# vahidk/EffectivePyTorch

PyTorch tutorials and best practices.

repo name | vahidk/EffectivePyTorch |

repo link | https://github.com/vahidk/EffectivePyTorch |

homepage | https://twitter.com/VahidK |

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stars (curr.) | 291 |

created | 2020-04-02 |

license | |

# Effective PyTorch

# Table of Contents

## Part I: PyTorch Fundamentals

- PyTorch basics
- Encapsulate your model with Modules
- Broadcasting the good and the ugly
- Take advantage of the overloaded operators
- Optimizing runtime with TorchScript
- Numerical stability in PyTorch

*To install PyTorch follow the instructions on the official website:*

```
pip install torch torchvision
```

*We aim to gradually expand this series by adding new articles and keep the content up to date with the latest releases of PyTorch API. If you have suggestions on how to improve this series or find the explanations ambiguous, feel free to create an issue, send patches, or reach out by email.*

# Part I: PyTorch Fundamentals

## PyTorch basics

PyTorch is one of the most popular libraries for numerical computation and currently is amongst the most widely used libraries for performing machine learning research. In many ways PyTorch is similar to NumPy, with the additional benefit that PyTorch allows you to perform your computations on CPUs, GPUs, and TPUs without any material change to your code. PyTorch also makes it easy to distribute your computation across multiple devices or machines. One of the most important features of PyTorch is automatic differentiation. It allows computing the gradients of your functions analytically in an efficient manner which is crucial for training machine learning models using gradient descent method. Our goal here is to provide a gentle introduction to PyTorch and discuss best practices for using PyTorch.

The first thing to learn about PyTorch is the concept of Tensors. Tensors are simply multidimensional arrays. A PyTorch Tensor is very similar to a NumPy array with some ~~magical~~ additional functionality.

A tensor can store a scalar value:

```
import torch
a = torch.tensor(3)
print(a) # tensor(3)
```

or an array:

```
b = torch.tensor([1, 2])
print(b) # tensor([1, 2])
```

a matrix:

```
c = torch.zeros([2, 2])
print(c) # tensor([[0., 0.], [0., 0.]])
```

or any arbitrary dimensional tensor:

```
d = torch.rand([2, 2, 2])
```

Tensors can be used to perform algebraic operations efficiently. One of the most commonly used operations in machine learning applications is matrix multiplication. Say you want to multiply two random matrices of size 3x5 and 5x4, this can be done with the matrix multiplication (@) operation:

```
import torch
x = torch.randn([3, 5])
y = torch.randn([5, 4])
z = x @ y
print(z)
```

Similarly, to add two vectors, you can do:

```
z = x + y
```

To convert a tensor into a numpy array you can call Tensor’s numpy() method:

```
print(z.numpy())
```

And you can always convert a numpy array into a tensor by:

```
x = torch.tensor(np.random.normal([3, 5]))
```

### Automatic differentiation

The most important advantage of PyTorch over NumPy is its automatic differentiation functionality which is very useful in optimization applications such as optimizing parameters of a neural network. Let’s try to understand it with an example.

Say you have a composite function which is a chain of two functions: g(u(x)). To compute the derivative of g with respect to x we can use the chain rule which states that: dg/dx = dg/du * du/dx. PyTorch can analytically compute the derivatives for us.

To compute the derivatives in PyTorch first we create a tensor and set its requires_grad to true. We can use tensor operations to define our functions. We assume u is a quadratic function and g is a simple linear function:

```
x = torch.tensor(1.0, requires_grad=True)
def u(x):
return x * x
def g(u):
return -u
```

In this case our composite function is g(u(x)) = -x*x. So its derivative with respect to x is -2x. At point x=1, this is equal to -2.

