Note The code for this section is available for download here. In this section, we show how Theano can be used to implement the most basic classifier: the logistic regression. We start off with a quick primer of the model, which serves both as a refresher but also to anchor the notation and show how mathematical expressions are mapped onto Theano graphs. In the deepest of machine learning traditions, this tutorial will tackle the exciting problem of MNIST digit classification. The ModelLogistic regression is a probabilistic, linear classifier. It is parametrized by a weight matrix and a bias vector . Classification is done by projecting an input vector onto a set of hyperplanes, each of which corresponds to a class. The distance from the input to a hyperplane reflects the probability that the input is a member of the corresponding class. Mathematically, the probability that an input vector is a member of a class , a value of a stochastic variable , can be written as: The model’s prediction is the class whose probability is maximal, specifically: The code to do this in Theano is the following: # initialize with 0 the weights W as a matrix of shape (n_in, n_out) self.W = theano.shared( value=numpy.zeros( (n_in, n_out), dtype=theano.config.floatX ), name='W', borrow=True ) # initialize the baises b as a vector of n_out 0s self.b = theano.shared( value=numpy.zeros( (n_out,), dtype=theano.config.floatX ), name='b', borrow=True ) # symbolic expression for computing the matrix of classmembership # probabilities # Where: # W is a matrix where columnk represent the separation hyper plain for # classk # x is a matrix where rowj represents input training samplej # b is a vector where elementk represent the free parameter of hyper # plaink self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b) # symbolic description of how to compute prediction as class whose # probability is maximal self.y_pred = T.argmax(self.p_y_given_x, axis=1) Since the parameters of the model must maintain a persistent state throughout training, we allocate shared variables for . This declares them both as being symbolic Theano variables, but also initializes their contents. The dot and softmax operators are then used to compute the vector . The result p_y_given_x is a symbolic variable of vectortype. To get the actual model prediction, we can use the T.argmax operator, which will return the index at which p_y_given_x is maximal (i.e. the class with maximum probability). Now of course, the model we have defined so far does not do anything useful yet, since its parameters are still in their initial state. The following section will thus cover how to learn the optimal parameters. Defining a Loss FunctionLearning optimal model parameters involves minimizing a loss function. In the case of multiclass logistic regression, it is very common to use the negative loglikelihood as the loss. This is equivalent to maximizing the likelihood of the data set under the model parameterized by . Let us first start by defining the likelihood and loss : While entire books are dedicated to the topic of minimization, gradient descent is by far the simplest method for minimizing arbitrary nonlinear functions. This tutorial will use the method of stochastic gradient method with minibatches (MSGD). See Stochastic Gradient Descent for more details. The following Theano code defines the (symbolic) loss for a given minibatch: # y.shape[0] is (symbolically) the number of rows in y, i.e., # number of examples (call it n) in the minibatch # T.arange(y.shape[0]) is a symbolic vector which will contain # [0,1,2,... n1] T.log(self.p_y_given_x) is a matrix of # LogProbabilities (call it LP) with one row per example and # one column per class LP[T.arange(y.shape[0]),y] is a vector # v containing [LP[0,y[0]], LP[1,y[1]], LP[2,y[2]], ..., # LP[n1,y[n1]]] and T.mean(LP[T.arange(y.shape[0]),y]) is # the mean (across minibatch examples) of the elements in v, # i.e., the mean loglikelihood across the minibatch. return T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]), y]) Note Even though the loss is formally defined as the sum, over the data set, of individual error terms, in practice, we use the mean (T.mean) in the code. This allows for the learning rate choice to be less dependent of the minibatch size. Creating a LogisticRegression classWe now have all the tools we need to define a LogisticRegression class, which encapsulates the basic behaviour of logistic regression. The code is very similar to what we have covered so far, and should be self