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【Pytorch基础】反向传播算法

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 之前我们只讨论过线性模型$\hat{y} = w * x$的权重更新方法。实际上,它可以被看成一个最简单的神经网络:

 可以看出,得到预测之后直接就能将误差反馈给权重进而调整权重值,因为它仅仅只有一个结点。但是,如果是非常复杂的模型呢?比如:

上图是一个 4 隐层的神经网络,中间的的线条数就是其权重的个数,即

实际情况还可能更多。显然想用线性模型的算法传播算法来更新权重在这里是行不通的。

计算图

先把模型看成一个图,考虑数据在图内的传播算法。

 对于上图所示的模型,是一个两层(单隐层)的神经网络。假设其输入$X$的维度为$n$,隐层(中间层)输出$H$的维度为$m$,输出层$O$的维度为$n$其结构如下:

那么,对应的两个权重矩阵为$W{1 (m\times n)},W{2 (n\times m)}$。
$b_1,b_2$ 称为偏置。

问题

如果对上述模型进行变形可以发现,无论它有多少层,最终都会被简化为单层。

这样一来,意味着层数(权重数量)的增加对模型的效果没有起到积极意义。因此,我们要将每一层的输出作用于一个非线性的变化函数(如$Sigmoid(x_i)=\frac{1}{1+e^{-x_i}}$函数)。如此一来,就无法再对模型进行简化了。

如何计算(先算出损失,再反向传播调整权重)

1. 损失的计算(前馈 Forward):

前馈过程比较简单,将$x,w$带入模型$\hat{y} = f(x,w)$,再将$\hat{y}$传入一个非线性函数即可得到结果$Z$, 再与真实值比较计算损失$Loss$。

2. 误差的反向传播 (BackPropagation):

 当前馈计算得到损失$Loss$后,需要将损失一步一步分配到来源中去。这里要用到链式求导法则,即要计算$x,w$对于$Loss$的影响大小要先求$Z$对$Loss$的影响再乘以$x,w$对$Z$的影响大小,进而得到$\frac{\partial L}{\partial x},\frac{\partial L}{\partial w}$, 其中,$\frac{\partial L}{\partial w}$ 用来调整当前层的权重,而$\frac{\partial L}{\partial x}$被视为更后一层的误差继续向后传播。

计算图的计算过程示意:

 一般我们在前馈过程中计算中间梯度,误差方向传播时就可以直接用了。

Pytorch 中的前馈和反馈计算

 在Pytorch中,Tensor(类) 是一个在创建动态计算图中非常重要的组成部分。它可以存储标量,向量,矩阵(二维或高维),计算过程中的所有数值都可以保存在 Tensor 中。它包含两个重要成员 data 和 grad(tensor), 分别保存计算过程中用到的数据(如权重等)和梯度值(如$\frac{\partial loss}{\partial w}$)。

在Pytorch中构建计算图

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import torch as th
import matplotlib.pyplot as plt

# 准备数据集
x_data = [1.0,2.0,3.0]
y_data = [2.0,4.0,6.0]

w_list = []
loss_list = []

a = 0.01 # 学习率

# 初始化权重tensor
w = th.Tensor([1.0]) # 线性模型权重只有一个,多个可传入多维矩阵
w.requires_grad = True # 声明该权重需要计算梯度,否则计算过程中torch不会计算该变量的梯度

# 定义模型
def forward(x):
# 由于w为Tensor因此乘法被重载,x也自动转为tensor与之相乘(矩阵乘法)
# 又w需要计算梯度故返回的Tensor也需要计算梯度
return x * w

# 定义损失函数, 实际上该函数可以看成构建了一个计算图
def loss(x, y):
y_pred = forward(x)
return (y_pred - y) ** 2

print("Predict (Before training)", 4, forward(4).item()) #item函数使得里面的一个值编程标量

# 开始训练网络
for epoch in range(100):
# 采用随机梯度下降
for x, y in zip(x_data, y_data):
l = loss(x,y) # 前馈计算,创建一个计算图
l.backward() # 反向传播, 算出计算链路上所有需要计算梯度的变量的梯度,计算完成后释放计算图
print('\t grad:', x, y, w.grad.item())
w.data = w.data - a * w.grad.data # 更新权重,由于grad 也是一个tensor,因此需要取里面的data才能正常计算
w.grad.data.zero_() # 清零梯度,因为默认情况下用backward()函数计算的梯度会累加

w_list.append(w.data.item())
loss_list.append(l.item())
print("progress:",epoch,l.item())

print("predict (after training)",4, forward(4).item())

# 绘图(权重与平均损失的关系)
plt.plot(w_list, loss_list)
plt.ylabel('loss')
plt.xlabel('W')
plt.xlim(1.0,2)
plt.show()

输出的数据:

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Predict (Before training) 4 4.0
grad: 1.0 2.0 -2.0
grad: 2.0 4.0 -7.840000152587891
grad: 3.0 6.0 -16.228801727294922
progress: 0 7.315943717956543
grad: 1.0 2.0 -1.478623867034912
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progress: 1 3.9987640380859375
grad: 1.0 2.0 -1.0931644439697266
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progress: 2 2.1856532096862793
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progress: 3 1.1946394443511963
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progress: 4 0.6529689431190491
grad: 1.0 2.0 -0.4417421817779541
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progress: 5 0.35690122842788696
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progress: 6 0.195076122879982
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progress: 7 0.10662525147199631
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progress: 8 0.0582793727517128
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predict (after training) 4 7.999998569488525

收敛情况图像:

文章作者: Liam
文章链接: https://www.ccyh.xyz/p/38c9.html
版权声明: 本博客所有文章除特别声明外,均采用 CC BY-NC-SA 4.0 许可协议。转载请注明来自 Liam's Blog
ღ喜欢记得五星好评哦~
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