torch.autograd.grad finds the second derivative

1 用法介绍

 pytorch中torch.autograd.gradThe function is mainly used to calculate and return the sum of the gradients of the output with respect to the input,The specific parameters are as follows：

torch.tril(input, diagonal=0, *, out=None) $*$Tensor

• outputs(sequence of Tensor)：represents the output of the differential function
• inputs (sequence of Tensor)：Represents the input to the differential function
• grad_outputs (sequence of Tensor)：表示“向量-雅克比矩阵”的向量
• retain_graph (bool, optional)：Indicates whether the computation graph needs to be released,It needs to be set to when calculating the second derivativeTrue
• create_graph (bool, optional)：Indicates whether the gradient needs to be added to the computational graph,It will be set as required when computing higher-order derivatives or other calculationsTrue
• allow_unused (bool, optional)：Indicates whether to return only the gradient of the input,Instead of returning the gradients of other leaf nodes

2 实例讲解

 Mathematical examples of specific second-order derivatives analytical solutions are given below

给定一个向量$\mathbf{x}=(x_1,x_2{)}^{\top }$,可以得到向量$\mathbf{y}=(y_1,y_2{)}^{\top }=(x_1^2,x_2^2{)}^{\top }$.对向量$\mathbf{y}$The loss function can be obtained by averaging the elements of ${\mathrm{loss}}_1$为：$${\mathrm{loss}}_1(\mathbf{x})=\mathrm{mean}(\mathbf{y})=\frac{x_1^2+x_2^2}{2}$$向量$\mathbf{y}$The components of the elements are$\mathbf{x}$求偏导,Then add and average to get the loss function${\mathrm{loss}}_2$为$$\{\begin{array}{rl}{h}_1(\mathbf{x})& =\frac{\partial y_1}{\partial \mathbf{x}}=(2x_1,0{)}^{\top }\\ h_2(\mathbf{x})& =\frac{\partial y_2}{\partial \mathbf{x}}=(0,2x_2{)}^{\top }\end{array},{\mathrm{loss}}_2(\mathbf{x})=\mathrm{mean}(h_1({\mathbf{x}}_1)-h_2({\mathbf{x}}_2))=x_1-x_2$$将损失函数${\mathrm{loss}}_1$与损失函数${\mathrm{loss}}_2$相加可以得到$$\mathrm{loss}(\mathbf{x})={\mathrm{loss}}_1(\mathbf{x})+{\mathrm{loss}}_2(\mathbf{x})=\frac{x_1^2+x_2^2}{2}+x_1-x_2$$最终损失函数$\mathrm{loss}$对向量$\mathbf{x}$的偏导数为$$\frac{\partial \mathrm{loss}}{\partial \mathbf{x}}=(x_1+1,x_2-1{)}^{\top }$$

以下为用pytorchImplement the code example corresponding to the second derivative：

import torch

x = torch.tensor([5.0, 7.0], requires_grad=True)
y = x**2

loss1 = torch.mean(y)

h1 = torch.autograd.grad(y[0], x, retain_graph = True, create_graph=True)
h2 = torch.autograd.grad(y[1], x, retain_graph = True, create_graph=True)
loss2 = torch.mean(h1[0] - h2[0])

loss = loss1 + loss2

result = torch.autograd.grad(loss, x)
print(result)

当向量$\mathbf{x}$取值为$(5,7{)}^{\top }$时,The second derivative obtained from the mathematical analytical solution is $(6,6{)}^{\top }$,The experimental results of the corresponding code run are also $(6,6)$.