Let’s verify this. This can be done using grad function in PyTorch:

```
dgdx = torch.autograd.grad(g(u(x)), x)[0]
print(dgdx) # tensor(-2.)
```

### Curve fitting

To understand how powerful automatic differentiation can be let’s have a look at another example. Assume that we have samples from a curve (say f(x) = 5x^2 + 3) and we want to estimate f(x) based on these samples. We define a parametric function g(x, w) = w0 x^2 + w1 x + w2, which is a function of the input x and latent parameters w, our goal is then to find the latent parameters such that g(x, w) ≈ f(x). This can be done by minimizing the following loss function: L(w) = ∑ (f(x) - g(x, w))^2. Although there’s a closed form solution for this simple problem, we opt to use a more general approach that can be applied to any arbitrary differentiable function, and that is using stochastic gradient descent. We simply compute the average gradient of L(w) with respect to w over a set of sample points and move in the opposite direction.

Here’s how it can be done in PyTorch:

```
import numpy as np
import torch
# Assuming we know that the desired function is a polynomial of 2nd degree, we
# allocate a vector of size 3 to hold the coefficients and initialize it with
# random noise.
w = torch.tensor(torch.randn([3, 1]), requires_grad=True)
# We use the Adam optimizer with learning rate set to 0.1 to minimize the loss.
opt = torch.optim.Adam([w], 0.1)
def model(x):
# We define yhat to be our estimate of y.
f = torch.stack([x * x, x, torch.ones_like(x)], 1)
yhat = torch.squeeze(f @ w, 1)
return yhat
def compute_loss(y, yhat):
# The loss is defined to be the mean squared error distance between our
# estimate of y and its true value.
loss = torch.nn.functional.mse_loss(yhat, y)
return loss
def generate_data():
# Generate some training data based on the true function
x = torch.rand(100) * 20 - 10
y = 5 * x * x + 3
return x, y
def train_step():
x, y = generate_data()
yhat = model(x)
loss = compute_loss(y, yhat)
opt.zero_grad()
loss.backward()
opt.step()
for _ in range(1000):
train_step()
print(w.detach().numpy())
```

By running this piece of code you should see a result close to this:

```
[4.9924135, 0.00040895029, 3.4504161]
```

Which is a relatively close approximation to our parameters.

This is just tip of the iceberg for what PyTorch can do. Many problems such as optimizing large neural networks with millions of parameters can be implemented efficiently in PyTorch in just a few lines of code. PyTorch takes care of scaling across multiple devices, and threads, and supports a variety of platforms.

## Encapsulate your model with Modules

In the previous example we used bare bone tensors and tensor oeprations to build our model. To make your code slightly more organized it’s recommended to use PyTorch’s modules. A module is simply a container for your parameters and encapsulates model operations. For example say you want to represent a linear model y = ax + b. This model can be represented with the following code:

```
import torch
class Net(torch.nn.Module):
def __init__(self):
super().__init__()
self.a = torch.nn.Parameter(torch.rand(1))
self.b = torch.nn.Parameter(torch.rand(1))
def forward(self, x):
yhat = self.a * x + self.b
return yhat
```

To use this model in practice you instantiate the module and simply call it like a function:

```
x = torch.arange(100, dtype=torch.float32)
net = Net()
y = net(x)
```

Parameters are essentially tensors with requires_grad set to true. It’s convenient to use parameters because you can simply retrieve them all with module’s parameters() method:

```
for p in net.parameters():
print(p)
```

Now, say you have an unknown function y = 5x^2 + 3 + some noise, and you want to optimize the parameters of your model to fit this function. You can start by sampling some points from your function:

```
x = torch.arange(100, dtype=torch.float32) / 100
y = 5 * x + 3 + torch.rand(100) * 0.3
```

Similar to the previous example, you can define a loss function and optimize the parameters of your model as follows:

```
criterion = torch.nn.MSELoss()
optimizer = torch.optim.SGD(net.parameters(), lr=0.01)
for i in range(10000):
net.zero_grad()
yhat = net(x)
loss = criterion(yhat, y)
loss.backward()
optimizer.step()
print(net.a, net.b) # Should be close to 5 and 3
```

PyTorch comes with a number of predefined modules. One such module is torch.nn.Linear which is a more general form of a linear function than what we defined above. We can rewrite our module above using torch.nn.Linear like this:

```
class Net(torch.nn.Module):
def __init__(self):
super().__init__()
self.linear = torch.nn.Linear(1, 1)
def forward(self, x):
yhat = self.linear(x.unsqueeze(1)).squeeze(1)
return yhat
```

Note that we used squeeze and unsqueeze since torch.nn.Linear operates on batch of vectors as opposed to scalars.

