Training Neural Networks

This is from Udacity's Deep Learning Repository which supports their Deep Learning Nanodegree.

The network we built in the previous part isn't so smart, it doesn't know anything about our handwritten digits. Neural networks with non-linear activations work like universal function approximators. There is some function that maps your input to the output. For example, images of handwritten digits to class probabilities. The power of neural networks is that we can train them to approximate this function, and basically any function given enough data and compute time.

At first the network is naive, it doesn't know the function mapping the inputs to the outputs. We train the network by showing it examples of real data, then adjusting the network parameters such that it approximates this function.

To find these parameters, we need to know how poorly the network is predicting the real outputs. For this we calculate a loss function (also called the cost), a measure of our prediction error. For example, the mean squared loss is often used in regression and binary classification problems

\[ \large \ell = \frac{1}{2n}\sum_i^n{\left(y_i - \hat{y}_i\right)^2} \]

where \(n\) is the number of training examples, \(y_i\) are the true labels, and \(\hat{y}_i\) are the predicted labels.

By minimizing this loss with respect to the network parameters, we can find configurations where the loss is at a minimum and the network is able to predict the correct labels with high accuracy. We find this minimum using a process called gradient descent. The gradient is the slope of the loss function and points in the direction of fastest change. To get to the minimum in the least amount of time, we then want to follow the gradient (downwards). You can think of this like descending a mountain by following the steepest slope to the base.

For single layer networks, gradient descent is straightforward to implement. However, it's more complicated for deeper, multilayer neural networks like the one we've built. Complicated enough that it took about 30 years before researchers figured out how to train multilayer networks.

Training multilayer networks is done through backpropagation which is really just an application of the chain rule from calculus. It's easiest to understand if we think of our two layer network as a graph representation.

Let's start by seeing how we calculate the loss with PyTorch. Through the nn module, PyTorch provides losses such as the cross-entropy loss (nn.CrossEntropyLoss). You'll usually see the loss assigned to criterion. As noted in the last part, with a classification problem such as MNIST, we're using the softmax function to predict class probabilities. With a softmax output, you want to use cross-entropy as the loss. To actually calculate the loss, you first define the criterion then pass in the output of your network and the correct labels.

There is something really important to note here. Looking at the documentation for nn.CrossEntropyLoss:

This criterion combines nn.LogSoftmax() and nn.NLLLoss() in one single class.

The input is expected to contain scores for each class.

This means we need to pass in the raw output of our network into the loss, not the output of the softmax function. This raw output is usually called the logits or scores. We use the logits because softmax gives you probabilities which will often be very close to zero or one but floating-point numbers can't accurately represent values near zero or one (read more here). It's usually best to avoid doing calculations with probabilities, typically we use log-probabilities.

We're going to build a feed-forward network using the pipeline-style of network definition and then pass in a batch of image to examine the loss.

model = nn.Sequential(
    OrderedDict(
        input_to_hidden=nn.Linear(inputs, hidden_nodes_1),
        relu_1=nn.ReLU(),
        hidden_to_hidden=nn.Linear(hidden_nodes_1, hidden_nodes_2),
        relu_2=nn.ReLU(),
        hidden_to_output=nn.Linear(hidden_nodes_2, outputs),
        log_softmax=nn.LogSoftmax(dim=1)
    )
)

And now our loss.

criterion = nn.NLLLoss() 

Now we get the next batch of images.

images, labels = next(iter(data_loader))

And once again we flatten them.

images = images.view(images.shape[0], -1)

A forward pass on the batch.

logits = model(images)

Calculate the loss with the logits and the labels

loss = criterion(logits, labels)

print(loss)

tensor(2.3208, grad_fn=<NllLossBackward>)

So that's interesting, but what does it mean?

