模型测试
正如在训练模型一节解释的，我们还需要关注在分类过程中，实际分类错误的数目。上面的LogisticRegression 类增加了两个附加的方法，允许去得到每一个迷你批次里实际错误分类的样本书。
代码如下：
def errors(self, y):"""Return a float representing the number of errors in the minibatchover the total number of examples of the minibatch ; zero oneloss over the size of the minibatch:type y: theano.tensor.TensorType:param y: corresponds to a vector that gives for each example thecorrect label"""# check if y has same dimension of y_predif 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 datatypeif y.dtype.startswith('int'):# the T.neq operator returns a vector of 0s and 1s, where 1# represents a mistake in predictionreturn T.mean(T.neq(self.y_pred, y))else:raise NotImplementedError()
接下来我们创建一个test_model的方法，这样我们就能够来调用它，并返模型搞错的数目。马上我们还会看到，validate_model这个方法是我们提前停止实现的一个关键。
如果有不懂的请看提前停止这两个方法把迷你批的索引作为参数去计算，然后计算出来那个迷你批的错误分类数是多少。差别是test_model是从测试数据集里取数的，而validate_model是从validate数据集里取数的。
我们来看代码如下：
# compiling a Theano function that computes the mistakes that are made by# the model on a minibatchtest_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]})
好，我们把整个例子放在一起来看，最终成品如下：
""" 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 timeitimport numpyimport theano
import theano.tensor as Tclass LogisticRegression(object):"""Multi-class Logistic Regression ClassThe logistic regression is fully described by a weight matrix :math:`W`and bias vector :math:`b`. Classification is done by projecting datapoints onto a set of hyperplanes, the distance to which is used todetermine 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 thearchitecture (one minibatch):type n_in: int:param n_in: number of input units, the dimension of the space inwhich the datapoints lie:type n_out: int:param n_out: number of output units, the dimension of the space inwhich 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 0sself.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-kself.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 maximalself.y_pred = T.argmax(self.p_y_given_x, axis=1)# end-snippet-1# parameters of the modelself.params = [self.W, self.b]# keep track of model inputself.input = inputdef negative_log_likelihood(self, y):"""Return the mean of the negative log-likelihood of the predictionof 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 thecorrect labelNote: we use the mean instead of the sum so thatthe 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-2def errors(self, y):"""Return a float representing the number of errors in the minibatchover the total number of examples of the minibatch ; zero oneloss over the size of the minibatch:type y: theano.tensor.TensorType:param y: corresponds to a vector that gives for each example thecorrect label"""# check if y has same dimension of y_predif 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 datatypeif y.dtype.startswith('int'):# the T.neq operator returns a vector of 0s and 1s, where 1# represents a mistake in predictionreturn 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 presentdata_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_pathif (not os.path.isfile(dataset)) and data_file == 'mnist.pkl.gz':from six.moves import urlliborigin = ('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 datasetwith 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 variablesThe reason we store our dataset in shared variables is to allowTheano to copy it into the GPU memory (when code is run on GPU).Since copying data into the GPU is slow, copying a minibatch everytimeis needed (the default behaviour if the data is not in a sharedvariable) would lead to a large decrease in performance."""data_x, data_y = data_xyshared_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 issuereturn 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 rvaldef 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-linearmodelThis is demonstrated on MNIST.:type learning_rate: float:param learning_rate: learning rate used (factor for the stochasticgradient):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 fromhttp://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 testingn_train_batches = train_set_x.get_value(borrow=True).shape[0] // batch_sizen_valid_batches = valid_set_x.get_value(borrow=True).shape[0] // batch_sizen_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 dataindex = 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 imagesy = T.ivector('y') # labels, presented as 1D vector of [int] labels# construct the logistic regression class# Each MNIST image has size 28*28classifier = 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 formatcost = classifier.negative_log_likelihood(y)# compiling a Theano function that computes the mistakes that are made by# the model on a minibatchtest_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 parameterspatience = 5000 # look as this many examples regardlesspatience_increase = 2 # wait this much longer when a new best is# foundimprovement_threshold = 0.995 # a relative improvement of this much is# considered significantvalidation_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 epochbest_validation_loss = numpy.inftest_score = 0.start_time = timeit.default_timer()done_looping = Falseepoch = 0while (epoch < n_epochs) and (not done_looping):epoch = epoch + 1for minibatch_index in range(n_train_batches):minibatch_avg_cost = train_model(minibatch_index)# iteration numberiter = (epoch - 1) * n_train_batches + minibatch_indexif (iter + 1) % validation_frequency == 0:# compute zero-one loss on validation setvalidation_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 nowif this_validation_loss < best_validation_loss:#improve patience if loss improvement is good enoughif 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 settest_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 modelwith open('best_model.pkl', 'wb') as f:pickle.dump(classifier, f)if patience <= iter:done_looping = Truebreakend_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 itto predict labels."""# load the saved modelclassifier = pickle.load(open('best_model.pkl'))# compile a predictor functionpredict_model = theano.function(inputs=[classifier.input],outputs=classifier.y_pred)# We can test it on some examples from test testdataset='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()你可以通过一下命令行来训练模型：
python code/logistic_sgd.py输出应该是这样子的:
...
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
在英特尔双核8400 3.00G赫兹的机器上，耗费大约1.936 轮/秒 ，一共用了75轮，达到了7.4%的错误率。 在gpu上， 耗费大约是 10轮/秒 , 我们使用了600作为迷你批的大小。
使用训练好的模型来预测
当新的比较低的验证错误出现的时候，代码会保存模型，我们可以重新读取次模型，然后去预测新的数据。预测函数的代码如下：
def predict():"""An example of how to load a trained model and use itto predict labels."""# load the saved modelclassifier = pickle.load(open('best_model.pkl'))# compile a predictor functionpredict_model = theano.function(inputs=[classifier.input],outputs=classifier.y_pred)# We can test it on some examples from test testdataset='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)脚注
针对那些小的数据集，以及简单的模型，更加成熟稳定的算法会更有效，这个代码展示了联合梯度的具体方法。