By default calling paramters() on a module will return the paramters of all its submodules:

```
net = Net()
for p in net.parameters():
print(p)
```

There are some predefined modules that act as a container for other modules. The most commonly used container module is torch.nn.Sequential. As its name implies it’s used to to stack multiple modules (or layers) on top of each other. For example to stack two Linear layers with a ReLU nonlinearity in between you can do:

```
model = torch.nn.Sequential(
torch.nn.Linear(64, 32),
torch.nn.ReLU(),
torch.nn.Linear(32, 10),
)
```

## Optimizing runtime with TorchScript

PyTorch is optimized to perform operations on large tensors. Doing many operations on small tensors is quite inefficient in PyTorch. So, whenever possible you should rewrite your computations in batch form to reduce overhead and improve performance. If there’s no way you can manually batch your operations, using TorchScript may improve your code’s performance. TorchScript is simply a subset of Python functions that are recognized by PyTorch. PyTorch can automatically optimize your TorchScript code using its just in time (jit) compiler and reduce some overheads.

Let’s look at an example. A very common operation in ML applications is “batch gather”. This operation can simply written as output[i] = input[i, index[i]]. This can be simply implemented in PyTorch as follows:

```
import torch
def batch_gather(tensor, indices):
output = []
for i in range(tensor.size(0)):
output += [tensor[i][indices[i]]]
return torch.stack(output)
```

To implement the same function using TorchScript simply use the torch.jit.script decorator:

```
@torch.jit.script
def batch_gather_jit(tensor, indices):
output = []
for i in range(tensor.size(0)):
output += [tensor[i][indices[i]]]
return torch.stack(output)
```

On my tests this is about 10% faster.

But nothing beats manually batching your operations. A vectorized implementation in my tests is 100 times faster:

```
def batch_gather_vec(tensor, indices):
shape = list(tensor.shape)
flat_first = torch.reshape(
tensor, [shape[0] * shape[1]] + shape[2:])
offset = torch.reshape(
torch.arange(shape[0]).cuda() * shape[1],
[shape[0]] + [1] * (len(indices.shape) - 1))
output = flat_first[indices + offset]
return output
```

## Broadcasting the good and the ugly

PyTorch supports broadcasting elementwise operations. Normally when you want to perform operations like addition and multiplication, you need to make sure that shapes of the operands match, e.g. you can’t add a tensor of shape [3, 2] to a tensor of shape [3, 4]. But there’s a special case and that’s when you have a singular dimension. PyTorch implicitly tiles the tensor across its singular dimensions to match the shape of the other operand. So it’s valid to add a tensor of shape [3, 2] to a tensor of shape [3, 1]

```
import torch
a = torch.tensor([[1., 2.], [3., 4.]])
b = torch.tensor([[1.], [2.]])
# c = a + b.repeat([1, 2])
c = a + b
print(c)
```

Broadcasting allows us to perform implicit tiling which makes the code shorter, and more memory efficient, since we don’t need to store the result of the tiling operation. One neat place that this can be used is when combining features of varying length. In order to concatenate features of varying length we commonly tile the input tensors, concatenate the result and apply some nonlinearity. This is a common pattern across a variety of neural network architectures:

```
a = torch.rand([5, 3, 5])
b = torch.rand([5, 1, 6])
linear = torch.nn.Linear(11, 10)
# concat a and b and apply nonlinearity
tiled_b = b.repeat([1, 3, 1])
c = torch.cat([a, tiled_b], 2)
d = torch.nn.functional.relu(linear(c))
print(d.shape) # torch.Size([5, 3, 10])
```

But this can be done more efficiently with broadcasting. We use the fact that f(m(x + y)) is equal to f(mx + my). So we can do the linear operations separately and use broadcasting to do implicit concatenation:

```
a = torch.rand([5, 3, 5])
b = torch.rand([5, 1, 6])
linear1 = torch.nn.Linear(5, 10)
linear2 = torch.nn.Linear(6, 10)
pa = linear1(a)
pb = linear2(b)
d = torch.nn.functional.relu(pa + pb)
print(d.shape) # torch.Size([5, 3, 10])
```

