# Learning PyTorch with Examples¶

**Author**: Justin Johnson

Note

This is one of our older PyTorch tutorials. You can view our latest beginner content in Learn the Basics.

This tutorial introduces the fundamental concepts of PyTorch through self-contained examples.

At its core, PyTorch provides two main features:

An n-dimensional Tensor, similar to numpy but can run on GPUs

Automatic differentiation for building and training neural networks

We will use a problem of fitting \(y=\sin(x)\) with a third order polynomial as our running example. The network will have four parameters, and will be trained with gradient descent to fit random data by minimizing the Euclidean distance between the network output and the true output.

Note

You can browse the individual examples at the end of this page.

Table of Contents

## Tensors¶

### Warm-up: numpy¶

Before introducing PyTorch, we will first implement the network using numpy.

Numpy provides an n-dimensional array object, and many functions for manipulating these arrays. Numpy is a generic framework for scientific computing; it does not know anything about computation graphs, or deep learning, or gradients. However we can easily use numpy to fit a third order polynomial to sine function by manually implementing the forward and backward passes through the network using numpy operations:

```
# -*- coding: utf-8 -*-
import numpy as np
import math
# Create random input and output data
x = np.linspace(-math.pi, math.pi, 2000)
y = np.sin(x)
# Randomly initialize weights
a = np.random.randn()
b = np.random.randn()
c = np.random.randn()
d = np.random.randn()
learning_rate = 1e-6
for t in range(2000):
# Forward pass: compute predicted y
# y = a + b x + c x^2 + d x^3
y_pred = a + b * x + c * x ** 2 + d * x ** 3
# Compute and print loss
loss = np.square(y_pred - y).sum()
if t % 100 == 99:
print(t, loss)
# Backprop to compute gradients of a, b, c, d with respect to loss
grad_y_pred = 2.0 * (y_pred - y)
grad_a = grad_y_pred.sum()
grad_b = (grad_y_pred * x).sum()
grad_c = (grad_y_pred * x ** 2).sum()
grad_d = (grad_y_pred * x ** 3).sum()
# Update weights
a -= learning_rate * grad_a
b -= learning_rate * grad_b
c -= learning_rate * grad_c
d -= learning_rate * grad_d
print(f'Result: y = {a} + {b} x + {c} x^2 + {d} x^3')
```

### PyTorch: Tensors¶

Numpy is a great framework, but it cannot utilize GPUs to accelerate its numerical computations. For modern deep neural networks, GPUs often provide speedups of 50x or greater, so unfortunately numpy won’t be enough for modern deep learning.

Here we introduce the most fundamental PyTorch concept: the **Tensor**.
A PyTorch Tensor is conceptually identical to a numpy array: a Tensor is
an n-dimensional array, and PyTorch provides many functions for
operating on these Tensors. Behind the scenes, Tensors can keep track of
a computational graph and gradients, but they’re also useful as a
generic tool for scientific computing.

Also unlike numpy, PyTorch Tensors can utilize GPUs to accelerate their numeric computations. To run a PyTorch Tensor on GPU, you simply need to specify the correct device.

