# Stacked Denoising Autoencoders (SdA)¶

Note

This section assumes you have already read through Classifying MNIST digits using Logistic Regression and Multilayer Perceptron. Additionally it uses the following Theano functions and concepts: T.tanh, shared variables, basic arithmetic ops, T.grad, Random numbers, floatX. If you intend to run the code on GPU also read GPU.

Note

The code for this section is available for download here.

The Stacked Denoising Autoencoder (SdA) is an extension of the stacked autoencoder [Bengio07] and it was introduced in [Vincent08].

This tutorial builds on the previous tutorial Denoising Autoencoders. Especially if you do not have experience with autoencoders, we recommend reading it before going any further.

## Stacked Autoencoders¶

Denoising autoencoders can be stacked to form a deep network by
feeding the latent representation (output code)
of the denoising autoencoder found on the layer
below as input to the current layer. The **unsupervised pre-training** of such an
architecture is done one layer at a time. Each layer is trained as
a denoising autoencoder by minimizing the error in reconstructing its input
(which is the output code of the previous layer).
Once the first layers
are trained, we can train the -th layer because we can now
compute the code or latent representation from the layer below.

Once all layers are pre-trained, the network goes through a second stage
of training called **fine-tuning**. Here we consider **supervised fine-tuning**
where we want to minimize prediction error on a supervised task.
For this, we first add a logistic regression
layer on top of the network (more precisely on the output code of the
output layer). We then
train the entire network as we would train a multilayer
perceptron. At this point, we only consider the encoding parts of
each auto-encoder.
This stage is supervised, since now we use the target class during
training. (See the Multilayer Perceptron for details on the multilayer perceptron.)

This can be easily implemented in Theano, using the class defined previously for a denoising autoencoder. We can see the stacked denoising autoencoder as having two facades: a list of autoencoders, and an MLP. During pre-training we use the first facade, i.e., we treat our model as a list of autoencoders, and train each autoencoder seperately. In the second stage of training, we use the second facade. These two facades are linked because:

- the autoencoders and the sigmoid layers of the MLP share parameters, and
- the latent representations computed by intermediate layers of the MLP are fed as input to the autoencoders.

```
class SdA(object):
"""Stacked denoising auto-encoder class (SdA)
A stacked denoising autoencoder model is obtained by stacking several
dAs. The hidden layer of the dA at layer `i` becomes the input of
the dA at layer `i+1`. The first layer dA gets as input the input of
the SdA, and the hidden layer of the last dA represents the output.
Note that after pretraining, the SdA is dealt with as a normal MLP,
the dAs are only used to initialize the weights.
"""
def __init__(
self,
numpy_rng,
theano_rng=None,
n_ins=784,
hidden_layers_sizes=[500, 500],
n_outs=10,
corruption_levels=[0.1, 0.1]
):
""" This class is made to support a variable number of layers.
:type numpy_rng: numpy.random.RandomState
:param numpy_rng: numpy random number generator used to draw initial
weights
:type theano_rng: theano.tensor.shared_randomstreams.RandomStreams
:param theano_rng: Theano random generator; if None is given one is
generated based on a seed drawn from `rng`
:type n_ins: int
:param n_ins: dimension of the input to the sdA
:type hidden_layers_sizes: list of ints
:param hidden_layers_sizes: intermediate layers size, must contain
at least one value
:type n_outs: int
:param n_outs: dimension of the output of the network
:type corruption_levels: list of float
:param corruption_levels: amount of corruption to use for each
layer
"""
self.sigmoid_layers = []
self.dA_layers = []
self.params = []
self.n_layers = len(hidden_layers_sizes)
assert self.n_layers > 0
if not theano_rng:
theano_rng = RandomStreams(numpy_rng.randint(2 ** 30))
# allocate symbolic variables for the data
self.x = T.matrix('x') # the data is presented as rasterized images
self.y = T.ivector('y') # the labels are presented as 1D vector of
# [int] labels
```

`self.sigmoid_layers`

will store the sigmoid layers of the MLP facade, while
`self.dA_layers`

will store the denoising autoencoder associated with the layers of the MLP.

Next, we construct `n_layers`

sigmoid layers and `n_layers`

denoising
autoencoders, where `n_layers`

is the depth of our model. We use the
`HiddenLayer`

class introduced in Multilayer Perceptron, with one
modification: we replace the `tanh`

