""" 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. References: - textbooks: "Pattern Recognition and Machine Learning" - Christopher M. Bishop, section 4.3.2 """ from __future__ import print_function __docformat__ = 'restructedtext en' import six.moves.cPickle as pickle import gzip import os import sys import timeit 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 """ # start-snippet-1 # 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 biases 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 class-membership # probabilities # Where: # W is a matrix where column-k represent the separation hyperplane for # class-k # x is a matrix where row-j represents input training sample-j # b is a vector where element-k represent the free parameter of # hyperplane-k 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) # end-snippet-1 # parameters of the model self.params = [self.W, self.b] # keep track of model input self.input = input 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 """ # start-snippet-2 # 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]) # end-snippet-2 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': from six.moves import urllib origin = ( 'http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz' ) print('Downloading data from %s' % origin) urllib.request.urlretrieve(origin, dataset) print('... loading data') # Load the dataset with gzip.open(dataset, 'rb') as f: try: train_set, valid_set, test_set = pickle.load(f, encoding='latin1') except: train_set, valid_set, test_set = pickle.load(f) # train_set, valid_set, test_set format: tuple(input, target) # input is a numpy.ndarray of 2 dimensions (a matrix) # where each row corresponds to an example. target is a # numpy.ndarray of 1 dimension (vector) that has the same length as # the number of rows in the input. It should give the 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 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 # 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) # start-snippet-3 # 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] } ) # end-snippet-3 ############### # 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_validation_loss = numpy.inf test_score = 0. start_time = timeit.default_timer() done_looping = False epoch = 0 while (epoch < n_epochs) and (not done_looping): epoch = epoch + 1 for minibatch_index in range(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 range(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 range(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. ) ) # save the best model with open('best_model.pkl', 'wb') as f: pickle.dump(classifier, f) if patience <= iter: done_looping = True break end_time = timeit.default_timer() 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(('The code for file ' + os.path.split(__file__)[1] + ' ran for %.1fs' % ((end_time - start_time))), file=sys.stderr) def predict(): """ An example of how to load a trained model and use it to predict labels. """ # load the saved model classifier = pickle.load(open('best_model.pkl')) # compile a predictor function predict_model = theano.function( inputs=[classifier.input], outputs=classifier.y_pred) # We can test it on some examples from test test dataset='mnist.pkl.gz' datasets = load_data(dataset) test_set_x, test_set_y = datasets[2] test_set_x = test_set_x.get_value() predicted_values = predict_model(test_set_x[:10]) print("Predicted values for the first 10 examples in test set:") print(predicted_values) if __name__ == '__main__': sgd_optimization_mnist()