.. _chapter_basic_gan:
Generative Adversarial Networks
===============================
Throughout most of this book, we’ve talked about how to make
predictions. In some form or another, we used deep neural networks
learned mappings from data points to labels. This kind of learning is
called discriminative learning, as in, we’d like to be able to
discriminate between photos cats and photos of dogs. Classifiers and
regressors are both examples of discriminative learning. And neural
networks trained by backpropagation have upended everything we thought
we knew about discriminative learning on large complicated datasets.
Classification accuracies on high-res images has gone from useless to
human-level (with some caveats) in just 5-6 years. We’ll spare you
another spiel about all the other discriminative tasks where deep neural
networks do astoundingly well.
But there’s more to machine learning than just solving discriminative
tasks. For example, given a large dataset, without any labels, we might
want to learn a model that concisely captures the characteristics of
this data. Given such a model, we could sample synthetic data points
that resemble the distribution of the training data. For example, given
a large corpus of photographs of faces, we might want to be able to
generate a new photorealistic image that looks like it might plausibly
have come from the same dataset. This kind of learning is called
generative modeling.
Until recently, we had no method that could synthesize novel
photorealistic images. But the success of deep neural networks for
discriminative learning opened up new possibilities. One big trend over
the last three years has been the application of discriminative deep
nets to overcome challenges in problems that we don’t generally think of
as supervised learning problems. The recurrent neural network language
models are one example of using a discriminative network (trained to
predict the next character) that once trained can act as a generative
model.
In 2014, a breakthrough paper introduced Generative adversarial networks
(GANs) :cite:`Goodfellow.Pouget-Abadie.Mirza.ea.2014`, a clever new
way to leverage the power of discriminative models to get good
generative models. At their heart, GANs rely on the idea that a data
generator is good if we cannot tell fake data apart from real data. In
statistics, this is called a two-sample test - a test to answer the
question whether datasets :math:`X=\{x_1,\ldots,x_n\}` and
:math:`X'=\{x'_1,\ldots,x'_n\}` were drawn from the same distribution.
The main difference between most statistics papers and GANs is that the
latter use this idea in a constructive way. In other words, rather than
just training a model to say “hey, these two datasets don’t look like
they came from the same distribution”, they use the `two-sample
test `__ to
provide training signal to a generative model. This allows us to improve
the data generator until it generates something that resembles the real
data. At the very least, it needs to fool the classifier. And if our
classifier is a state of the art deep neural network.
.. _fig_gan:
.. figure:: ../img/gan.svg
Generative Adversarial Networks
The GANs architecture is illustrated in :numref:`fig_gan`. As you can
see, there are two pieces to GANs - first off, we need a device (say, a
deep network but it really could be anything, such as a game rendering
engine) that might potentially be able to generate data that looks just
like the real thing. If we are dealing with images, this needs to
generate images. If we’re dealing with speech, it needs to generate
audio sequences, and so on. We call this the generator network. The
second component is the discriminator network. It attempts to
distinguish fake and real data from each other. Both networks are in
competition with each other. The generator network attempts to fool the
discriminator network. At that point, the discriminator network adapts
to the new fake data. This information, in turn is used to improve the
generator network, and so on.
The discriminator is a binary classifier to distinguish if the input
:math:`x` is real (from real data) or fake (from the generator).
Typically, the discriminator outputs a scalar prediction
:math:`o\in\mathbb R` for input :math:`\mathbf x`, such as using a dense
layer with hidden size 1, and then applies sigmoid function to obtain
the predicted probability :math:`D(\mathbf x) = 1/(1+e^{-o})`. Assume
the label :math:`y` for true data is :math:`1` and :math:`0` for fake
data. We train the discriminator to minimize the cross-entropy loss,
i.e.
.. math:: \min - y \log D(\mathbf x) - (1-y)\log(1-D(\mathbf x)),
For the generator, it first draws some parameter
:math:`\mathbf z\in\mathbb R^d` from a source of randomness, e.g. a
normal distribution :math:`\mathbf z\sim\mathcal(0,1)`. We often call
:math:`\mathbf z` the latent variable. It then applies a function to
generate :math:`\mathbf x'=G(\mathbf z)`. The goal of the generator is
to fool the discriminator to classify :math:`\mathbf x'` as true data.
In other words, we update the parameters of the generator to maximize
the cross-entropy loss when :math:`y=0`, i.e.
.. math:: \max - \log(1-D(\mathbf x')).
If the discriminator does a perfect job, then
:math:`D(\mathbf x')\approx 1` so the above loss near 0, which results
the gradients are too small to make a good progress for the
discriminator. So commonly we minimize the following loss
.. math:: \max \log(D(\mathbf x')),
which is just feed :math:`\mathbf x'` into the discriminator but giving
label :math:`y=1`.
Many of the GANs applications are in the context of images. As a
demonstration purpose, we’re going to content ourselves with fitting a
much simpler distribution first. We will illustrate what happens if we
use GANs to build the world’s most inefficient estimator of parameters
for a Gaussian. Let’s get started.
.. code:: python
%matplotlib inline
import d2l
from mxnet import nd, gluon, autograd, init
from mxnet.gluon import nn
Generate some “real” data
-------------------------
Since this is going to be the world’s lamest example, we simply generate
data drawn from a Gaussian.
.. code:: python
X = nd.random.normal(shape=(1000, 2))
A = nd.array([[1, 2], [-0.1, 0.5]])
b = nd.array([1, 2])
data = nd.dot(X, A) + b
Let’s see what we got. This should be a Gaussian shifted in some rather
arbitrary way with mean :math:`b` and covariance matrix :math:`A^TA`.
