 # 👽 Computing hessian of a simple autograd function

Dear community, does anyone managed to compute a hessian of an arbitrary autograd function?

Let’s consider simple quadratic function: f(x) = x.T A x, it’s gradient is (A + A.T)x and hessian is just (A + A.T).

I have two code snippets, first works:

``````import torch
from torch import Tensor
from torch.autograd import Variable
from torch.autograd import grad
from torch import nn

torch.manual_seed(623)

x = Variable(torch.ones(2,1), requires_grad=True)
A = torch.FloatTensor([[1,2],[3,4]])

print(A)
print(x)

f = x.view(-1) @ A @ x
print(f)

x_1grad, = grad(f, x, create_graph=True)
print(x_1grad)
print(A @ x + A.t() @ x)

x_2grad0, = grad(x_1grad, x, create_graph=True)
x_2grad1, = grad(x_1grad, x, create_graph=True)

Hessian = torch.cat((x_2grad0, x_2grad1), dim=1)
print(Hessian)

print(A + A.t())
``````

while the second does not work.

``````import torch
from torch import Tensor
from torch.autograd import Variable
from torch.autograd import grad
from torch import nn

torch.manual_seed(623)

class Quadro(torch.autograd.Function):

def __init__(self, A):
self.A = torch.FloatTensor(A)

def forward(self, input):

self.save_for_backward(input)
return input.t() @ self.A @ input

def backward(self, grad_output):

input, = self.saved_tensors
input = torch.autograd.Variable(input, requires_grad = True)
grad_input = torch.mul(grad_output, (self.A + self.A.t()) @ input)
return grad_input

x = Variable(torch.ones(2,1), requires_grad=True)
A = torch.FloatTensor([[1,2],[3,4]])

print(A)
print(x)

qq = Quadro(A)
f = qq(x)
print(f)

x_1grad, = grad(f, x, create_graph=True, retain_graph=True)
print(x_1grad)
print(A @ x + A.t() @ x)

x_2grad0, = grad(x_1grad, x, create_graph=True)
x_2grad1, = grad(x_1grad, x, create_graph=True)

Hessian = torch.cat((x_2grad0, x_2grad1), dim=1)
print(Hessian)

print(A + A.t())
``````

The problem is that I cant differentiate the first derivative, since it has no grad_fn attribute, but I don’t know how to fix it.

P.S. I completely understand, that PyTorch framework was mostly done to train neural networks, not such simple things, but I would be happy, if someone could help me.

What error are you getting? I think you need to use retain_graph=True otherwise with one time gradient calculation the buffer will be zeroed.