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Pytorch Tutorial【Chapter 2. Autograd】

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Pytorch Tutorial

文章目录

Pytorch Tutorial
- Chapter 2. Autograd
- - 1. Review Matrix Calculus
  - - 1.1 Definition向量对向量求导
    - 1.2 Definition标量对向量求导
    - 1.3 Definition标量对矩阵求导
  - 2.关于autograd的说明
  - 3. grad的计算
  - - 3.1 Manual手动计算
    - 3.2 backward()自动计算
- Reference

Chapter 2. Autograd

1. Review Matrix Calculus

1.1 Definition向量对向量求导

Define the derivative of a function mapping $f:\mathbb{R}^n\to\mathbb{R}^m$ as the $n\times m$ matrix of partial derivatives. That is, if $x\in\mathbb{R}^n,f(x)\in\mathbb{R}^m$ , the derivative of $f$ with respect to $x$ is defined as
$\begin{bmatrix} \frac{\partial f}{\partial x} \end{bmatrix}_{ij} = \frac{\partial f_i}{\partial x_i}$
Let
$\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}, f(x) = \begin{bmatrix} f_1(x) \\ f_2(x) \\ \vdots \\ f_m(x) \end{bmatrix}$

then we define the Jacobian Matrix

$\frac{\partial f}{\partial x} = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_2}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_1} \\ \frac{\partial f_1}{\partial x_2} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_2} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_1}{\partial x_n} & \frac{\partial f_2}{\partial x_n} & \cdots & \frac{\partial f_m}{\partial x_n} \\ \end{bmatrix}$

1.2 Definition标量对向量求导

If $f$ is scalar, one has

$\frac{\partial f}{\partial x} = \begin{bmatrix} \frac{\partial f}{\partial x_1} \\ \frac{\partial f}{\partial x_2} \\ \vdots \\ \frac{\partial f}{\partial x_n} \\ \end{bmatrix}$
这其实是一种分母布局

1.3 Definition标量对矩阵求导

Now we give some results on the derivative of scalar functions of a matrix. Let $X=[x_{ij}]$ be a matrix of order $m\times n$ and let $y = f (X)$ be a scalar function of $X$ . The derivative of $y$ with respect to $X$ , denoted by $\frac{\partial y}{\partial X}$ , is defined as the following matrix of order $m\times n$ ,
$\frac{\partial y}{\partial X} = \begin{bmatrix} \frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{12}} & \cdots & \frac{\partial y}{\partial x_{1n}} \\ \frac{\partial y}{\partial x_{21}} & \frac{\partial y}{\partial x_{22}} & \cdots & \frac{\partial y}{\partial x_{2n}} \\ \vdots & \vdots & \ddots & \vdots& \\ \frac{\partial y}{\partial x_{m1}} & \frac{\partial y}{\partial x_{m2}} & \cdots & \frac{\partial y}{\partial x_{mn}} \end{bmatrix} = \Big[\frac{\partial y}{\partial x_{ij}} \Big]$

2.关于autograd的说明

torch.Tensor 是包的核心类。如果将其属性 tensor.requires_grad 设置为 True，则会开始跟踪针对 tensor 的所有操作。完成计算后，您可以调用 tensor.backward() 来自动计算所有梯度。该张量的梯度将累积到 tensor.grad 属性中。

要停止 tensor 历史记录的跟踪，您可以调用 tensor.detach()，它将其与计算历史记录分离，并防止将来的计算被跟踪。

要停止跟踪历史记录（和使用内存），您还可以将代码块使用 with torch.no_grad(): 包装起来。在评估模型时，这是特别有用，因为模型在训练阶段具有 requires_grad = True 的可训练参数有利于调参，但在评估阶段我们不需要梯度。

还有一个类对于 autograd 实现非常重要那就是 Function。Tensor 和 Function 互相连接并构建一个非循环图，它保存整个完整的计算过程的历史信息。每个张量都有一个 tensor.grad_fn 属性保存着创建了张量的 Function 的引用，（如果用户自己创建张量，则 grad_fn=None）。

如果你想计算导数，你可以调用 tensor.backward()。如果 Tensor 是标量（即它包含一个元素数据），则不需要指定任何参数backward()，但是如果它有更多元素，则需要指定一个gradient 参数来指定张量的形状。

最后的计算结果保存在tensor.grad属性里

使用tensor.requires_grad在初始化时，设置跟踪梯度

import torch
import numpy as np

x = torch.ones(2,2, requires_grad=True)
print(x)

