Pk對zk的求導，以及Pk對zj的求導請參考https://blog.csdn.net/u013066730/article/details/86231215

前向程式碼：

for (int i = 0; i < outer_num_; ++i) {
    for (int j = 0; j < inner_num_; j++) {
      const int label_value = static_cast<int>(label[i * inner_num_ + j]);
      if (has_ignore_label_ && label_value == ignore_label_) {
        continue;
      }
      DCHECK_GE(label_value, 0);
      DCHECK_LT(label_value, channels);
      const int index = i * dim + label_value * inner_num_ + j;
      // FL(p_t) = -(1 - p_t) ^ gamma * log(p_t)
      // loss -= std::max(power_prob_data[index] * log_prob_data[index],
      //                      Dtype(log(Dtype(FLT_MIN))));
      loss -= power_prob_data[index] * log_prob_data[index];
      ++count;
    }
  }

  // prob
  top[0]->mutable_cpu_data()[0] = loss / get_normalizer(normalization_, count);
  
 

反向程式碼：

    for (int i = 0; i < outer_num_; ++i) {
      for (int j = 0; j < inner_num_; ++j) {
        // label
        const int label_value = static_cast<int>(label[i * inner_num_ + j]);
        
        // ignore label
        if (has_ignore_label_ && label_value == ignore_label_) {
          for (int c = 0; c < channels; ++c) {
            bottom_diff[i * dim + c * inner_num_ + j] = 0;
          }
          continue;
        }

        // the gradient from FL w.r.t p_t, here ignore the `sign`
        int ind_i  = i * dim + label_value * inner_num_ + j; // index of ground-truth label
        Dtype grad = 0 - gamma_ * (power_prob_data[ind_i] / std::max(1 - prob_data[ind_i], eps)) 
                                * log_prob_data[ind_i] * prob_data[ind_i]
                       + power_prob_data[ind_i];
        // the gradient w.r.t input data x
        for (int c = 0; c < channels; ++c) {
          int ind_j = i * dim + c * inner_num_ + j;
          if(c == label_value) {
            CHECK_EQ(ind_i, ind_j);
            // if i == j, (here i,j are refered for derivative of softmax)
            bottom_diff[ind_j] = grad * (prob_data[ind_i] - 1);
          } else {
            // if i != j, (here i,j are refered for derivative of softmax)
            bottom_diff[ind_j] = grad * prob_data[ind_j];
          }
        }
        // count                    
        ++count;
      }
    }
    // Scale gradient
    Dtype loss_weight = top[0]->cpu_diff()[0] / get_normalizer(normalization_, count);
    caffe_scal(prob_.count(), loss_weight, bottom_diff);