In fact this piece of code is pretty general and can be applied to tensors of arbitrary shape as long as broadcasting between tensors is possible:

```
class Merge(torch.nn.Module):
def __init__(self, in_features1, in_features2, out_features, activation=None):
super().__init__()
self.linear1 = torch.nn.Linear(in_features1, out_features)
self.linear2 = torch.nn.Linear(in_features2, out_features)
self.activation = activation
def forward(self, a, b):
pa = self.linear1(a)
pb = self.linear2(b)
c = pa + pb
if self.activation is not None:
c = self.activation(c)
return c
```

So far we discussed the good part of broadcasting. But what’s the ugly part you may ask? Implicit assumptions almost always make debugging harder to do. Consider the following example:

```
a = torch.tensor([[1.], [2.]])
b = torch.tensor([1., 2.])
c = torch.sum(a + b)
print(c)
```

What do you think the value of c would be after evaluation? If you guessed 6, that’s wrong. It’s going to be 12. This is because when rank of two tensors don’t match, PyTorch automatically expands the first dimension of the tensor with lower rank before the elementwise operation, so the result of addition would be [[2, 3], [3, 4]], and the reducing over all parameters would give us 12.

The way to avoid this problem is to be as explicit as possible. Had we specified which dimension we would want to reduce across, catching this bug would have been much easier:

```
a = torch.tensor([[1.], [2.]])
b = torch.tensor([1., 2.])
c = torch.sum(a + b, 0)
print(c)
```

Here the value of c would be [5, 7], and we immediately would guess based on the shape of the result that there’s something wrong. A general rule of thumb is to always specify the dimensions in reduction operations and when using torch.squeeze.

## Take advantage of the overloaded operators

Just like NumPy, PyTorch overloads a number of python operators to make PyTorch code shorter and more readable.

The slicing op is one of the overloaded operators that can make indexing tensors very easy:

```
z = x[begin:end] # z = torch.narrow(0, begin, end-begin)
```

Be very careful when using this op though. The slicing op, like any other op, has some overhead. Because it’s a common op and innocent looking it may get overused a lot which may lead to inefficiencies. To understand how inefficient this op can be let’s look at an example. We want to manually perform reduction across the rows of a matrix:

```
import torch
import time
x = torch.rand([500, 10])
z = torch.zeros([10])
start = time.time()
for i in range(500):
z += x[i]
print("Took %f seconds." % (time.time() - start))
```

This runs quite slow and the reason is that we are calling the slice op 500 times, which adds a lot of overhead. A better choice would have been to use torch.unbind op to slice the matrix into a list of vectors all at once:

```
z = torch.zeros([10])
for x_i in torch.unbind(x):
z += x_i
```

This is significantly (~30% on my machine) faster.

Of course, the right way to do this simple reduction is to use torch.sum op to this in one op:

```
z = torch.sum(x, dim=0)
```

which is extremely fast (~100x faster on my machine).

PyTorch also overloads a range of arithmetic and logical operators:

```
z = -x # z = torch.neg(x)
z = x + y # z = torch.add(x, y)
z = x - y
z = x * y # z = torch.mul(x, y)
z = x / y # z = torch.div(x, y)
z = x // y
z = x % y
z = x ** y # z = torch.pow(x, y)
z = x @ y # z = torch.matmul(x, y)
z = x > y
z = x >= y
z = x < y
z = x <= y
z = abs(x) # z = torch.abs(x)
z = x & y
z = x | y
z = x ^ y # z = torch.logical_xor(x, y)
z = ~x # z = torch.logical_not(x)
z = x == y # z = torch.eq(x, y)
z = x != y # z = torch.ne(x, y)
```

You can also use the augmented version of these ops. For example `x += y`

and `x **= 2`

are also valid.

Note that Python doesn’t allow overloading “and”, “or”, and “not” keywords.

## Numerical stability in PyTorch

When using any numerical computation library such as NumPy or PyTorch, it’s important to note that writing mathematically correct code doesn’t necessarily lead to correct results. You also need to make sure that the computations are stable.

