How PyTorch differentiates on non-scalar variable?

autograd
1

I am new to PyTorch and Neural Network in general. I am following this tutorial: A Gentle Introduction to torch.autograd — PyTorch Tutorials 2.2.0+cu121 documentation

There are two examples of differentiation using backward(): one for scalar and one for non-scalar variable.

The scalar example:
imgur: the simple image sharer .png

The non-scalar example:
back2

x = torch.randn(3) # input is taken randomly
x = Variable(x, requires_grad=True)

y = x * 2

c = 0
while y.data.norm() < 1000:
y = y * 2
c += 1

gradients = torch.FloatTensor([0.1, 1.0, 0.0001]) # specifying gradient because input is non-scalar
y.backward(gradients)

print( c )
print(x.grad)

Output:

9

102.4000
1024.0000
0.1024

I tried to understand it as I did for scalar example:

But, I can’t figure out how it works.
I get exact same output for different values of c regardless of the input values:

8

51.2000
512.0000
0.0512

10

204.8000
2048.0000
0.2048

I know the math behind getting 4.5000 4.5000 4.5000 4.5000 for scalar example (shown above). Can you explain the math behind getting 102.4000 1024.0000 0.1024 for non-scalar example. Please explain how it is calculating y.backward(gradients)

I have also asked this question on stackexchange, but no one seems to answer it:

2

They are plenty of sources on the forum, you should do some search before asking. For example, see discussion here

3

we all know the chain rule.

dz/dx =  dz/dy * dy/dx

here
x.grad is dz/dx
[0.1, 1.0, 0.0001 ] are dz/dy
y = 1024*x so dy/dx =1024
…

hope it maybe helpful

4

Thanks.
Could you please explain why is y = 1024 * x?
If y = (y * 2)^c, and c = 9, I am getting 102.4000 1024.0000 0.1024
But, when c = 8, I am getting 51.2000 512.0000 0.0512.
And when c = 10 I am getting 204.8000 2048.0000 0.2048
Why is that?

5 
y = x*2
y = y*2
y = y*2
.....

repeat it c times
and y = x*(2^(c+1))

6

Thank you so much!, you’re the best. I updated my solution:

back3
back3562×541 25.6 KB

1 Like