explanatory. class LogisticRegression(object): """Multiclass Logistic Regression Class The logistic regression is fully described by a weight matrix :math:`W` and bias vector :math:`b`. Classification is done by projecting data points onto a set of hyperplanes, the distance to which is used to determine a class membership probability. """ def __init__(self, input, n_in, n_out): """ Initialize the parameters of the logistic regression :type input: theano.tensor.TensorType :param input: symbolic variable that describes the input of the architecture (one minibatch) :type n_in: int :param n_in: number of input units, the dimension of the space in which the datapoints lie :type n_out: int :param n_out: number of output units, the dimension of the space in which the labels lie """ # startsnippet1 # initialize with 0 the weights W as a matrix of shape (n_in, n_out) self.W = theano.shared( value=numpy.zeros( (n_in, n_out), dtype=theano.config.floatX ), name='W', borrow=True ) # initialize the baises b as a vector of n_out 0s self.b = theano.shared( value=numpy.zeros( (n_out,), dtype=theano.config.floatX ), name='b', borrow=True ) # symbolic expression for computing the matrix of classmembership # probabilities # Where: # W is a matrix where columnk represent the separation hyper plain for # classk # x is a matrix where rowj represents input training samplej # b is a vector where elementk represent the free parameter of hyper # plaink self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b) output layer # symbolic description of how to compute prediction as class whose # probability is maximal self.y_pred = T.argmax(self.p_y_given_x, axis=1) # endsnippet1 # parameters of the model self.params = [self.W, self.b] def negative_log_likelihood(self, y): """Return the mean of the negative loglikelihood of the prediction of this model under a given target distribution. .. math:: \frac{1}{\mathcal{D}} \mathcal{L} (\theta=\{W,b\}, \mathcal{D}) = \frac{1}{\mathcal{D}} \sum_{i=0}^{\mathcal{D}} \log(P(Y=y^{(i)}x^{(i)}, W,b)) \\ \ell (\theta=\{W,b\}, \mathcal{D}) :type y: theano.tensor.TensorType :param y: corresponds to a vector that gives for each example the correct label Note: we use the mean instead of the sum so that the learning rate is less dependent on the batch size """ # startsnippet2 # y.shape[0] is (symbolically) the number of rows in y, i.e., # number of examples (call it n) in the minibatch # T.arange(y.shape[0]) is a symbolic vector which will contain # [0,1,2,... n1] T.log(self.p_y_given_x) is a matrix of # LogProbabilities (call it LP) with one row per example and # one column per class LP[T.arange(y.shape[0]),y] is a vector # v containing [LP[0,y[0]], LP[1,y[1]], LP[2,y[2]], ..., # LP[n1,y[n1]]] and T.mean(LP[T.arange(y.shape[0]),y]) is # the mean (across minibatch examples) of the elements in v, # i.e., the mean loglikelihood across the minibatch. return T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]), y]) # endsnippet2 def errors(self, y): """Return a float representing the number of errors in the minibatch over the total number of examples of the minibatch ; zero one loss over the size of the minibatch :type y: theano.tensor.TensorType :param y: corresponds to a vector that gives for each example the correct label """ # check if y has same dimension of y_pred if y.ndim != self.y_pred.ndim: raise TypeError( 'y should have the same shape as self.y_pred', ('y', y.type, 'y_pred', self.y_pred.type) ) # check if y is of the correct datatype if y.dtype.startswith('int'): # the T.neq operator returns a vector of 0s and 1s, where 1 # represents a mistake in prediction return T.mean(T.neq(self.y_pred, y)) else: raise NotImplementedError() We instantiate this class as follows: # generate symbolic variables for input (x and y represent a # minibatch) x = T.matrix('x') # data, presented as rasterized images y = T.ivector('y') # labels, presented as 1D vector of [int] labels # construct the logistic regression class # Each MNIST image has size 28*28 classifier = LogisticRegression(input=x, n_in=28 * 28, n_out=10) We start by allocating symbolic variables for the training inputs and their corresponding classes . Note that x and y are defined outside the scope of the LogisticRegression object. Since the class requires the input to build its graph, it is passed as a parameter of the __init__ function. This is useful in case you want to connect instances of such classes to form a deep network. The output of one layer can be passed as the input of the layer above. (This tutorial does not build a