结果如下

tensor([[1., 1.],[1., 1.]], requires_grad=True)

设置了跟踪梯度的tensor,将会出现tensor.grad_fn的属性，用于记录上次计算的Function

y = torch.add(x, 1)
print(y)
print(y.grad_fn)

结果如下

tensor([[2., 2.],[2., 2.]], grad_fn=<AddBackward0>)
<AddBackward0 object at 0x0000020D723EBE80>

tensor.requires_grad_(True / False) 会改变张量的 requires_grad 标记。如果没有提供相应的参数输入的标记默认为 False。

a = torch.randn(2,2)
a = (a * 3) / (a-1)
print(a)
a.requires_grad_(True)
print(a)
a = a + 1
print(a)

tensor([[  0.0646, -46.3478],[  5.6683,  -0.8896]])
tensor([[  0.0646, -46.3478],[  5.6683,  -0.8896]], requires_grad=True)
tensor([[  1.0646, -45.3478],[  6.6683,   0.1104]], grad_fn=<AddBackward0>)

3. grad的计算

3.1 Manual手动计算

可以使用函数torch.autograd.grad()来手动计算梯度，详细可参考此处

例如计算 $y = x_1^2 + x_2^2 + x_1x_2$ 的梯度

x1 = torch.tensor(3., requires_grad=True)
x2 = torch.tensor(1., requires_grad=True)
y = x1**2+x2**2+x1*x2# 求一阶导数
# torch.autograd.grad(y, x1,retain_graph=True, create_graph=True)
x1_1 = torch.autograd.grad(y, x1, retain_graph=True, create_graph=True)[0]
x2_1 = torch.autograd.grad(y, x2, retain_graph=True, create_graph=True)[0]
print(x1_1,x2_1)# 求二阶混合偏导数
x1_11 = torch.autograd.grad(x1_1, x1)[0]
x1_12 = torch.autograd.grad(x1_1, x2)[0]
x2_21 = torch.autograd.grad(x2_1, x1)[0]
x2_22 = torch.autograd.grad(x2_1, x2)[0]
print(x1_11,x1_12,x2_21,x2_22)

结果如下

tensor(7., grad_fn=<AddBackward0>) tensor(5., grad_fn=<AddBackward0>)
tensor(2.) tensor(1.) tensor(1.) tensor(2.)

3.2 backward()自动计算

当输出是标量scalar函数时
考虑如下的计算问题

x = torch.ones(2,2, requires_grad=True)
y = x + 2
print(y)
z = y * y * 3
out = z.mean()
print(z, out)  #输出out是一个标量
out.backward()
print(x.grad)

输出是

tensor([[3., 3.],[3., 3.]], grad_fn=<AddBackward0>)
tensor([[27., 27.],[27., 27.]], grad_fn=<MulBackward0>) tensor(27., grad_fn=<MeanBackward0>)
tensor([[4.5000, 4.5000],[4.5000, 4.5000]])

$=\begin{bmatrix} x_1 & x_2 \\ x_3 & x_4 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$

中间变量是

$=\begin{bmatrix} z_1 & z_2 \\ z_3 & z_4 \end{bmatrix} = \begin{bmatrix} 3(x_1+2)^2 & 3(x_1+2)^2 \\ 3(x_1+2)^2 & 3(x_1+2)^2 \end{bmatrix}$

最后获得是输出是

$\begin{aligned} \text{out} & = \frac{1}{4}\sum_{i=1} z_i = \frac{1}{4}(z_1+z_2+z_3+z_4) \\ & = \frac{1}{4}(3(x_1+2)^2+3(x_2+2)^2+3(x_3+2)^2+3(x_4+2)^2) \\ & = f(\mathrm{x}) \end{aligned}$

其中将矩阵 $X$ 和矩阵 $Z$ 中的所有元素拼接为向量

$\mathrm{x} = [x_1,x_2,x_3,x_4]^T \\ \mathrm{z} = [z_1,z_2,z_3,z_4]^T$

我们利用矩阵求导的链式法则

$\frac{\partial f}{\partial \mathrm{x}} = \frac{f(\mathrm{x})}{\partial \mathrm{x}} = \frac{\partial \mathrm{z}}{\partial \mathrm{x}} \frac{\partial f(\mathrm{x})}{\partial \mathrm{z}}$

