起因:希望深度学习输入时小的扰动不会影响结果,我们会在输入端加一些噪声,让模型自己去适应这种扰动,从而提升整体的鲁棒性,CV领域可以直接在图像输入添加噪声,NLP领域因为输入都是one-hot形式,无法直接添加噪声,我们可以考虑在embedding之后的词向量上添加扰动,或者直接在embedding矩阵中添加扰动(FGM方法)。
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?细微的噪声加入图层,对结果影响巨大
? ?这里取 为loss的梯度,正向梯度loss下降,反向梯度loss上升。
? ? ? ? ? ? ? ? 其中 为模型参数
?
模型参数更新( ?为已经计算出来的常数):?
![\begin{align*} \frac{\partial f(x+\Delta x,\theta)}{\partial \theta}&=\frac{\partial f(x,\theta)}{\partial \theta}+\frac{\partial^{2}f(x,\theta)}{\partial x \partial \theta}\cdot\Delta x\\&=\frac{\partial}{\partial \theta}[f(x,\theta)+\frac{1}{2}\cdot(\frac{\partial f(x,\theta)}{\partial x})^{2}]\end{}](https://latex.codecogs.com/gif.latex?%5Cbegin%7Balign*%7D%20%5Cfrac%7B%5Cpartial%20f%28x+%5CDelta%20x%2C%5Ctheta%29%7D%7B%5Cpartial%20%5Ctheta%7D%26%3D%5Cfrac%7B%5Cpartial%20f%28x%2C%5Ctheta%29%7D%7B%5Cpartial%20%5Ctheta%7D+%5Cfrac%7B%5Cpartial%5E%7B2%7Df%28x%2C%5Ctheta%29%7D%7B%5Cpartial%20x%20%5Cpartial%20%5Ctheta%7D%5Ccdot%5CDelta%20x%5C%5C%26%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20%5Ctheta%7D%5Bf%28x%2C%5Ctheta%29+%5Cfrac%7B1%7D%7B2%7D%5Ccdot%28%5Cfrac%7B%5Cpartial%20f%28x%2C%5Ctheta%29%7D%7B%5Cpartial%20x%7D%29%5E%7B2%7D%5D%5Cend%7B%7D)
![-->\triangledown_{\theta} f(x=\Delta x,\theta)=\triangledown_{\theta}[f(x,\theta)+\frac{1}{2}(\triangledown_{x}f(x,\theta))^{2}]](https://latex.codecogs.com/gif.latex?--%3E%5Ctriangledown_%7B%5Ctheta%7D%20f%28x%3D%5CDelta%20x%2C%5Ctheta%29%3D%5Ctriangledown_%7B%5Ctheta%7D%5Bf%28x%2C%5Ctheta%29+%5Cfrac%7B1%7D%7B2%7D%28%5Ctriangledown_%7Bx%7Df%28x%2C%5Ctheta%29%29%5E%7B2%7D%5D)
?可以看出在输入中添加的扰动,近似等价于在loss函数中添加梯度惩罚项:

Pytorch简易实现:
self.optim.zero_grad()
loss.backward(retain_graph=True) # loss梯度回溯两次,要保留计算图
# ******************************
name_list = ['hidden_layer.0.weight'] # 需要提取梯度的层名
for name, para in model.named_parameters():
if name in name_list:
gp = (para.grad ** 2).sum()
loss += 0.5 * gp * epsilon
self.optim.zero_grad() # 梯度清零,在新得到的loss上进行梯度更新
loss.backward()
break
# ******************************** 围起来的是在原先基础上需要添加的代码
self.optim.step()
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