Here we use PyTorch Tensors to fit a third order polynomial to sine function. Like the numpy example above we need to manually implement the forward and backward passes through the network:

```
# -*- coding: utf-8 -*-
import torch
import math
dtype = torch.float
device = torch.device("cpu")
# device = torch.device("cuda:0") # Uncomment this to run on GPU
# Create random input and output data
x = torch.linspace(-math.pi, math.pi, 2000, device=device, dtype=dtype)
y = torch.sin(x)
# Randomly initialize weights
a = torch.randn((), device=device, dtype=dtype)
b = torch.randn((), device=device, dtype=dtype)
c = torch.randn((), device=device, dtype=dtype)
d = torch.randn((), device=device, dtype=dtype)
learning_rate = 1e-6
for t in range(2000):
# Forward pass: compute predicted y
y_pred = a + b * x + c * x ** 2 + d * x ** 3
# Compute and print loss
loss = (y_pred - y).pow(2).sum().item()
if t % 100 == 99:
print(t, loss)
# Backprop to compute gradients of a, b, c, d with respect to loss
grad_y_pred = 2.0 * (y_pred - y)
grad_a = grad_y_pred.sum()
grad_b = (grad_y_pred * x).sum()
grad_c = (grad_y_pred * x ** 2).sum()
grad_d = (grad_y_pred * x ** 3).sum()
# Update weights using gradient descent
a -= learning_rate * grad_a
b -= learning_rate * grad_b
c -= learning_rate * grad_c
d -= learning_rate * grad_d
print(f'Result: y = {a.item()} + {b.item()} x + {c.item()} x^2 + {d.item()} x^3')
```

## Autograd¶

### PyTorch: Tensors and autograd¶

In the above examples, we had to manually implement both the forward and backward passes of our neural network. Manually implementing the backward pass is not a big deal for a small two-layer network, but can quickly get very hairy for large complex networks.

Thankfully, we can use automatic
differentiation
to automate the computation of backward passes in neural networks. The
**autograd** package in PyTorch provides exactly this functionality.
When using autograd, the forward pass of your network will define a
**computational graph**; nodes in the graph will be Tensors, and edges
will be functions that produce output Tensors from input Tensors.
Backpropagating through this graph then allows you to easily compute
gradients.

This sounds complicated, it’s pretty simple to use in practice. Each Tensor
represents a node in a computational graph. If `x`

is a Tensor that has
`x.requires_grad=True`

then `x.grad`

is another Tensor holding the
gradient of `x`

with respect to some scalar value.

Here we use PyTorch Tensors and autograd to implement our fitting sine wave with third order polynomial example; now we no longer need to manually implement the backward pass through the network:

### PyTorch: Defining new autograd functions¶

Under the hood, each primitive autograd operator is really two functions
that operate on Tensors. The **forward** function computes output
Tensors from input Tensors. The **backward** function receives the
gradient of the output Tensors with respect to some scalar value, and
computes the gradient of the input Tensors with respect to that same
scalar value.

In PyTorch we can easily define our own autograd operator by defining a
subclass of `torch.autograd.Function`

and implementing the `forward`

and `backward`

functions. We can then use our new autograd operator by
constructing an instance and calling it like a function, passing
Tensors containing input data.

In this example we define our model as \(y=a+b P_3(c+dx)\) instead of \(y=a+bx+cx^2+dx^3\), where \(P_3(x)=\frac{1}{2}\left(5x^3-3x\right)\) is the Legendre polynomial of degree three. We write our own custom autograd function for computing forward and backward of \(P_3\), and use it to implement our model:

## nn module¶

### PyTorch: nn¶

Computational graphs and autograd are a very powerful paradigm for defining complex operators and automatically taking derivatives; however for large neural networks raw autograd can be a bit too low-level.

When building neural networks we frequently think of arranging the
computation into **layers**, some of which have **learnable parameters**
which will be optimized during learning.

In TensorFlow, packages like Keras, TensorFlow-Slim, and TFLearn provide higher-level abstractions over raw computational graphs that are useful for building neural networks.

In PyTorch, the `nn`

package serves this same purpose. The `nn`

package defines a set of **Modules**, which are roughly equivalent to
neural network layers. A Module receives input Tensors and computes
output Tensors, but may also hold internal state such as Tensors
containing learnable parameters. The `nn`

package also defines a set
of useful loss functions that are commonly used when training neural
networks.