再利用标量函数对矩阵导数的定义，则有

$\frac{\partial f}{\partial \mathrm{x}} = \begin{bmatrix} \frac{\partial z_1}{\partial x_1} & \frac{\partial z_2}{\partial x_1} & \frac{\partial z_3}{\partial x_1} & \frac{\partial z_4}{\partial x_1} \\ \frac{\partial z_1}{\partial x_2} & \frac{\partial z_2}{\partial x_2} & \frac{\partial z_3}{\partial x_2} & \frac{\partial z_4}{\partial x_2} \\ \frac{\partial z_1}{\partial x_3} & \frac{\partial z_2}{\partial x_3} & \frac{\partial z_3}{\partial x_3} & \frac{\partial z_4}{\partial x_3} \\ \frac{\partial z_1}{\partial x_4} & \frac{\partial z_2}{\partial x_4} & \frac{\partial z_3}{\partial x_4} & \frac{\partial z_4}{\partial x_4} \end{bmatrix} \begin{bmatrix} \frac{\partial f}{\partial z_1} \\ \frac{\partial f}{\partial z_2} \\ \frac{\partial f}{\partial z_3} \\ \frac{\partial f}{\partial z_4} \end{bmatrix}= \begin{bmatrix} 6(x_1+2) & 0 & 0 & 0 \\ 0 & 6(x_2+2) & 0 & 0 \\ 0 & 0 & 6(x_3+2) & 0 \\ 0 & 0 & 0 & 6(x_4+2) \end{bmatrix} \begin{bmatrix} \frac{1}{4} \\ \frac{1}{4} \\ \frac{1}{4} \\ \frac{1}{4} \end{bmatrix} = \begin{bmatrix} 4.5 \\ 4.5 \\ 4.5 \\ 4.5 \\ \end{bmatrix}$

所以最后获得关于的矩阵 $X$ 的导数为

$\frac{\partial f}{\partial X} = \begin{bmatrix} \frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_2} \\ \frac{\partial f}{\partial x_3} & \frac{\partial f}{\partial x_4} \end{bmatrix} = \begin{bmatrix} 4.5 & 4.5 \\ 4.5 & 4.5 \\ \end{bmatrix}$

当输出是张量tensor函数时

x = torch.tensor([[1.0, 2, 3],[4, 5, 6],[7, 8, 9]], requires_grad=True)
w = torch.tensor([[1.0, 2, 3],[4, 5, 6]], requires_grad=True)y = torch.matmul(x,w.T)
print(y)
print(torch.ones_like(y))
y.backward(gradient = torch.ones_like(y))
print(x.grad)

输出是

tensor([[ 14.,  32.],[ 32.,  77.],[ 50., 122.]], grad_fn=<MmBackward0>)
tensor([[1., 1.],[1., 1.],[1., 1.]])
tensor([[5., 7., 9.],[5., 7., 9.],[5., 7., 9.]])

$XW^T \\ \begin{bmatrix} y_{11} & y_{21} \\ y_{12} & y_{22} \\ y_{13} & y_{23} \\ \end{bmatrix} = \begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \\ \end{bmatrix} \begin{bmatrix} w_{11} & w_{21} \\ w_{12} & w_{22} \\ w_{13} & w_{23} \\ \end{bmatrix} \\ \begin{bmatrix} y_{11} & y_{21} \\ y_{12} & y_{22} \\ y_{13} & y_{23} \\ \end{bmatrix} = \begin{bmatrix} (x_{11}w_{11} + x_{12}w_{12} + x_{13}w_{13}) & (x_{11}w_{21} + x_{12}w_{22} + x_{13}w_{23}) \\ (x_{21}w_{11} + x_{22}w_{12} + x_{23}w_{13}) & (x_{21}w_{21} + x_{22}w_{22} + x_{23}w_{23}) \\ (x_{31}w_{11} + x_{32}w_{12} + x_{33}w_{13}) & (x_{31}w_{21} + x_{32}w_{22} + x_{33}w_{23}) \\ \end{bmatrix}$

gradient=torch.ones_like(y)用于指定矩阵 $Y$ 中每一项的权重都为1，由矩阵 $Y$ 中元素加权得到的scalar函数为

$\begin{aligned} f(\mathrm{x},\mathrm{w}) & = 1\times y_{11}+1\times y_{12}+1\times y_{13}+1\times y_{21}+1\times y_{22}+1\times y_{23}, \\ & \mathrm{x} = [x_{11}, x_{12}, x_{13}, x_{21}, x_{22}, x_{23}, x_{31}, x_{32}, x_{33}]^T \\ & \mathrm{w} = [w_{11}, w_{12}, w_{13}, w_{21}, w_{22}, w_{23}, w_{31}, w_{32}, w_{33}]^T \end{aligned}$

