 = C_i\). We can understand it as verso background class.