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Method 1: Create tensor with gradients

It is very similar to creating a tensor, all you need to do is to add an additional argument.

import torch

a = torch.ones((2, 2), requires_grad=True)
a

tensor([[ 1.,  1.],
[ 1.,  1.]])


This should return True otherwise you've not done it right.

a.requires_grad


True


Method 2: Create tensor with gradients

This allows you to create a tensor as usual then an additional line to allow it to accumulate gradients.

# Normal way of creating gradients
a = torch.ones((2, 2))


True


A tensor without gradients just for comparison

If you do not do either of the methods above, you'll realize you will get False for checking for gradients.

# Not a variable


no_gradient.requires_grad

False


# Behaves similarly to tensors
print(a + b)

tensor([[ 2.,  2.],
[ 2.,  2.]])

tensor([[ 2.,  2.],
[ 2.,  2.]])


As usual, the operations we learnt previously for tensors apply for tensors with gradients. Feel free to try divisions, mean or standard deviation!

print(a * b)
print(torch.mul(a, b))


tensor([[ 1.,  1.],
[ 1.,  1.]])
tensor([[ 1.,  1.],
[ 1.,  1.]])


### Manually and Automatically Calculating Gradients¶

What exactly is requires_grad? - Allows calculation of gradients w.r.t. the tensor that all allows gradients accumulation

y_i = 5(x_i+1)^2

Create tensor of size 2x1 filled with 1's that requires gradient

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

tensor([ 1.,  1.])


Simple linear equation with x tensor created

y_i\bigr\rvert_{x_i=1} = 5(1 + 1)^2 = 5(2)^2 = 5(4) = 20

We should get a value of 20 by replicating this simple equation

y = 5 * (x + 1) ** 2
y

tensor([ 20.,  20.])


Simple equation with y tensor

Backward should be called only on a scalar (i.e. 1-element tensor) or with gradient w.r.t. the variable

Let's reduce y to a scalar then...

o = \frac{1}{2}\sum_i y_i

As you can see above, we've a tensor filled with 20's, so average them would return 20

o = (1/2) * torch.sum(y)
o

tensor(20.)


Calculating first derivative

Recap y equation: $y_i = 5(x_i+1)^2$

Recap o equation: $o = \frac{1}{2}\sum_i y_i$

Substitute y into o equation: $o = \frac{1}{2} \sum_i 5(x_i+1)^2$

\frac{\partial o}{\partial x_i} = \frac{1}{2}[10(x_i+1)]
\frac{\partial o}{\partial x_i}\bigr\rvert_{x_i=1} = \frac{1}{2}[10(1 + 1)] = \frac{10}{2}(2) = 10

We should expect to get 10, and it's so simple to do this with PyTorch with the following line...

Get first derivative:

o.backward()


Print out first derivative:

x.grad

tensor([ 10.,  10.])


If x requires gradient and you create new objects with it, you get all gradients

print(x.requires_grad)

True
True
True


# Summary¶

We've learnt to...

Success

• Substitute equation with x values
• Reduce to scalar output, o through mean
• Calculate gradients with o.backward()
• Then access gradients of the x tensor with requires_grad through x.grad