In this chapter, we will learn how to use Theano to implement the most basic logarithmic regression classifier. First, we will briefly review a model. In the process, you can further understand how to combine mathematical expressions with Theano's graph model.

# 数学模型

Log-regressive model is a trial of linear probability classifier with two parameters, weight matrix WW and offset vector bb. The process of classification is to project data onto a set of high-dimensional hyperplanes. The distance between the data and the plane reflects the probability that it belongs to this category. The mathematical formula of this model can be expressed as:

P(Y=i|x,W,b)=softmaxi(Wx+b)=eWix+bi∑jeWjx+bjP(Y=i|x,W,b)=softmaxi(Wx+b)=eWix+bi∑jeWjx+bj

The output of the model is the predicted result. Its value is:

ypred=argmaxiP(Y=i|x,W,b)ypred=argmaxiP(Y=i|x,W,b)

In Theano, the following functions are implemented by the following function

# generate symbolic variables for input (x and y represent a # minibatch) x = T.fmatrix('x') y = T.lvector('y') # allocate shared variables model params b = theano.shared(numpy.zeros((10,)), name='b') W = theano.shared(numpy.zeros((784, 10)), name='W') # symbolic expression for computing the vector of # class-membership probabilities p_y_given_x = T.nnet.softmax(T.dot(x, W) + b) # compiled Theano function that returns the vector of class-membership # probabilities get_p_y_given_x = theano.function(inputs=[x], outputs=p_y_given_x) # print the probability of some example represented by x_value # x_value is not a symbolic variable but a numpy array describing the # datapoint print 'Probability that x is of class %i is %f' % (i, get_p_y_given_x(x_value)[i]) # symbolic description of how to compute prediction as class whose probability # is maximal y_pred = T.argmax(p_y_given_x, axis=1) # compiled theano function that returns this value classify = theano.function(inputs=[x], outputs=y_pred)

In the above code, The input variables xx, yy are first defined. Because the model maintains a stable state during training, the model parameters WW, bb are defined as shared variables. This definition not only declares variables, but also initializes their values. Next, the point multiplication and softmax operations are used to calculate the model output P(Y|x, W, b)P(Y|x, W, b). The result is stored in the variable p_y_given_x.

So far, we have only booked the calculation graph model that Theano runs. In order to get the true P(Y|x, W, b)P(Y|x, W, b) values, we need to create the function get_p_y_given_x with x as the parameter and the output value p_y_given_x. We can iterate through its values and get the probability that the data belongs to each category.

Now, let's end the creation of Theano map. In order to get the predicted results of the model, we use the T.argmax operator, which returns p_y_given_x to make a big worth index.

Similar, in order to get the predicted result for a given input, we define the function classify. The function takes the model input matrix xx as a parameter and the output as a column vector, indicating the prediction category of each instance.

Of course, this model has no use, because the model parameters are still in the initial state. In the following sections, we will learn how to train the model. The

# loss function

model training process is also the process of minimizing the loss function. In multi-class logarithmic regression models, a negative log-likelihood function is usually used as a parameter of the model. This is equivalent to maximizing the likelihood of training data in a model with θθ as a parameter. If we define the likelihood and loss function as follows:

L(θ={W,b},D)=∑i=0|D|log(P(Y=y(i)|x(i),W,b))ℓ(θ={W,b},D)=−L(θ={W,b},D)L(θ={W,b},D)=∑i=0|D|log(P(Y=y(i)|x(i),W,b))ℓ(θ={W,b},D)=−L(θ={W,b},D)

The following code demonstrates how to calculate the loss of a minbatch

loss = -T.mean(T.log(p_y_given_x)[T.arange(y.shape[0]), y]) # note on syntax: T.arange(y.shape[0]) is a vector of integers [0,1,2,...,len(y)]. # Indexing a matrix M by the two vectors [0,1,...,K], [a,b,...,k] returns the # elements M[0,a], M[1,b], ..., M[K,k] as a vector. Here, we use this # syntax to retrieve the log-probability of the correct labels, y.

