## Stacked Denoising Autoencoders (SdA)Note This section assumes the reader has already read through 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 ## Stacked AutoencodersThe denoising autoencoders can be stacked to form a deep network by
feeding the latent representation (output code)
of the denoising auto-encoder found on the layer
below as input to the current layer. The This can be easily implemented in Theano, using the class defined before for a denoising autoencoder. We can see the stacked denoising autoencoder as having two facades, one is a list of autoencoders, the other is 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 facedes are linked by the fact that the autoencoders and the sigmoid layers of the MLP share parameters, and the fact that autoencoders get as input latent representations of intermediate layers of the MLP. 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 n_layers_sizes: list of ints :param n_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
Next step, we construct for i in xrange(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 the logistic layer on top of the sigmoid
layers such that we have an MLP. We will
use the # 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 class also provides a method that generates training functions for
each of the denoising autoencoder associated with the different layers.
They are returned as a list, where element is a function that
implements one step of training the 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 In order to be able to change the corruption level or the learning rate during training we associate a Theano variable to 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.Param(corruption_level, default=0.2), theano.Param(learning_rate, default=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 In the same fashion we build a method for constructing function required
during finetuning ( a 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 xrange(n_valid_batches)] # Create a function that scans the entire test set def test_score(): return [test_score_i(i) for i in xrange(n_test_batches)] return train_fn, valid_score, test_score Note that the returned ## Putting it all togetherThe few lines of code below constructs 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 in training this network, a layer-wise pre-training and fine-tuning afterwards. 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 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 = time.clock() ## Pre-train layer-wise corruption_levels = [.1, .2, .3] for i in xrange(sda.n_layers): # go through pretraining epochs for epoch in xrange(pretraining_epochs): # go through the training set c = [] for batch_index in xrange(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 ' % (i, epoch), print numpy.mean(c) end_time = time.clock() print >> sys.stderr, ('The pretraining code for file ' + os.path.split(__file__)[1] + ' ran for %.2fm' % ((end_time - start_time) / 60.)) The fine-tuning loop is very similar with the one in the ## Running the CodeThe 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 level for the first layer is 0.1, for the second 0.2 and 0.3 for the third. The pretraining learning rate is was 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 TricksOne way to improve the running time of your code (given that 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 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-tunning the model. |

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