non-linearity with the
logistic function ).
We link the sigmoid layers to form an MLP, and construct
the denoising autoencoders such that each shares the weight matrix and the
bias of its encoding part with its corresponding sigmoid layer.

```
for i in range(self.n_layers):
# construct the sigmoidal layer
# the size of the input is either the number of hidden units of
# the layer below or the input size if we are on the first layer
if i == 0:
input_size = n_ins
else:
input_size = hidden_layers_sizes[i - 1]
# the input to this layer is either the activation of the hidden
# layer below or the input of the SdA if you are on the first
# layer
if i == 0:
layer_input = self.x
else:
layer_input = self.sigmoid_layers[-1].output
sigmoid_layer = HiddenLayer(rng=numpy_rng,
input=layer_input,
n_in=input_size,
n_out=hidden_layers_sizes[i],
activation=T.nnet.sigmoid)
# add the layer to our list of layers
self.sigmoid_layers.append(sigmoid_layer)
# its arguably a philosophical question...
# but we are going to only declare that the parameters of the
# sigmoid_layers are parameters of the StackedDAA
# the visible biases in the dA are parameters of those
# dA, but not the SdA
self.params.extend(sigmoid_layer.params)
# Construct a denoising autoencoder that shared weights with this
# layer
dA_layer = dA(numpy_rng=numpy_rng,
theano_rng=theano_rng,
input=layer_input,
n_visible=input_size,
n_hidden=hidden_layers_sizes[i],
W=sigmoid_layer.W,
bhid=sigmoid_layer.b)
self.dA_layers.append(dA_layer)
```

All we need now is to add a logistic layer on top of the sigmoid
layers such that we have an MLP. We will
use the `LogisticRegression`

class introduced in Classifying MNIST digits using Logistic Regression.

```
# We now need to add a logistic layer on top of the MLP
self.logLayer = LogisticRegression(
input=self.sigmoid_layers[-1].output,
n_in=hidden_layers_sizes[-1],
n_out=n_outs
)
self.params.extend(self.logLayer.params)
# construct a function that implements one step of finetunining
# compute the cost for second phase of training,
# defined as the negative log likelihood
self.finetune_cost = self.logLayer.negative_log_likelihood(self.y)
# compute the gradients with respect to the model parameters
# symbolic variable that points to the number of errors made on the
# minibatch given by self.x and self.y
self.errors = self.logLayer.errors(self.y)
```

The `SdA`

class also provides a method that generates training functions for
the denoising autoencoders in its layers.
They are returned as a list, where element is a function that
implements one step of training the `dA`

corresponding to layer
.

```
def pretraining_functions(self, train_set_x, batch_size):
''' Generates a list of functions, each of them implementing one
step in trainnig the dA corresponding to the layer with same index.
The function will require as input the minibatch index, and to train
a dA you just need to iterate, calling the corresponding function on
all minibatch indexes.
:type train_set_x: theano.tensor.TensorType
:param train_set_x: Shared variable that contains all datapoints used
for training the dA
:type batch_size: int
:param batch_size: size of a [mini]batch
:type learning_rate: float
:param learning_rate: learning rate used during training for any of
the dA layers
'''
# index to a [mini]batch
index = T.lscalar('index') # index to a minibatch
```

To be able to change the corruption level or the learning rate during training, we associate Theano variables with them.

```
corruption_level = T.scalar('corruption') # % of corruption to use
learning_rate = T.scalar('lr') # learning rate to use
# begining of a batch, given `index`
batch_begin = index * batch_size
# ending of a batch given `index`
batch_end = batch_begin + batch_size
pretrain_fns = []
for dA in self.dA_layers:
# get the cost and the updates list
cost, updates = dA.get_cost_updates(corruption_level,
learning_rate)
# compile the theano function
fn = theano.function(
inputs=[
index,
theano.In(corruption_level, value=0.2),
theano.In(learning_rate, value=0.1)
],
outputs=cost,
updates=updates,
givens={
self.x: train_set_x[batch_begin: batch_end]
}
)
# append `fn` to the list of functions
pretrain_fns.append(fn)
return pretrain_fns
```

Now any function `pretrain_fns[i]`

takes as arguments `index`

and
optionally `corruption`

—the corruption level or `lr`

—the
learning rate. Note that the names of the parameters are the names given
to the Theano variables when they are constructed, not the names of the
Python variables (`learning_rate`

or `corruption_level`

). Keep this
in mind when working with Theano.

In the same fashion we build a method for constructing the functions required
during finetuning (`train_fn`