这里不包括复合求导，可以直接计算

$\begin{aligned} \frac{\partial f}{\partial \mathrm{x}} & = \Big[\frac{\partial f}{\partial x_{11}}, \frac{\partial f}{\partial x_{12}}, \frac{\partial f}{\partial x_{13}}, \frac{\partial f}{\partial x_{21}}, \frac{\partial f}{\partial x_{22}}, \frac{\partial f}{\partial x_{23}}, \frac{\partial f}{\partial x_{31}}, \frac{\partial f}{\partial x_{32}}, \frac{\partial f}{\partial x_{33}} \Big]^T \\ \frac{\partial f}{\partial \mathrm{x}} & = \Big[ w_{11} + w_{21}, w_{12} + w_{22}, w_{13} + w_{23}, w_{11} + w_{21}, w_{12} + w_{22}, w_{13} + w_{23}, w_{11} + w_{21}, w_{12} + w_{22}, w_{13} + w_{23} \Big]^T \\ & = [5, 7, 9, 5, 7, 9, 5, 7, 9]^T \end{aligned}$
再写成矩阵的形式则有

$\frac{\partial f}{\partial X} = \begin{bmatrix} 5 & 7 & 9 \\ 5 & 7 & 9 \\ 5 & 7 & 9 \\ \end{bmatrix}$

再考虑一个更一般求二阶导的情况

x = torch.ones(3, requires_grad=True)
print(x)
y = x * 2
print(y)
z = y * 2
print(z)
v = torch.tensor([0.1, 1.0, 0.0001], dtype=torch.float)
z.backward(gradient=v)
print(x.grad)

结果如下

tensor([1., 1., 1.], requires_grad=True)
tensor([2., 2., 2.], grad_fn=<MulBackward0>)
tensor([4., 4., 4.], grad_fn=<MulBackward0>)
tensor([4.0000e-01, 4.0000e+00, 4.0000e-04])

其中
$\begin{aligned} \mathrm{x} & = [x_1,x_2,x_3]^T = [1,1,1]^T \\ \mathrm{y} = 2\mathrm{x} & = [y_1,y_2,y_3]^T = [2,2,2]^T \\ \mathrm{z} = 2\mathrm{y} & = [z_1,z_2,z_3]^T = [4,4,4]^T \end{aligned}$
若考虑gradient=torch.tensor([a1,a2,a3], dtype=torch.folat)，那么最终加权得到的scalar函数为
$f = a_1 z_1 + a_2 z_2 + a_3 z_3$
那么对 $\mathrm{x}$ 求偏导则有
$\begin{aligned} \frac{\partial f}{\partial \mathrm{x}} & = \frac{\partial \mathrm{y}}{\partial \mathrm{x}} \frac{\partial f}{\partial \mathrm{y}} \\ & = \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_2}{\partial x_1} & \frac{\partial y_3}{\partial x_1} \\ \frac{\partial y_1}{\partial x_2} & \frac{\partial y_2}{\partial x_2} & \frac{\partial y_3}{\partial x_2} \\ \frac{\partial y_1}{\partial x_3} & \frac{\partial y_2}{\partial x_3} & \frac{\partial y_3}{\partial x_3} \\ \end{bmatrix} \begin{bmatrix} \frac{\partial f}{\partial y_1} \\ \frac{\partial f}{\partial y_2} \\ \frac{\partial f}{\partial y_3} \\ \end{bmatrix} \\ & = \begin{bmatrix} 2 & & \\ & 2 & \\ & & 2 \end{bmatrix} \begin{bmatrix} 2 a_1 \\ 2 a_2 \\ 2 a_3 \\ \end{bmatrix} \\ & = [4a_1, 4a_2, 4a_3]^T \end{aligned}$

一些启示

Reference

参考教程1
参考教程2

Pytorch Tutorial

文章目录

Chapter 2. Autograd

1. Review Matrix Calculus

1.1 Definition向量对向量求导

1.2 Definition标量对向量求导

1.3 Definition标量对矩阵求导

2.关于autograd的说明

3. grad的计算

3.1 Manual手动计算

3.2 backward()自动计算

Reference

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