用Theano实现最基本的对数回归分类器
在本章节中，我们会学习如何用Theano实现最基本的对数回归分类器。首先，我们会简单的复习一个这个模型，在这个过程中，大家可以进一步的了解如何把数学表达式和Theano的图模型结合起来。
数学模型
对数回归模型是试过线性概率分类器，它有两个参数，权重矩阵W和偏移向量b.分类的过程是把数据投影到一组高维超平面上，数据和平面的距离反应了它属于这个类别的概率。这个模型的数学公式可以表示为：
P(Y=i|x,W,b)=softmaxi(Wx+b)=eWix+bi∑jeWjx+bj
模型的输出即为预测的结果，它的值为：
ypred=argmaxiP(Y=i|x,W,b)
在Theano中，通过以下函数实现如下功能
# 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)
在以上代码中，首先定义了输入变量x,y. 因为模型在训练过程中要保持一个稳定的状态，模型参数W,b定义成共享变量，这种定义不仅可以声明变量，还会初始化他们的值。接下来，点乘和softmax操作用来计算模型输出P(Y|x,W,b). 结果保存在变量p_y_given_x中。
到目前为止，我们仅仅订了Theano运行的计算图模型。为了得到真实的P(Y|x,W,b)值，我们需要创建函数 get_p_y_given_x, 它以x为参数，输出值为p_y_given_x。我们可以遍历它的值，并得到数据属于每一个类别的概率。
现在，让我们结束Theano图的创建。为了得到模型的预测结果，我们用T.argmax操作符，这个操作返回p_y_given_x中做大值得索引。
类似的，为了得到给定输入的预测结果，我们定义函数classify。该函数以模型输入矩阵x为参数，输出为列向量，表示了每个实例的预测类别。
当然，这个模型还没有任何用途，因为模型参数还处于初始状态。下面的章节中，我们将学习如何训练模型 。
损失函数
模型的训练过程也就是最小化损失函数的过程。 在多类别的对数回归模型中，通常采用负对数似然函数作为模型的参数。这相当于在以θ为参数的模型中，最大化训练数据的似然。如果我们定义似然和损失函数如下：
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)
下面的代码演示了如何计算一个minbatch的损失
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.
创建LogisticRegression类
现在我们已经有了LogisticRegression类的所有功能。该类的代码如下，这些代码涵盖了我们之前学习的所有功能。
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])
这个类可以通过以下方式实例化：
# 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)
最后，定义损失函数：
cost = classifier.negative_log_likelihood(y)
模型的训练
为了在编程语言里面实现MSGD，我们需要手动计算模型的微分。如果模型比较复杂的话，计算过程会变得非常困难。
在Theano中，这个工作可以通过函数自动的完成，实例代码如下：
# 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和g_b是符号变量，他们可以用在计算图模型中。下面代码演示了执行一步梯度下降算法的过程：
# 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这列表里面包含了对每一个变量的随机梯度算法下面的更新操作。givens 字典里面包含数据和计算图模型中变量的映射关系。整个train_model定义了：
- 输入：为通过index索引的mini-batch，其数据定义为x，相应的label表示为y.
- 返回值，为相应的损失
- 每次函数调用的时候，首先通过index检索相应的参数 x， y， 然后计算在这个minbatch上面的函数损失，并应用定义在updates 列表中的操作更新参数。
函数 train_model(index) 调用的时候，它会计算并返回近似的损失，并执行一步MSGD操作。整个学习过程包括一系列的在该数据集上的循环，也就是是一个反复的调用这个函数的过程
模型的测试
正如第一节介绍的，我们对模型的测试主要是关心它的错误分类的数据的数量，而不仅仅是似然函数。因此类 LogisticRegression 中需要一个成员函数，用于建立返回测试数据上面的误分数据的数目符号图（symbolic graph）。 代码如下：
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))
接下来我们定义函数 test_model和validte_model， 以便于得到这个函数的值。 validate_model是实现前期结束的关键（见前一节）。两个函数的功能都是对已一个给定的batch，计算其误分类的实例的数目。两个函数的区别在于，它们一个运行在测试数据上，一个运行在验证数据上。相应的函数代码如下：
# 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]})
综合所有功能
如果把以上所有的功能整合在一起，就得到如下的代码：
""" 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()
这段程序采用SGD逻辑回归算法学习分类器，在DeepLearningTutorials文件夹中，可以通过以下命令调用：
python code/logistic_sgd.py