multilayer network, but this code will be reused in future tutorials that do.) Finally, we define a (symbolic) cost variable to minimize, using the instance method classifier.negative_log_likelihood. # the cost we minimize during training is the negative log likelihood of # the model in symbolic format cost = classifier.negative_log_likelihood(y) Note that x is an implicit symbolic input to the definition of cost, because the symbolic variables of classifier were defined in terms of x at initialization. Learning the ModelTo implement MSGD in most programming languages (C/C++, Matlab, Python), one would start by manually deriving the expressions for the gradient of the loss with respect to the parameters: in this case , and , This can get pretty tricky for complex models, as expressions for can get fairly complex, especially when taking into account problems of numerical stability. With Theano, this work is greatly simplified. It performs automatic differentiation and applies certain math transforms to improve numerical stability. To get the gradients and in Theano, simply do the following: g_W = T.grad(cost=cost, wrt=classifier.W) g_b = T.grad(cost=cost, wrt=classifier.b) g_W and g_b are symbolic variables, which can be used as part of a computation graph. The function train_model, which performs one step of gradient descent, can then be defined as follows: # specify how to update the parameters of the model as a list of # (variable, update expression) pairs. updates = [(classifier.W, classifier.W  learning_rate * g_W), (classifier.b, classifier.b  learning_rate * g_b)] b is initialized as zero but is updated along with w # compiling a Theano function `train_model` that returns the cost, but in # the same time updates the parameter of the model based on the rules # defined in `updates` train_model = theano.function( inputs=[index], outputs=cost, updates=updates, givens={ how to determine x and y from index. x: train_set_x[index * batch_size: (index + 1) * batch_size], y: train_set_y[index * batch_size: (index + 1) * batch_size] } ) updates is a list of pairs. In each pair, the first element is the symbolic variable to be updated in the step, and the second element is the symbolic function for calculating its new value. Similarly, givens is a dictionary whose keys are symbolic variables and whose values specify their replacements during the step. The function train_model is then defined such that:
Each time train_model(index) is called, it will thus compute and return the cost of a minibatch, while also performing a step of MSGD. The entire learning algorithm thus consists in looping over all examples in the dataset, considering all the examples in one minibatch at a time, and repeatedly calling the train_model function. Testing the modelAs explained in Learning a Classifier, when testing the model we are interested in the number of misclassified examples (and not only in the likelihood). The LogisticRegression class therefore has an extra instance method, which builds the symbolic graph for retrieving the number of misclassified examples in each minibatch. The code is as follows: def errors(self, y): """Return a float representing the number of errors in the minibatch over the total number of examples of the minibatch ; zero one loss over the size of the minibatch :type y: theano.tensor.TensorType :param y: corresponds to a vector that gives for each example the correct label """ # check if y has same dimension of y_pred if y.ndim != self.y_pred.ndim: raise TypeError( 'y should have the same shape as self.y_pred', ('y', y.type, 'y_pred', self.y_pred.type) ) # check if y is of the correct datatype if y.dtype.startswith('int'): # the T.neq operator returns a vector of 0s and 1s, where 1 # represents a mistake in prediction return T.mean(T.neq(self.y_pred, y)) else: raise NotImplementedError() We then create a function test_model and a function validate_model, which we can call to retrieve this value. As you will see shortly, validate_model is key to our earlystopping implementation (see EarlyStopping). These functions take a minibatch index and compute, for the examples in that minibatch, the number that were misclassified by the model. The only difference between them is that test_model draws its minibatches from the testing set, while validate_model draws its from the validation set. # compiling a Theano function that computes the mistakes that are made by # the model on a minibatch test_model = theano.function( inputs=[index], outputs=classifier.errors(y), givens={ x: test_set_x[index * batch_size: (index + 1) * batch_size], y: test_set_y[index * batch_size: (index + 1) * batch_size] } ) validate_model = theano.function( inputs=[index], outputs=classifier.errors(y), givens={ x: valid_set_x[index * batch_size: (index + 1) * batch_size], y: valid_set_y[index * batch_size: (index + 1) * batch_size] } ) Putting it All TogetherThe finished product is as follows. """ This tutorial introduces logistic regression using Theano and stochastic gradient descent. Logistic regression is a probabilistic, linear classifier. It is parametrized by a weight matrix :math:`W` and a bias vector :math:`b`. Classification is done by projecting data points onto a set of hyperplanes, the distance to which is used to determine a class membership probability. Mathematically, this can be written as: .. math:: P(Y=ix, W,b) &= softmax_i(W x + b) \\ &= \frac {e^{W_i x + b_i}} {\sum_j e^{W_j x + b_j}} The output of the model or prediction is then done by taking the argmax of the vector whose i'th element is P(Y=ix). .. math:: y_{pred} = argmax_i P(Y=ix,W,b) This tutorial presents a stochastic gradient descent optimization method suitable for large datasets. References:  textbooks: "Pattern Recognition and Machine Learning"  Christopher M. Bishop, section 4.3.2 """ __docformat__ = 'restructedtext en' import cPickle import gzip import os import sys import time import numpy import theano import theano.tensor as T class LogisticRegression(object): """Multiclass Logistic Regression Class The logistic regression is fully described by a weight matrix :math:`W` and bias vector :math:`b`. Classification is done by projecting data points onto a set of hyperplanes, the distance to which is used to determine a class membership probability. """ def __init__(self, input, n_in, n_out): """ Initialize the parameters of the logistic regression :type input: theano.tensor.TensorType :param input: symbolic variable that describes the input of the architecture (one minibatch) :type n_in: int :param n_in: number of input units, the dimension of the space in which the datapoints lie :type n_out: int :param n_out: number of output units, the dimension of the space in which the labels lie """ # startsnippet1 # initialize with 0 the weights W as a matrix of shape (n_in, n_out) self.W = theano.shared( value=numpy.zeros( (n_in, n_out), dtype=theano.config.floatX ), name='W', borrow=True ) # initialize the baises b as a vector of n_out 0s self.b = theano.shared( value=numpy.zeros( (n_out,), dtype=theano.config.floatX ), name='b', borrow=True ) # symbolic expression for computing the matrix of classmembership # probabilities # Where: # W is a matrix where columnk represent the separation hyper plain for # classk # x is a matrix where rowj represents input training samplej # b is a vector where elementk represent the free parameter of hyper # plaink self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b) # symbolic description of how to compute prediction as class whose # probability is maximal self.y_pred = T.argmax(self.p_y_given_x, axis=1) # endsnippet1 # parameters of the model self.params = [self.W, self.b] def negative_log_likelihood(self, y): """Return the mean of the negative loglikelihood of the prediction of this model under a given target distribution. .. math:: \frac{1}{\mathcal{D}} \mathcal{L} (\theta=\{W,b\}, \mathcal{D}) = \frac{1}{\mathcal{D}} \sum_{i=0}^{\mathcal{D}} \log(P(Y=y^{(i)}x^{(i)}, W,b)) \\ \ell (\theta=\{W,b\}, \mathcal{D}) :type y: theano.tensor.TensorType :param y: corresponds to a vector that gives for each example the correct label Note: we use the mean instead of the sum so that the learning rate is less dependent on the batch size """ # startsnippet2 # y.shape[0] is (symbolically) the number of rows in y, i.e., # number of examples (call it n) in the minibatch # T.arange(y.shape[0]) is a symbolic vector which will contain # [0,1,2,... n1] T.log(self.p_y_given_x) is a matrix of # LogProbabilities (call it LP) with one row per example and # one column per class LP[T.arange(y.shape[0]),y] is a vector # v containing [LP[0,y[0]], LP[1,y[1]], LP[2,y[2]], ..., # LP[n1,y[n1]]] and T.mean(LP[T.arange(y.shape[0]),y]) is # the mean (across minibatch examples) of the elements in v, # i.e., the mean loglikelihood across the minibatch. return T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]), y]) # endsnippet2 def errors(self, y): """Return a float representing the number of errors in the minibatch over the total number of examples of the minibatch ; zero one loss over the size of the minibatch :type y: theano.tensor.TensorType :param y: corresponds to a vector that gives for each example the correct label """ # check if y has same dimension of y_pred if y.ndim != self.y_pred.ndim: raise TypeError( 'y should have the same shape as self.y_pred', ('y', y.type, 'y_pred', self.y_pred.type) ) # check if y is of the correct datatype if y.dtype.startswith('int'): # the T.neq operator returns a vector of 0s and 1s, where 1 # represents a mistake in prediction return T.mean(T.neq(self.y_pred, y)) else: raise NotImplementedError() def load_data(dataset): ''' Loads the dataset :type dataset: string :param dataset: the path to the dataset (here MNIST) ''' ############# # LOAD DATA # ############# # Download the MNIST dataset if it is not present data_dir, data_file = os.path.split(dataset) if data_dir == "" and not os.path.isfile(dataset): # Check if dataset is in the data directory. new_path = os.path.join( os.path.split(__file__)[0], "..", "data", dataset ) if os.path.isfile(new_path) or data_file == 'mnist.pkl.gz': dataset = new_path if (not os.path.isfile(dataset)) and data_file == 'mnist.pkl.gz': import urllib origin = ( 'http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz' ) print 'Downloading data from %s' % origin urllib.urlretrieve(origin, dataset) print '... loading data' # Load the dataset f = gzip.open(dataset, 'rb') train_set, valid_set, test_set = cPickle.load(f) f.close() #train_set, valid_set, test_set format: tuple(input, target) #input is an numpy.ndarray of 2 dimensions (a matrix) #witch row's correspond to an example. target is a #numpy.ndarray of 1 dimensions (vector)) that have the same length as #the number of rows in the input. It should give the target #target to the example with the same index in the input. def shared_dataset(data_xy, borrow=True): """ Function that loads the dataset into shared variables The reason we store our dataset in shared variables is to allow Theano to copy it into the GPU memory (when code is run on GPU). Since copying data into the GPU is slow, copying a minibatch everytime is needed (the default behaviour if the data is not in a shared variable) would lead to a large decrease in performance. """ data_x, data_y = data_xy shared_x = theano.shared(numpy.asarray(data_x, dtype=theano.config.floatX), borrow=borrow) shared_y = theano.shared(numpy.asarray(data_y, dtype=theano.config.floatX), borrow=borrow) # When storing data on the GPU it has to be stored as floats # therefore we will store the labels as ``floatX`` as well # (``shared_y`` does exactly that). But during our computations # we need them as ints (we use labels as index, and if they are # floats it doesn't make sense) therefore instead of returning # ``shared_y`` we will have to cast it to int. This little hack # lets ous get around this issue return shared_x, T.cast(shared_y, 'int32') test_set_x, test_set_y = shared_dataset(test_set) valid_set_x, valid_set_y = shared_dataset(valid_set) train_set_x, train_set_y = shared_dataset(train_set) rval = [(train_set_x, train_set_y), (valid_set_x, valid_set_y), (test_set_x, test_set_y)] return rval def sgd_optimization_mnist(learning_rate=0.13, n_epochs=1000, dataset='mnist.pkl.gz', batch_size=600): """ Demonstrate stochastic gradient descent optimization of a loglinear model This is demonstrated on MNIST. :type learning_rate: float :param learning_rate: learning rate used (factor for the stochastic gradient) :type n_epochs: int :param n_epochs: maximal number of epochs to run the optimizer :type dataset: string :param dataset: the path of the MNIST dataset file from http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz """ datasets = load_data(dataset) 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_train_batches = train_set_x.get_value(borrow=True).shape[0] / batch_size n_valid_batches = valid_set_x.get_value(borrow=True).shape[0] / batch_size n_test_batches = test_set_x.get_value(borrow=True).shape[0] / batch_size ###################### # BUILD ACTUAL MODEL # ###################### print '... building the model' # allocate symbolic variables for the data index = T.lscalar() # index to a [mini]batch # generate symbolic variables for input (x and y represent a # minibatch) x = T.matrix('x') # data, presented as rasterized images y = T.ivector('y') # labels, presented as 1D vector of [int] labels # construct the logistic regression class # Each MNIST image has size 28*28 classifier = LogisticRegression(input=x, n_in=28 * 28, n_out=10) # the cost we minimize during training is the negative log likelihood of # the model