.. code:: python
d2l.set_figsize((3.5, 2.5))
#d2l.plt.figure(figsize=())
d2l.plt.scatter(data[:100,0].asnumpy(), data[:100,1].asnumpy());
print("The covariance matrix is", nd.dot(A.T,A))
.. parsed-literal::
:class: output
The covariance matrix is
[[1.01 1.95]
[1.95 4.25]]
.. image:: output_gan_4e4dd7_5_1.svg
.. code:: python
batch_size = 8
data_iter = d2l.load_array((data,), batch_size)
Generator
---------
Our generator network will be the simplest network possible - a single
layer linear model. This is since we’ll be driving that linear network
with a Gaussian data generator. Hence, it literally only needs to learn
the parameters to fake things perfectly.
.. code:: python
net_G = nn.Sequential()
net_G.add(nn.Dense(2))
Discriminator
-------------
For the discriminator we will be a bit more discriminating: we will use
an MLP with 3 layers to make things a bit more interesting.
.. code:: python
net_D = nn.Sequential()
net_D.add(nn.Dense(5, activation='tanh'),
nn.Dense(3, activation='tanh'),
nn.Dense(1))
Training
--------
First we define a function to update the discriminator.
.. code:: python
# Save to the d2l package.
def update_D(X, Z, net_D, net_G, loss, trainer_D):
"""Update discriminator"""
batch_size = X.shape[0]
ones = nd.ones((batch_size,), ctx=X.context)
zeros = nd.zeros((batch_size,), ctx=X.context)
with autograd.record():
real_Y = net_D(X)
fake_X = net_G(Z)
# Don't need to compute gradient for net_G, detach it from
# computing gradients.
fake_Y = net_D(fake_X.detach())
loss_D = (loss(real_Y, ones) + loss(fake_Y, zeros)) / 2
loss_D.backward()
trainer_D.step(batch_size)
return loss_D.sum().asscalar()
The generator is updated similarly. Here we reuse the cross-entropy loss
but change the label of the fake data from :math:`0` to :math:`1`.
.. code:: python
# Save to the d2l package.
def update_G(Z, net_D, net_G, loss, trainer_G): # saved in d2l
"""Update generator"""
batch_size = Z.shape[0]
ones = nd.ones((batch_size,), ctx=Z.context)
with autograd.record():
# We could reuse fake_X from update_D to save computation.
fake_X = net_G(Z)
# Recomputing fake_Y is needed since net_D is changed.
fake_Y = net_D(fake_X)
loss_G = loss(fake_Y, ones)
loss_G.backward()
trainer_G.step(batch_size)
return loss_G.sum().asscalar()
Both the discriminator and the generator performs a binary logistic
regression with the cross-entropy loss. We use Adam to smooth the
training process. In each iteration, we first update the discriminator
and then the generator. We visualize both losses and generated examples.
.. code:: python
def train(net_D, net_G, data_iter, num_epochs, lr_D, lr_G, latent_dim, data):
loss = gluon.loss.SigmoidBCELoss()
net_D.initialize(init=init.Normal(0.02), force_reinit=True)
net_G.initialize(init=init.Normal(0.02), force_reinit=True)
trainer_D = gluon.Trainer(net_D.collect_params(),
'adam', {'learning_rate': lr_D})
trainer_G = gluon.Trainer(net_G.collect_params(),
'adam', {'learning_rate': lr_G})
animator = d2l.Animator(xlabel='epoch', ylabel='loss',
xlim=[1, num_epochs], nrows=2, figsize=(5,5),
legend=['generator', 'discriminator'])
animator.fig.subplots_adjust(hspace=0.3)
for epoch in range(1, num_epochs+1):
# Train one epoch
timer = d2l.Timer()
metric = d2l.Accumulator(3) # loss_D, loss_G, num_examples
for X in data_iter:
batch_size = X.shape[0]
Z = nd.random.normal(0, 1, shape=(batch_size, latent_dim))
metric.add(update_D(X, Z, net_D, net_G, loss, trainer_D),
update_G(Z, net_D, net_G, loss, trainer_G),
batch_size)
# Visualize generated examples
Z = nd.random.normal(0, 1, shape=(100, latent_dim))
fake_X = net_G(Z).asnumpy()
animator.axes[1].cla()
animator.axes[1].scatter(data[:,0], data[:,1])
animator.axes[1].scatter(fake_X[:,0], fake_X[:,1])
animator.axes[1].legend(['real', 'generated'])
# Show the losses
loss_D, loss_G = metric[0]/metric[2], metric[1]/metric[2]
animator.add(epoch, (loss_D, loss_G))
print('loss_D %.3f, loss_G %.3f, %d examples/sec' % (
loss_D, loss_G, metric[2]/timer.stop()))
Now we specify the hyper-parameters to fit the Gaussian distribution.
.. code:: python
lr_D, lr_G, latent_dim, num_epochs = 0.05, 0.005, 2, 20
train(net_D, net_G, data_iter, num_epochs, lr_D, lr_G,
latent_dim, data[:100].asnumpy())
.. parsed-literal::
:class: output
loss_D 0.693, loss_G 0.693, 831 examples/sec
.. image:: output_gan_4e4dd7_18_1.svg
Summary
-------
- Generative adversarial networks (GANs) composes of two deep networks,
the generator and the discriminator.
- The generator generates the image as much closer to the true image as
possible to fool the discriminator, via maximizing the cross-entropy
loss, i.e., :math:`\max \log(D(\mathbf{x'}))`.
- The discriminator tries to distinguish the generated images from the
true images, via minimizing the cross-entropy loss, i.e.,
:math:`\min - y \log D(\mathbf{x}) - (1-y)\log(1-D(\mathbf{x}))`.
Reference
---------