Pytorch 自定义激活函数前向与反向传播 Tanh

看完这篇，你基本上可以自定义前向与反向传播，可以自己定义自己的算子

文章目录

- Tanh
- - 公式
  - 求导过程
  - 优点：
  - 缺点：
  - 自定义Tanh
  - 与Torch定义的比较
  - 可视化

import matplotlib
import matplotlib.pyplot as plt
import numpy as np
import torch
import torch.nn as nn
import torch.nn.functional as F%matplotlib inlineplt.rcParams['figure.figsize'] = (7, 3.5)
plt.rcParams['figure.dpi'] = 150
plt.rcParams['axes.unicode_minus'] = False  #解决坐标轴负数的铅显示问题

Tanh

公式

$tanh⁡(x)=sinh⁡(x)cosh⁡(x)=ex−e−xex+e−x\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

$tanh⁡(x)=2σ(2x)−1\tanh(x) = 2 \sigma(2x) - 1$

求导过程

$tanh⁡′(x)=(ex−e−xex+e−x)′=[(ex−e−x)(ex+e−x)−1]′=(ex+e−x)(ex+e−x)−1+(ex−e−x)(−1)(ex+e−x)−2(ex−e−x)=1−(ex−e−x)2(ex+e−x)−2=1−(ex−e−x)2(ex+e−x)2=1−tanh⁡2(x)\begin{aligned} \tanh'(x) =& \big(\frac{e^x - e^{-x}}{e^x + e^{-x}}\big)' \\ =& \big[(e^x - e^{-x})(e^x + e^{-x})^{-1}\big]' \\ =& (e^x + e^{-x})(e^x + e^{-x})^{-1} + (e^x - e^{-x})(-1)(e^x + e^{-x})^{-2} (e^x - e^{-x}) \\ =& 1-(e^x - e^{-x})^2(e^x + e^{-x})^{-2} \\ =& 1 - \frac{(e^x - e^{-x})^2}{(e^x + e^{-x})^2} \\ =& 1- \tanh^2(x) \\ \end{aligned}$

优点：

Tanh也称为双切正切函数，取值范围为[-1,1]。tanh在特征相差明显时的效果会很好，在循环过程中会不断扩大特征效果。与 sigmoid 的区别是，tanh 是 0 均值的，因此实际应用中 tanh 会比 sigmoid 更好。文献 [LeCun, Y., et al., Backpropagation applied to handwritten zip code recognition. Neural computation, 1989. 1(4): p. 541-551.] 中提到tanh 网络的收敛速度要比sigmoid快，因为tanh 的输出均值比 sigmoid 更接近 0，SGD会更接近 natural gradient[4]（一种二次优化技术），从而降低所需的迭代次数。非常优秀，几乎适合所有的场景

缺点：

该导数在正负饱和区的梯度都会接近于0值，会造成梯度消失。还有其更复杂的幂运算。

自定义Tanh

class SelfDefinedTanh(torch.autograd.Function):@staticmethoddef forward(ctx, inp):exp_x = torch.exp(inp)exp_x_ = torch.exp(-inp)result = torch.divide((exp_x - exp_x_), (exp_x + exp_x_))ctx.save_for_backward(result)return result@staticmethoddef backward(ctx, grad_output):# ctx.saved_tensors is tuple (tensors, grad_fn)result, = ctx.saved_tensorsreturn grad_output * (1 - result.pow(2))class Tanh(nn.Module):def __init__(self):super().__init__()def forward(self, x):out = SelfDefinedTanh.apply(x)return out

def tanh_sigmoid(x):"""according to the equation"""# 2 * torch.sigmoid(2 * x) -1 return torch.mul(torch.sigmoid(torch.mul(x, 2)), 2) - 1

与Torch定义的比较

# self defined
torch.manual_seed(0)tanh = Tanh()  # SelfDefinedTanh
inp = torch.randn(5, requires_grad=True)
out = tanh((inp + 1).pow(2))print(f'Out is\n{out}')out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nFirst call\n{inp.grad}")out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nSecond call\n{inp.grad}")inp.grad.zero_()
out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nCall after zeroing gradients\n{inp.grad}")

Out is
tensor([1.0000, 0.4615, 0.8831, 0.9855, 0.0071],grad_fn=<SelfDefinedTanhBackward>)First call
tensor([ 5.0889e-05,  1.1121e+00, -5.1911e-01,  9.0267e-02, -1.6904e-01])Second call
tensor([ 1.0178e-04,  2.2243e+00, -1.0382e+00,  1.8053e-01, -3.3807e-01])Call after zeroing gradients
tensor([ 5.0889e-05,  1.1121e+00, -5.1911e-01,  9.0267e-02, -1.6904e-01])

# self defined tanh_sigmoid
torch.manual_seed(0)inp = torch.randn(5, requires_grad=True)
out = tanh_sigmoid((inp + 1).pow(2))print(f'Out is\n{out}')out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nFirst call\n{inp.grad}")out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nSecond call\n{inp.grad}")inp.grad.zero_()
out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nCall after zeroing gradients\n{inp.grad}")

Out is
tensor([1.0000, 0.4615, 0.8831, 0.9855, 0.0071], grad_fn=<SubBackward0>)First call
tensor([ 5.0889e-05,  1.1121e+00, -5.1911e-01,  9.0267e-02, -1.6904e-01])Second call
tensor([ 1.0178e-04,  2.2243e+00, -1.0382e+00,  1.8053e-01, -3.3807e-01])Call after zeroing gradients
tensor([ 5.0889e-05,  1.1121e+00, -5.1911e-01,  9.0267e-02, -1.6904e-01])

# torch defined
torch.manual_seed(0)inp = torch.randn(5, requires_grad=True)
out = torch.tanh((inp + 1).pow(2))print(f'Out is\n{out}')out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nFirst call\n{inp.grad}")out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nSecond call\n{inp.grad}")inp.grad.zero_()
out.backward(torch.ones_like(inp), retain_graph=True)
print(f"\nCall after zeroing gradients\n{inp.grad}")

Out is
tensor([1.0000, 0.4615, 0.8831, 0.9855, 0.0071], grad_fn=<TanhBackward>)First call
tensor([ 5.0283e-05,  1.1121e+00, -5.1911e-01,  9.0267e-02, -1.6904e-01])Second call
tensor([ 1.0057e-04,  2.2243e+00, -1.0382e+00,  1.8053e-01, -3.3807e-01])Call after zeroing gradients
tensor([ 5.0283e-05,  1.1121e+00, -5.1911e-01,  9.0267e-02, -1.6904e-01])

从上3个结果，可以看出，不管是经过sigmoid来计算，还是公式定义都可以得到一样的output与gradient。但在输入的值较大时，torch应该是减去一个小值，使得梯度更小。

可视化

# visualization
inp = torch.arange(-8, 8, 0.1, requires_grad=True)
out = tanh(inp)
out.sum().backward()inp_grad = inp.gradplt.plot(inp.detach().numpy(),out.detach().numpy(),label=r"$\tanh(x)$",alpha=0.7)
plt.plot(inp.detach().numpy(),inp_grad.numpy(),label=r"$\tanh'(x)$",alpha=0.5)
plt.grid()
plt.legend()
plt.show()