in symbolic format cost = classifier.negative_log_likelihood(y) # compiling a Theano function that computes the mistakes that are made by # the model on a minibatch test_model = theano.function( inputs=[index], outputs=classifier.errors(y), givens={ x: test_set_x[index * batch_size: (index + 1) * batch_size], y: test_set_y[index * batch_size: (index + 1) * batch_size] } ) validate_model = theano.function( inputs=[index], outputs=classifier.errors(y), givens={ x: valid_set_x[index * batch_size: (index + 1) * batch_size], y: valid_set_y[index * batch_size: (index + 1) * batch_size] } ) # compute the gradient of cost with respect to theta = (W,b) g_W = T.grad(cost=cost, wrt=classifier.W) g_b = T.grad(cost=cost, wrt=classifier.b) # startsnippet3 # specify how to update the parameters of the model as a list of # (variable, update expression) pairs. updates = [(classifier.W, classifier.W  learning_rate * g_W), (classifier.b, classifier.b  learning_rate * g_b)] # compiling a Theano function `train_model` that returns the cost, but in # the same time updates the parameter of the model based on the rules # defined in `updates` train_model = theano.function( inputs=[index], outputs=cost, updates=updates, givens={ x: train_set_x[index * batch_size: (index + 1) * batch_size], y: train_set_y[index * batch_size: (index + 1) * batch_size] } ) # endsnippet3 ############### # TRAIN MODEL # ############### print '... training the model' # earlystopping parameters patience = 5000 # look as this many examples regardless patience_increase = 2 # wait this much longer when a new best is # found improvement_threshold = 0.995 # a relative improvement of this much is # considered significant validation_frequency = min(n_train_batches, patience / 2) # go through this many # minibatche before checking the network # on the validation set; in this case we # check every epoch best_validation_loss = numpy.inf test_score = 0. start_time = time.clock() done_looping = False epoch = 0 while (epoch < n_epochs) and (not done_looping): epoch = epoch + 1 for minibatch_index in xrange(n_train_batches): minibatch_avg_cost = train_model(minibatch_index) # iteration number iter = (epoch  1) * n_train_batches + minibatch_index if (iter + 1) % validation_frequency == 0: # compute zeroone loss on validation set validation_losses = [validate_model(i) for i in xrange(n_valid_batches)] this_validation_loss = numpy.mean(validation_losses) print( 'epoch %i, minibatch %i/%i, validation error %f %%' % ( epoch, minibatch_index + 1, n_train_batches, this_validation_loss * 100. ) ) # if we got the best validation score until now if this_validation_loss < best_validation_loss: #improve patience if loss improvement is good enough if this_validation_loss < best_validation_loss * \ improvement_threshold: patience = max(patience, iter * patience_increase) best_validation_loss = this_validation_loss # test it on the test set test_losses = [test_model(i) for i in xrange(n_test_batches)] test_score = numpy.mean(test_losses) print( ( ' epoch %i, minibatch %i/%i, test error of' ' best model %f %%' ) % ( epoch, minibatch_index + 1, n_train_batches, test_score * 100. ) ) if patience <= iter: done_looping = True break end_time = time.clock() print( ( 'Optimization complete with best validation score of %f %%,' 'with test performance %f %%' ) % (best_validation_loss * 100., test_score * 100.) ) print 'The code run for %d epochs, with %f epochs/sec' % ( epoch, 1. * epoch / (end_time  start_time)) print >> sys.stderr, ('The code for file ' + os.path.split(__file__)[1] + ' ran for %.1fs' % ((end_time  start_time))) if __name__ == '__main__': sgd_optimization_mnist() The user can learn to classify MNIST digits with SGD logistic regression, by typing, from within the DeepLearningTutorials folder: python code/logistic_sgd.py The output one should expect is of the form : ... epoch 72, minibatch 83/83, validation error 7.510417 % epoch 72, minibatch 83/83, test error of best model 7.510417 % epoch 73, minibatch 83/83, validation error 7.500000 % epoch 73, minibatch 83/83, test error of best model 7.489583 % Optimization complete with best validation score of 7.500000 %,with test performance 7.489583 % The code run for 74 epochs, with 1.936983 epochs/sec On an Intel(R) Core(TM)2 Duo CPU E8400 @ 3.00 Ghz the code runs with approximately 1.936 epochs/sec and it took 75 epochs to reach a test error of 7.489%. On the GPU the code does almost 10.0 epochs/sec. For this instance we used a batch size of 600. Footnotes

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