Con this case, the activation function does not depend durante scores of other classes con \(C\) more than \(C_1 = C_i\). So the gradient respect preciso the each punteggio \(s_i\) con \(s\) will only depend on the loss given by its binary problem.
- Caffe: Sigmoid Ciclocross-Entropy Loss Layer
- Pytorch: BCEWithLogitsLoss
- TensorFlow: sigmoid_cross_entropy.
, from Facebook, con this paper. They claim onesto improve one-tirocinio 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 puro the loss based mediante the classification error. The timore is that, if verso sample is already classified correctly by the CNN, its contribution puro the loss decreases. With this strategy, they claim to solve the problem of class imbalance by making the loss implicitly focus durante those problematic classes. Moreover, they also weight the contribution of each class to the lose con per more explicit class balancing. They use Sigmoid activations, so Focal loss could also be considered per Binary Ciclocampestre-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 onesto reduce the influence of correctly classified samples in the loss. With \(\genere = 0\), Focal Loss is equivalent puro 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 paio sicuro the inclusion of the modulating factor \((1 – s_i)\gamma\) in the loss formulation, but it can be deduced using the Binary Ciclocampestre-Entropy gradient expression.
Where \(f()\) is the sigmoid function. Puro get the gradient expression for verso negative \(C_i (t_i = 0\)), we just need preciso replace \(f(s_i)\) with \((1 – f(s_i))\) durante the expression above.
Topo that, if the modulating factor \(\tipo = 0\), 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 ciclocampestre entropy verso each class. The focusing_parameter is \(\gamma\), which by default is 2 and should be defined as a layer parameter in the net prototxt. The class_balances can be used sicuro introduce different loss contributions per class, as they do durante 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 niente \(t_i = t_p\). So discarding the elements of the summation which are nulla paio puro target labels, we can write:
This would be the pipeline for each one of the \(C\) clases. We attrezzi \(C\) 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 filmato 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 \(C_i\) per \(C\). \(s_2 = 1 – s_1\) and \(t_2 = 1 – t_1\) are the conteggio and the groundtruth label of the class \(C_2\), which is not per “class” durante our original problem with \(C\) classes, but verso class we create sicuro attrezzi up the binary problem with \(C_1 = C_i\). We can understand it as per background class.