Now that we know how to calculate a loss, how do we use it to perform backpropagation? Torch provides a module, autograd, for automatically calculating the gradients of tensors. We can use it to calculate the gradients of all our parameters with respect to the loss. Autograd works by keeping track of operations performed on tensors, then going backwards through those operations, calculating gradients along the way. To make sure PyTorch keeps track of operations on a tensor and calculates the gradients, you need to set requires_grad = True on a tensor. You can do this at creation with the requires_grad keyword, or at any time with x.requires_grad_(True).

You can turn off gradients for a block of code with the torch.no_grad() content:

x = torch.zeros(1, requires_grad=True)
>>> with torch.no_grad():
...     y = x * 2
>>> y.requires_grad
False

Also, you can turn on or off gradients altogether with torch.set_grad_enabled(True|False).

The gradients are computed with respect to some variable z with z.backward(). This does a backward pass through the operations that created z.

x = torch.randn(2,2, requires_grad=True)
print(x)

tensor([[-0.7567, -0.2352],
        [-0.9346,  0.3097]], requires_grad=True)

y = x**2
print(y)

tensor([[0.5726, 0.0553],
        [0.8735, 0.0959]], grad_fn=<PowBackward0>)

We can see the operation that created y, a power operation PowBackward0.

grad_fn shows the function that generated this variable

print(y.grad_fn)

<PowBackward0 object at 0x7f591c505c50>

The autgrad module keeps track of these operations and knows how to calculate the gradient for each one. In this way, it's able to calculate the gradients for a chain of operations, with respect to any one tensor. Let's reduce the tensor y to a scalar value, the mean.

z = y.mean()
print(z)

tensor(0.3993, grad_fn=<MeanBackward1>)

You can check the gradients for x and y but they are empty currently.

print(x.grad)

None

To calculate the gradients, you need to run the .backward method on a Variable, z for example. This will calculate the gradient for z with respect to x

\[ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}\left[\frac{1}{n}\sum_i^n x_i^2\right] = \frac{x}{2} \]

z.backward()
print(x.grad)
print(x/2)

tensor([[-0.3783, -0.1176],
        [-0.4673,  0.1548]])
tensor([[-0.3783, -0.1176],
        [-0.4673,  0.1548]], grad_fn=<DivBackward0>)

These gradients calculations are particularly useful for neural networks. For training we need the gradients of the weights with respect to the cost. With PyTorch, we run data forward through the network to calculate the loss, then, go backwards to calculate the gradients with respect to the loss. Once we have the gradients we can make a gradient descent step.

When we create a network with PyTorch, all of the parameters are initialized with requires_grad = True. This means that when we calculate the loss and call loss.backward(), the gradients for the parameters are calculated. These gradients are used to update the weights with gradient descent. Below you can see an example of calculating the gradients using a backwards pass.

Get the next batch.

images, labels = next(iter(data_loader))
images = images.view(images.shape[0], -1)

Now get the logits and loss for the batch.

logits = model(images)
loss = criterion(logits, labels)

This is what the weights from the input layer to the first hidden layer look like before and after the backward-pass.

print('Before backward pass: \n{}\n'.format(model.input_to_hidden.weight.grad))

loss.backward()

print('After backward pass: \n', model.input_to_hidden.weight.grad)

Before backward pass: 
None

After backward pass: 
 tensor([[ 0.0001,  0.0001,  0.0001,  ...,  0.0001,  0.0001,  0.0001],
        [ 0.0011,  0.0011,  0.0011,  ...,  0.0011,  0.0011,  0.0011],
        [ 0.0004,  0.0004,  0.0004,  ...,  0.0004,  0.0004,  0.0004],
        ...,
        [ 0.0001,  0.0001,  0.0001,  ...,  0.0001,  0.0001,  0.0001],
        [ 0.0003,  0.0003,  0.0003,  ...,  0.0003,  0.0003,  0.0003],
        [-0.0005, -0.0005, -0.0005,  ..., -0.0005, -0.0005, -0.0005]])

There's one last piece we need to start training, an optimizer that we'll use to update the weights with the gradients. We get these from PyTorch's optim package(). For example we can use stochastic gradient descent with optim.SGD. You can see how to define an optimizer below.