, `valid_score`

and
`test_score`

).

```
def build_finetune_functions(self, datasets, batch_size, learning_rate):
'''Generates a function `train` that implements one step of
finetuning, a function `validate` that computes the error on
a batch from the validation set, and a function `test` that
computes the error on a batch from the testing set
:type datasets: list of pairs of theano.tensor.TensorType
:param datasets: It is a list that contain all the datasets;
the has to contain three pairs, `train`,
`valid`, `test` in this order, where each pair
is formed of two Theano variables, one for the
datapoints, the other for the labels
:type batch_size: int
:param batch_size: size of a minibatch
:type learning_rate: float
:param learning_rate: learning rate used during finetune stage
'''
(train_set_x, train_set_y) = datasets[0]
(valid_set_x, valid_set_y) = datasets[1]
(test_set_x, test_set_y) = datasets[2]
# compute number of minibatches for training, validation and testing
n_valid_batches = valid_set_x.get_value(borrow=True).shape[0]
n_valid_batches //= batch_size
n_test_batches = test_set_x.get_value(borrow=True).shape[0]
n_test_batches //= batch_size
index = T.lscalar('index') # index to a [mini]batch
# compute the gradients with respect to the model parameters
gparams = T.grad(self.finetune_cost, self.params)
# compute list of fine-tuning updates
updates = [
(param, param - gparam * learning_rate)
for param, gparam in zip(self.params, gparams)
]
train_fn = theano.function(
inputs=[index],
outputs=self.finetune_cost,
updates=updates,
givens={
self.x: train_set_x[
index * batch_size: (index + 1) * batch_size
],
self.y: train_set_y[
index * batch_size: (index + 1) * batch_size
]
},
name='train'
)
test_score_i = theano.function(
[index],
self.errors,
givens={
self.x: test_set_x[
index * batch_size: (index + 1) * batch_size
],
self.y: test_set_y[
index * batch_size: (index + 1) * batch_size
]
},
name='test'
)
valid_score_i = theano.function(
[index],
self.errors,
givens={
self.x: valid_set_x[
index * batch_size: (index + 1) * batch_size
],
self.y: valid_set_y[
index * batch_size: (index + 1) * batch_size
]
},
name='valid'
)
# Create a function that scans the entire validation set
def valid_score():
return [valid_score_i(i) for i in range(n_valid_batches)]
# Create a function that scans the entire test set
def test_score():
return [test_score_i(i) for i in range(n_test_batches)]
return train_fn, valid_score, test_score
```

Note that `valid_score`

and `test_score`

are not Theano
functions, but rather Python functions that loop over the entire
validation set and the entire test set, respectively, producing a list of the losses
over these sets.

## Putting it all together¶

The few lines of code below construct the stacked denoising autoencoder:

```
numpy_rng = numpy.random.RandomState(89677)
print('... building the model')
# construct the stacked denoising autoencoder class
sda = SdA(
numpy_rng=numpy_rng,
n_ins=28 * 28,
hidden_layers_sizes=[1000, 1000, 1000],
n_outs=10
)
```

There are two stages of training for this network: layer-wise pre-training followed by fine-tuning.

For the pre-training stage, we will loop over all the layers of the
network. For each layer we will use the compiled Theano function that
implements a SGD step towards optimizing the weights for reducing
the reconstruction cost of that layer. This function will be applied
to the training set for a fixed number of epochs given by
`pretraining_epochs`

.

```
#########################
# PRETRAINING THE MODEL #
#########################
print('... getting the pretraining functions')
pretraining_fns = sda.pretraining_functions(train_set_x=train_set_x,
batch_size=batch_size)
print('... pre-training the model')
start_time = timeit.default_timer()
## Pre-train layer-wise
corruption_levels = [.1, .2, .3]
for i in range(sda.n_layers):
# go through pretraining epochs
for epoch in range(pretraining_epochs):
# go through the training set
c = []
for batch_index in range(n_train_batches):
c.append(pretraining_fns[i](index=batch_index,
corruption=corruption_levels[i],
lr=pretrain_lr))
print('Pre-training layer %i, epoch %d, cost %f' % (i, epoch, numpy.mean(c, dtype='float64')))
end_time = timeit.default_timer()
print(('The pretraining code for file ' +
os.path.split(__file__)[1] +
' ran for %.2fm' % ((end_time - start_time) / 60.)), file=sys.stderr)
```

The fine-tuning loop is very similar to the one in the Multilayer Perceptron. The
only difference is that it uses the functions given by
`build_finetune_functions`

.

## Running the Code¶

The user can run the code by calling:

```
python code/SdA.py
```

By default the code runs 15 pre-training epochs for each layer, with a batch size of 1. The corruption levels are 0.1 for the first layer, 0.2 for the second, and 0.3 for the third. The pretraining learning rate is 0.001 and the finetuning learning rate is 0.1. Pre-training takes 585.01 minutes, with an average of 13 minutes per epoch. Fine-tuning is completed after 36 epochs in 444.2 minutes, with an average of 12.34 minutes per epoch. The final validation score is 1.39% with a testing score of 1.3%. These results were obtained on a machine with an Intel Xeon E5430 @ 2.66GHz CPU, with a single-threaded GotoBLAS.

## Tips and Tricks¶

One way to improve the running time of your code (assuming you have sufficient memory available), is to compute how the network, up to layer , transforms your data. Namely, you start by training your first layer dA. Once it is trained, you can compute the hidden units values for every datapoint in your dataset and store this as a new dataset that you will use to train the dA corresponding to layer 2. Once you have trained the dA for layer 2, you compute, in a similar fashion, the dataset for layer 3 and so on. You can see now, that at this point, the dAs are trained individually, and they just provide (one to the other) a non-linear transformation of the input. Once all dAs are trained, you can start fine-tuning the model.