## Create a LogisticRegression class

Now we have all the features of the LogisticRegression class. The code for this class is as follows, and the code covers all the features we have learned before.

class LogisticRegression(object): 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 (e.g., one minibatch of input images) :type n_in: int :param n_in: number of input units, the dimension of the space in which the datapoint lies :type n_out: int :param n_out: number of output units, the dimension of the space in which the target lies """ # 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' ) # 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' ) # compute vector of class-membership probabilities in symbolic form self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b) # compute prediction as class whose probability is maximal in # symbolic form self.y_pred=T.argmax(self.p_y_given_x, axis=1) def negative_log_likelihood(self, y): """Return the mean of the negative log-likelihood 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}) :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 """ return -T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]), y])

This class can be instantiated in the following way:

# allocate symbolic variables for the data x = T.fmatrix() # the data is presented as rasterized images (each being a 1-D row vector in x) y = T.lvector() # the labels are presented as 1D vector of [long int] labels # construct the logistic regression class classifier = LogisticRegression( input=x.reshape((batch_size, 28 * 28)), n_in=28 * 28, n_out=10)

Finally, define the loss function:

cost = classifier.negative_log_likelihood(y)

# 模型的训练

In order to implement MSGD in the programming language, we need to manually calculate the differential of the model. If the model is more complicated, the calculation process becomes very difficult.

In Theano, this work can be done automatically by function. The example code is as follows:

# compute the gradient of cost with respect to theta = (W,b) g_W = T.grad(cost, classifier.W) g_b = T.grad(cost, classifier.b)

g_W and g_b are symbol variables, which can be used in the calculation graph model. The following code demonstrates the process of performing a one-step gradient descent algorithm:

# 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) # 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]})

update This list contains the update operations for the random gradient algorithm for each variable. The givens dictionary contains the mapping between data and variables in the calculation graph model. The entire train_model is defined:

- Input: is the mini-batch indexed by index, its data is defined as xx, the corresponding label is represented as yy.
- return value, for the corresponding loss
- each time When the function is called, the corresponding parameters xx, yy are first retrieved by index, then the function loss on the minbatch is calculated, and the operation parameters defined in the updates list are applied.

function train_model(index) When called, it calculates and returns the approximate loss and performs a one-step MSGD operation. The whole learning process consists of a series of loops on the dataset, that is, a process of repeatedly calling this function

# 模型的测试

As introduced in the first section, our test of the model is mainly It is the number of data that cares about its misclassification, not just the likelihood function. Therefore, a member function is required in the class LogisticRegression to establish a symbolic graph of the number of misclassified data returned on the test data. The code is as follows:

class LogisticRegression(object): 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 """ return T.mean(T.neq(self.y_pred, y))

Next we define the functions test_model and validte_model, in order to get the value of this function. Validate_model is the key to the end of the previous phase (see previous section). The function of both functions is to count the number of instances of a given batch that have misclassified it. The difference between the two functions is that they run on the test data and one on the verification data. The corresponding function code is as follows:

# 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]})

# 综合All functions

If you integrate all the above functions, you will get the following code:

""" 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=i|x, 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=i|x). .. math:: y_{pred} = argmax_i P(Y=i|x,W,b) This tutorial presents a stochastic gradient descent optimization method suitable for large datasets, and a conjugate gradient optimization method that is suitable for smaller 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): """Multi-class 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 """ # 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) # compute vector of class-membership probabilities in symbolic form self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b) # compute prediction as class whose probability is maximal in # symbolic form self.y_pred = T.argmax(self.p_y_given_x, axis=1) # parameters of the model self.params = [self.W, self.b] def negative_log_likelihood(self, y): """Return the mean of the negative log-likelihood 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 """ # 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,... n-1] T.log(self.p_y_given_x) is a matrix of # Log-Probabilities (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[n-1,y[n-1]]] 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 log-likelihood across the minibatch. return -T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]), y]) 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', target.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 (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='../data/mnist.pkl.gz', batch_size=600): """ Demonstrate stochastic gradient descent optimization of a log-linear 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 x = T.matrix('x') # the data is presented as rasterized images y = T.ivector('y') # the labels are 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) # 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]}) ############### # TRAIN MODEL # ############### print '... training the model' # early-stopping 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_params = None 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 zero-one 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()

This program uses SGD logistic regression algorithm to learn the classifier. In the DeepLearningTutorials folder, you can call it with the following command:

python code/logistic_sgd.py