Optimizers require the parameters to optimize and a learning rate.

optimizer = optim.SGD(model.parameters(), lr=0.01)

Now we know how to use all the individual parts so it's time to see how they work together. Let's consider just one learning step before looping through all the data. The general process with PyTorch:

1. Make a forward pass through the network
2. Use the network output to calculate the loss
3. Perform a backward pass through the network with loss.backward() to calculate the gradients
4. Take a step with the optimizer to update the weights

Below I'll go through one training step and print out the weights and gradients so you can see how it changes. Note the line of code: optimizer.zero_grad(). When you do multiple backwards passes with the same parameters, the gradients are accumulated. This means that you need to zero the gradients on each training pass or you'll retain gradients from previous training batches.

Here's the weights for the first set of edges in the network before we start:

print('Initial weights - ', model.input_to_hidden.weight)

Initial weights -  Parameter containing:
tensor([[ 0.0170,  0.0055, -0.0258,  ..., -0.0295, -0.0028,  0.0312],
        [ 0.0246,  0.0314,  0.0259,  ..., -0.0091, -0.0276, -0.0238],
        [ 0.0336, -0.0133,  0.0045,  ..., -0.0284,  0.0278,  0.0029],
        ...,
        [-0.0085, -0.0300,  0.0222,  ...,  0.0066, -0.0162,  0.0062],
        [-0.0303, -0.0324, -0.0237,  ..., -0.0230,  0.0137, -0.0268],
        [-0.0327,  0.0012,  0.0174,  ...,  0.0311,  0.0058,  0.0034]],
       requires_grad=True)

images, labels = next(iter(data_loader))
images.resize_(64, 784)

Clear the gradients.

optimizer.zero_grad()

Make a forward pass, then a backward pass, then update the weights and check the gradient.

output = model.forward(images)
loss = criterion(output, labels)
loss.backward()
print('Gradient -', model.input_to_hidden.weight.grad)

Gradient - tensor([[-0.0076, -0.0076, -0.0076,  ..., -0.0076, -0.0076, -0.0076],
        [-0.0006, -0.0006, -0.0006,  ..., -0.0006, -0.0006, -0.0006],
        [-0.0014, -0.0014, -0.0014,  ..., -0.0014, -0.0014, -0.0014],
        ...,
        [-0.0028, -0.0028, -0.0028,  ..., -0.0028, -0.0028, -0.0028],
        [-0.0012, -0.0012, -0.0012,  ..., -0.0012, -0.0012, -0.0012],
        [ 0.0027,  0.0027,  0.0027,  ...,  0.0027,  0.0027,  0.0027]])

Now take an update step and check out the new weights.

optimizer.step()
print('Updated weights - ', model.input_to_hidden.weight)

Updated weights -  Parameter containing:
tensor([[ 0.0171,  0.0056, -0.0257,  ..., -0.0294, -0.0027,  0.0313],
        [ 0.0246,  0.0314,  0.0259,  ..., -0.0091, -0.0276, -0.0238],
        [ 0.0336, -0.0133,  0.0045,  ..., -0.0284,  0.0278,  0.0029],
        ...,
        [-0.0084, -0.0300,  0.0223,  ...,  0.0066, -0.0161,  0.0062],
        [-0.0303, -0.0324, -0.0237,  ..., -0.0229,  0.0137, -0.0268],
        [-0.0327,  0.0011,  0.0173,  ...,  0.0310,  0.0058,  0.0034]],
       requires_grad=True)

If you compare it to the first weights you'll notice that the first cell is the same, but many of the others have very small changes made to them. The first steps in the descent.

Now we'll put this algorithm into a loop so we can go through all the images. First some nomenclature - one pass through the entire dataset is called an epoch. So we're going to loop through data_loader to get our training batches. For each batch, we'll do a training pass where we calculate the loss, do a backwards pass, and update the weights. Then we'll start all over again with the batches until we're out of epochs.