In this example we use the `nn`

package to implement our polynomial model
network:

```
# -*- coding: utf-8 -*-
import torch
import math
# Create Tensors to hold input and outputs.
x = torch.linspace(-math.pi, math.pi, 2000)
y = torch.sin(x)
# For this example, the output y is a linear function of (x, x^2, x^3), so
# we can consider it as a linear layer neural network. Let's prepare the
# tensor (x, x^2, x^3).
p = torch.tensor([1, 2, 3])
xx = x.unsqueeze(-1).pow(p)
# In the above code, x.unsqueeze(-1) has shape (2000, 1), and p has shape
# (3,), for this case, broadcasting semantics will apply to obtain a tensor
# of shape (2000, 3)
# Use the nn package to define our model as a sequence of layers. nn.Sequential
# is a Module which contains other Modules, and applies them in sequence to
# produce its output. The Linear Module computes output from input using a
# linear function, and holds internal Tensors for its weight and bias.
# The Flatten layer flatens the output of the linear layer to a 1D tensor,
# to match the shape of `y`.
model = torch.nn.Sequential(
torch.nn.Linear(3, 1),
torch.nn.Flatten(0, 1)
)
# The nn package also contains definitions of popular loss functions; in this
# case we will use Mean Squared Error (MSE) as our loss function.
loss_fn = torch.nn.MSELoss(reduction='sum')
learning_rate = 1e-6
for t in range(2000):
# Forward pass: compute predicted y by passing x to the model. Module objects
# override the __call__ operator so you can call them like functions. When
# doing so you pass a Tensor of input data to the Module and it produces
# a Tensor of output data.
y_pred = model(xx)
# Compute and print loss. We pass Tensors containing the predicted and true
# values of y, and the loss function returns a Tensor containing the
# loss.
loss = loss_fn(y_pred, y)
if t % 100 == 99:
print(t, loss.item())
# Zero the gradients before running the backward pass.
model.zero_grad()
# Backward pass: compute gradient of the loss with respect to all the learnable
# parameters of the model. Internally, the parameters of each Module are stored
# in Tensors with requires_grad=True, so this call will compute gradients for
# all learnable parameters in the model.
loss.backward()
# Update the weights using gradient descent. Each parameter is a Tensor, so
# we can access its gradients like we did before.
with torch.no_grad():
for param in model.parameters():
param -= learning_rate * param.grad
# You can access the first layer of `model` like accessing the first item of a list
linear_layer = model[0]
# For linear layer, its parameters are stored as `weight` and `bias`.
print(f'Result: y = {linear_layer.bias.item()} + {linear_layer.weight[:, 0].item()} x + {linear_layer.weight[:, 1].item()} x^2 + {linear_layer.weight[:, 2].item()} x^3')
```

### PyTorch: optim¶

Up to this point we have updated the weights of our models by manually
mutating the Tensors holding learnable parameters with `torch.no_grad()`

.
This is not a huge burden for simple optimization algorithms like stochastic
gradient descent, but in practice we often train neural networks using more
sophisticated optimizers like AdaGrad, RMSProp, Adam, etc.

The `optim`

package in PyTorch abstracts the idea of an optimization
algorithm and provides implementations of commonly used optimization
algorithms.

In this example we will use the `nn`

package to define our model as
before, but we will optimize the model using the RMSprop algorithm provided
by the `optim`

package:

### PyTorch: Custom nn Modules¶

Sometimes you will want to specify models that are more complex than a
sequence of existing Modules; for these cases you can define your own
Modules by subclassing `nn.Module`

and defining a `forward`

which
receives input Tensors and produces output Tensors using other
modules or other autograd operations on Tensors.

In this example we implement our third order polynomial as a custom Module subclass:

### PyTorch: Control Flow + Weight Sharing¶

As an example of dynamic graphs and weight sharing, we implement a very strange model: a third-fifth order polynomial that on each forward pass chooses a random number between 3 and 5 and uses that many orders, reusing the same weights multiple times to compute the fourth and fifth order.

For this model we can use normal Python flow control to implement the loop, and we can implement weight sharing by simply reusing the same parameter multiple times when defining the forward pass.

We can easily implement this model as a Module subclass: