caffe softmaxLoss 梯度计算

softmaxLoss的梯度计算结果最后可以化简为qi - pi （qi是softmax对zi的偏导，pi是label的值 取 0 或 1）
详见Deep Learning P221    6.5.9 Differentiation outside the Deep Learning Community 
 
 
For example, suppose we have variables p1,p2,...,pn representing probabilities and variables z1 , z2 , . . . , zn representing unnormalized log probabilities. Suppose
...
 
construct a cross-entropy loss J = − sum( pi log qi). A human

mathematician can observe that the derivative of J with respect to zi takes a very

simple form: qi − pi .
 
知道这个trick之后  红框的表达式就不难理解了
bottom_diff[i * dim + label_value * inner_num + j] -= 1   前面还有一句   caffe_copy(prob_.count(), prob_data, bottom_diff);
首先向bottom_diff拷入prob_data, 也就是softmax输出的qi
dim : 表示一个num/batch里一个样本的大小，为 channel * Height * Width
inner_num：表示一个特征图/channel/一个label的大小，为Height * Width，这里就是1*1了
label_val = label[i * inner_num_ + j] 就是 p[label_val] = 1 (只有一个)   p[others_label_val] = 0
i * dim + label_value * inner_num + j 就是一个样本里，对应label_val的那个channel的index
对这个index执行 q[index] -= p[label_val]
对这个样本里的其它channel的index默认执行q[index] -= p[other_label_val]