Theano是一个可以让你定义，优化，计算数学表达式的一个python库。多种机器学习方法可以很方便的用Theano实现。下面是用Theano实现logistic regression （逻辑回归）的例子。 摘自Theano官网的tutorial里面。
主要步骤分为下面几步
1. 定义逻辑回归模型
2. 定义建立数据集方法
3.定义训练逻辑回归模型方法
4. 训练
代码下载地址： http://deeplearning.net/tutorial/logreg.html#logreg
"""This tutorial introduces logistic regression using Theano and stochasticgradient descent.Logistic regression is a probabilistic, linear classifier. It is parametrizedby a weight matrix :math:`W` and a bias vector :math:`b`. Classification isdone by projecting data points onto a set of hyperplanes, the distance towhich 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 ofthe 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 methodsuitable for large datasets.References: - textbooks: "Pattern Recognition and Machine Learning" - Christopher M. Bishop, section 4.3.2"""__docformat__ = 'restructedtext en'import cPickleimport gzipimport osimport sysimport timeimport numpyimport theanoimport theano.tensor as T#定义逻辑回归类。里面主要定义了：1） 逻辑回归模型的参数 W和b 2）两个损失函数 negative log likelihood 和0-1 损失# 3）其他辅助变量。 类属概率和最大类概率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 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 class-membership # probabilities # Where: # W is a matrix where column-k represent the separation hyper plain 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 hyper # plain-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] 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()#读取数据集。 主要是把数据分为训练，验证和测试数据。每一类包括feature 和 label。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#定义了如何利用数据集优化逻辑回归模型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 # ###################### #建立模型。1）定义一个逻辑回归模型，2）定义损失函数。3)然后定义检验，验证方法。 #4)计算逻辑回归模型的梯度信息 5）定义训练方法 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 # ############### #开始训练逻辑回归模型。 把每一个epoch分为若干mini-batch， 对于每一个mini-batch的数据，训练逻辑回归模型。对于满足 #一定条件的mini-batch， 调用验证函数计算验证集的错误率。如果错误率是显著的， 则把patience增大。当预定义的epoch数量 #训练完成或者patience小于已经进行的训练循环次数 （mini-batch），则整个训练过程结束。输出得到的最好validation错误率 # 和相应的test 错误率 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 = 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()