Let’s start with a simple example. From primary school we know that x * y / y is equal to x for any non zero value of x. But let’s see if that’s always true in practice:

```
import numpy as np
x = np.float32(1)
y = np.float32(1e-50) # y would be stored as zero
z = x * y / y
print(z) # prints nan
```

The reason for the incorrect result is that y is simply too small for float32 type. A similar problem occurs when y is too large:

```
y = np.float32(1e39) # y would be stored as inf
z = x * y / y
print(z) # prints nan
```

The smallest positive value that float32 type can represent is 1.4013e-45 and anything below that would be stored as zero. Also, any number beyond 3.40282e+38, would be stored as inf.

```
print(np.nextafter(np.float32(0), np.float32(1))) # prints 1.4013e-45
print(np.finfo(np.float32).max) # print 3.40282e+38
```

To make sure that your computations are stable, you want to avoid values with small or very large absolute value. This may sound very obvious, but these kind of problems can become extremely hard to debug especially when doing gradient descent in PyTorch. This is because you not only need to make sure that all the values in the forward pass are within the valid range of your data types, but also you need to make sure of the same for the backward pass (during gradient computation).

Let’s look at a real example. We want to compute the softmax over a vector of logits. A naive implementation would look something like this:

```
import torch
def unstable_softmax(logits):
exp = torch.exp(logits)
return exp / torch.sum(exp)
print(unstable_softmax(torch.tensor([1000., 0.])).numpy()) # prints [ nan, 0.]
```

Note that computing the exponential of logits for relatively small numbers results to gigantic results that are out of float32 range. The largest valid logit for our naive softmax implementation is ln(3.40282e+38) = 88.7, anything beyond that leads to a nan outcome.

But how can we make this more stable? The solution is rather simple. It’s easy to see that exp(x - c) / ∑ exp(x - c) = exp(x) / ∑ exp(x). Therefore we can subtract any constant from the logits and the result would remain the same. We choose this constant to be the maximum of logits. This way the domain of the exponential function would be limited to [-inf, 0], and consequently its range would be [0.0, 1.0] which is desirable:

```
import torch
def softmax(logits):
exp = torch.exp(logits - torch.reduce_max(logits))
return exp / torch.sum(exp)
print(softmax(torch.tensor([1000., 0.])).numpy()) # prints [ 1., 0.]
```

Let’s look at a more complicated case. Consider we have a classification problem. We use the softmax function to produce probabilities from our logits. We then define our loss function to be the cross entropy between our predictions and the labels. Recall that cross entropy for a categorical distribution can be simply defined as xe(p, q) = -∑ p_i log(q_i). So a naive implementation of the cross entropy would look like this:

```
def unstable_softmax_cross_entropy(labels, logits):
logits = torch.log(softmax(logits))
return -torch.sum(labels * logits)
labels = torch.tensor([0.5, 0.5])
logits = torch.tensor([1000., 0.])
xe = unstable_softmax_cross_entropy(labels, logits)
print(xe.numpy()) # prints inf
```

Note that in this implementation as the softmax output approaches zero, the log’s output approaches infinity which causes instability in our computation. We can rewrite this by expanding the softmax and doing some simplifications:

```
def softmax_cross_entropy(labels, logits, dim=-1):
scaled_logits = logits - torch.max(logits)
normalized_logits = scaled_logits - torch.logsumexp(scaled_logits, dim)
return -torch.sum(labels * normalized_logits)
labels = torch.tensor([0.5, 0.5])
logits = torch.tensor([1000., 0.])
xe = softmax_cross_entropy(labels, logits)
print(xe.numpy()) # prints 500.0
```

We can also verify that the gradients are also computed correctly:

```
logits.requires_grad_(True)
xe = softmax_cross_entropy(labels, logits)
g = torch.autograd.grad(xe, logits)[0]
print(g.numpy()) # prints [0.5, -0.5]
```

which is correct.

Let me remind again that extra care must be taken when doing gradient descent to make sure that the range of your functions as well as the gradients for each layer are within a valid range. Exponential and logarithmic functions when used naively are especially problematic because they can map small numbers to enormous ones and the other way around.