Introduce
Pytorch Autograd库 (自动求导机制) 是训练神经网络时，反向误差传播(BP)算法的核心。

本文通过logistic回归模型来介绍Pytorch的自动求导机制。首先，本文介绍了tensor与求导相关的属性。其次，通过logistic回归模型来帮助理解BP算法中的前向传播以及反向传播中的导数计算。

以下均为初学者笔记。

Tensor Attributes Related to Derivation
note: 以下用x代表创建的tensor张量。

x.requires_grad：True or False，用来指明该张量在反向传播过程中是否需要求导。
with torch.no_grad():：当我们在做模型评估的时候是不需要求导的，可以嵌套一层with torch.no_grad()以减少可能的计算和内存开销。
x.grad：返回损失函数对该张量求偏导的值，在调用backward()之后才有。
x.grad_fn：存储计算图上某中间节点进行的操作，如加减乘除等，用于指导反向传播时loss对该节点的求偏导计算。
x.is_leaf：True or False，用于判断某个张量在计算图中是否是叶子张量。叶子张量我个人认为可以理解为目标函数中非中间因变量（中间函数、一般是运算得到的张量），如神经网络中的权值参数w就是叶子张量（一般是手动创建的张量）。
x.detach()：返回tensor的数据以及requires_grad属性，且返回的tensor与原始tensor共享存储空间，即一个改变会导致另外一个改变。因此，如果我们在backward之前对x.detach()返回的张量进行改变会导致原始x的改变，从而导致求导错误，但是这时系统会报错提醒。
（note：虽然x.data也与x.detach()作用相似，但是x.data不被Autograd系统追踪，因此如果遇到上述问题并不会报错。推荐使用x.detach()）
x.item()：如果张量只包含一个元素，即标量（0维张量，scalar），可以用x.item()返回，通常loss只包含一个数值，因此常用loss.item()。
x.tolist()：如果张量只包含多个元素，可以用x.tolist()转换成python list返回。
Build Logistic regression Model
假设有一个损失函数如下（Logistic回归）：

z=w1x1+w2x2+b
yp=sigmoid(z)
Loss(yp,yt)=−1n∑i=1n(ytlog(yp)+(1−yt)log(1−yp))
摘自吴恩达机器学习

由损失函数构建简单计算图模型如下：

现在我们通过上述例子来理解前向传播和反向传播。在上述简单的神经网络模型中，我们需要对权值参数w1，w2以及阈值参数b进行更新。神经网络训练的总体过程如下：先由输入层逐级前向传播计算loss输出，再有输出层loss反向计算各层梯度传播误差，以此更新各层的权值参数w和阈值参数b。

在该模型中我们需要求出loss对w1、w2以及b的偏导，以此利用SGD更新各参数。对于根据链式法则的逐级求导过程不再赘述，吴恩达机器学习SGD部分有详细的计算过程以及解释。

现在我们利用pytorch实现logistic回归模型，并手动实现参数更新。

import torch
import numpy as np

可能网络上有许多资料在Tensor外再额外封装一层Variable，然而从Pytorch0.4版本以后，Variable已经合并进Tensor了，因此以后的代码可以直接用Tensor,不必要再封装一层Variable。

读入数据 x_t,y_t

x_t = torch.tensor(np.array([[1,1],[1,0],[0,1],[0,0]]),requires_grad=False,dtype=torch.float)
y_t = torch.tensor([[0],[1],[0],[1]],requires_grad=False,dtype=torch.float)
print(x_t.size())

定义权值参数w和阈值参数b

w = torch.randn([2,1], requires_grad=True,dtype=torch.float)
b = torch.zeros(1, requires_grad=True,dtype=torch.float)
print(w.size())

构建逻辑回归模型

def logistic_model(x_t):
a = torch.matmul(x_t,w) + b
return torch.sigmoid(a)

y_p = logistic_model(x_t)

计算误差

def get_loss(y_p, y_t):
return -torch.mean(y_t * torch.log(y_p)+(1-y_t) * torch.log(1-y_p))

loss = get_loss(y_p, y_t)
print(loss)

自动求导

loss.backward()

note：如果loss为一个标量（一般都是），那么loss.backward()不需要指定任何参数。然而如果有多个损失，即loss为一个向量tensor([loss1, loss2,loss3])，则需要指定一个gradient参数，它是与loss形状匹配的张量，如tensor([1.0,1.0,0.1]),里面数字我个人理解为代表各损失的权重。

查看 w 和 b 的梯度

print(w.grad)
print(b.grad)

更新一次参数

w.data = w.data - 1e-2 * w.grad.data
b.data = b.data - 1e-2 * b.grad.data

‘’’
note:
存在两个问题：

1. 如果没有前面先更新一次参数，后面直接进行迭代更新的话，会报错，具体原因也没搞懂。
2. 利用pycharm运行pytorch代码，调用了backward()之后，程序运行完成进程并不会终止，需要手动到任务管理器中kill进程，具体原因也不清楚。
   ‘’’

epoch

for e in range(10000): # 进行 10000 次更新
y_p = logistic_model(x_t)
loss = get_loss(y_p, y_t)

w.grad.zero_()  # 归零梯度
b.grad.zero_()  # 归零梯度
loss.backward()w.data = w.data - 1e-2 * w.grad.data  # 更新 w
b.data = b.data - 1e-2 * b.grad.data  # 更新 b
print('epoch: {}, loss: {}'.format(e, loss.data.item()))

print(w)
print(b)

‘’’
每500次迭代打印出输出结果，我们看到损失函数在迭代中逐步下降：
epoch: 0, loss: 0.9426676034927368
epoch: 500, loss: 0.5936437249183655
epoch: 1000, loss: 0.4318988025188446
epoch: 1500, loss: 0.33194077014923096
epoch: 2000, loss: 0.265964150428772
epoch: 2500, loss: 0.22003984451293945
epoch: 3000, loss: 0.18663322925567627
epoch: 3500, loss: 0.1614413857460022
epoch: 4000, loss: 0.14187511801719666
epoch: 4500, loss: 0.12630191445350647
epoch: 5000, loss: 0.11365044862031937
epoch: 5500, loss: 0.10319262742996216
epoch: 6000, loss: 0.09441888332366943
epoch: 6500, loss: 0.08696318417787552
epoch: 7000, loss: 0.08055643737316132
epoch: 7500, loss: 0.07499672472476959
epoch: 8000, loss: 0.07013023644685745
epoch: 8500, loss: 0.06583743542432785
epoch: 9000, loss: 0.06202460825443268
epoch: 9500, loss: 0.05861698091030121
至此，手动实现梯度下降，logistic模型搭建完成，之后将尝试利用pytorch框架搭建神经网络。
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