theano学习指南3(翻译)-多层感知器模型
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本节要用Theano实现的结构是一个隐层的多层感知器模型(MLP)。MLP可以看成一种对数回归器,其中输入通过非线性转移矩阵
(本节只要介绍了MLP的实现,对神经网络的背景知识介绍不多,感兴趣的朋友可以进一步阅读相应教程 - 译者注)
MLP模型
MLP模型可以用以下的图来表示:
单隐层的MLP定义了一个映射:
,其中
其中
向量
模型的输出向量为
为了训练MLP模型,我们用随机梯度下降算法学习所有参数,包括
从对数回归模型到多层感知器
本节我们专注于单层的MLP模型。在此,我们首先实现一个表示隐层的类。为了构建MLP模型,我们需要在此之上构建一个对数回归层。
class HiddenLayer(object): def __init__(self, rng, input, n_in, n_out, activation=T.tanh): """ Typical hidden layer of a MLP: units are fully-connected and have sigmoidal activation function. Weight matrix W is of shape (n_in,n_out) and the bias vector b is of shape (n_out,). NOTE : The nonlinearity used here is tanh Hidden unit activation is given by: tanh(dot(input,W) + b) :type rng: numpy.random.RandomState :param rng: a random number generator used to initialize weights :type input: theano.tensor.dmatrix :param input: a symbolic tensor of shape (n_examples, n_in) :type n_in: int :param n_in: dimensionality of input :type n_out: int :param n_out: number of hidden units :type activation: theano.Op or function :param activation: Non linearity to be applied in the hidden layer """ self.input = input
隐层权重的初始值需要从一个和激活函数相关的对称区间上面均匀采样得到。对于tanh函数,采样区间应该为
W_values = numpy.asarray(rng.uniform( low=-numpy.sqrt(6. / (n_in + n_out)), high=numpy.sqrt(6. / (n_in + n_out)), size=(n_in, n_out)), dtype=theano.config.floatX)if activation == theano.tensor.nnet.sigmoid: W_values *= 4self.W = theano.shared(value=W_values, name='W')b_values = numpy.zeros((n_out,), dtype=theano.config.floatX)self.b = theano.shared(value=b_values, name='b')
这里我们要注意到,隐层的激活函数为一个非线性函数。函数缺省为 tanh,但是很多情况下,我们可能用下面的函数
self.output = activation(T.dot(input, self.W) + self.b)# parameters of the modelself.params = [self.W, self.b]
结合理论知识,这里其实是计算了隐层的输出:
class MLP(object): """Multi-Layer Perceptron Class A multilayer perceptron is a feedforward artificial neural network model that has one layer or more of hidden units and nonlinear activations. Intermediate layers usually have as activation function tanh or the sigmoid function (defined here by a ``HiddenLayer`` class) while the top layer is a softamx layer (defined here by a ``LogisticRegression`` class). """ def __init__(self, rng, input, n_in, n_hidden, n_out): """Initialize the parameters for the multilayer perceptron :type rng: numpy.random.RandomState :param rng: a random number generator used to initialize weights :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_hidden: int :param n_hidden: number of hidden units :type n_out: int :param n_out: number of output units, the dimension of the space in which the labels lie """ # Since we are dealing with a one hidden layer MLP, this will # translate into a Hidden Layer connected to the LogisticRegression # layer self.hiddenLayer = HiddenLayer(rng = rng, input = input, n_in = n_in, n_out = n_hidden, activation = T.tanh) # The logistic regression layer gets as input the hidden units # of the hidden layer self.logRegressionLayer = LogisticRegression( input=self.hiddenLayer.output, n_in=n_hidden, n_out=n_out)
在本节中,我们仍然采用
# L1 norm ; one regularization option is to enforce L1 norm to# be smallself.L1 = abs(self.hiddenLayer.W).sum() \ + abs(self.logRegressionLayer.W).sum()# square of L2 norm ; one regularization option is to enforce# square of L2 norm to be smallself.L2_sqr = (self.hiddenLayer.W ** 2).sum() \ + (self.logRegressionLayer.W ** 2).sum()# negative log likelihood of the MLP is given by the negative# log likelihood of the output of the model, computed in the# logistic regression layerself.negative_log_likelihood = self.logRegressionLayer.negative_log_likelihood# same holds for the function computing the number of errorsself.errors = self.logRegressionLayer.errors# the parameters of the model are the parameters of the two layer it is# made out ofself.params = self.hiddenLayer.params + self.logRegressionLayer.params
和之前一样,我们用在mini-batch上面的随机梯度下降算法训练模型。这里的区别在于,我们修改损失函数并包括规范化项。 L1_reg 和 L2_reg 为超参数,用以控制规范化项在整个损失函数中的比重。计算损失的函数如下:
# the cost we minimize during training is the negative log likelihood of# the model plus the regularization terms (L1 and L2); cost is expressed# here symbolicallycost = classifier.negative_log_likelihood(y) \ + L1_reg * L1 \ + L2_reg * L2_sqr
接下来,模型参数通过梯度更新。这段代码和之前的基本上一样,除了参数多少的差别。
# compute the gradient of cost with respect to theta (stored in params)# the resulting gradients will be stored in a list gparamsgparams = []for param in classifier.params: gparam = T.grad(cost, param) gparams.append(gparam)# specify how to update the parameters of the model as a list of# (variable, update expression) pairsupdates = []# given two list the zip A = [a1, a2, a3, a4] and B = [b1, b2, b3, b4] of# same length, zip generates a list C of same size, where each element# is a pair formed from the two lists :# C = [(a1, b1), (a2, b2), (a3, b3) , (a4, b4)]for param, gparam in zip(classifier.params, gparams): updates.append((param, param - learning_rate * gparam))# compiling a Theano function `train_model` that returns the cost, butx# 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]})
功能综合
基于以上基本概念,写一个MLP的类变成了一件非常容易的事情。下面的代码演示了它是如何运作的,其原理和我们之前的对数回归分类器基本一致。
"""This tutorial introduces the multilayer perceptron using Theano. A multilayer perceptron is a logistic regressor whereinstead of feeding the input to the logistic regression you insert aintermediate layer, called the hidden layer, that has a nonlinearactivation function (usually tanh or sigmoid) . One can use many suchhidden layers making the architecture deep. The tutorial will also tacklethe problem of MNIST digit classification... math:: f(x) = G( b^{(2)} + W^{(2)}( s( b^{(1)} + W^{(1)} x))),References: - textbooks: "Pattern Recognition and Machine Learning" - Christopher M. Bishop, section 5"""__docformat__ = 'restructedtext en'import cPickleimport gzipimport osimport sysimport timeimport numpyimport theanoimport theano.tensor as Tfrom logistic_sgd import LogisticRegression, load_dataclass HiddenLayer(object): def __init__(self, rng, input, n_in, n_out, W=None, b=None, activation=T.tanh): """ Typical hidden layer of a MLP: units are fully-connected and have sigmoidal activation function. Weight matrix W is of shape (n_in,n_out) and the bias vector b is of shape (n_out,). NOTE : The nonlinearity used here is tanh Hidden unit activation is given by: tanh(dot(input,W) + b) :type rng: numpy.random.RandomState :param rng: a random number generator used to initialize weights :type input: theano.tensor.dmatrix :param input: a symbolic tensor of shape (n_examples, n_in) :type n_in: int :param n_in: dimensionality of input :type n_out: int :param n_out: number of hidden units :type activation: theano.Op or function :param activation: Non linearity to be applied in the hidden layer """ self.input = input # `W` is initialized with `W_values` which is uniformely sampled # from sqrt(-6./(n_in+n_hidden)) and sqrt(6./(n_in+n_hidden)) # for tanh activation function # the output of uniform if converted using asarray to dtype # theano.config.floatX so that the code is runable on GPU # Note : optimal initialization of weights is dependent on the # activation function used (among other things). # For example, results presented in [Xavier10] suggest that you # should use 4 times larger initial weights for sigmoid # compared to tanh # We have no info for other function, so we use the same as # tanh. if W is None: W_values = numpy.asarray(rng.uniform( low=-numpy.sqrt(6. / (n_in + n_out)), high=numpy.sqrt(6. / (n_in + n_out)), size=(n_in, n_out)), dtype=theano.config.floatX) if activation == theano.tensor.nnet.sigmoid: W_values *= 4 W = theano.shared(value=W_values, name='W', borrow=True) if b is None: b_values = numpy.zeros((n_out,), dtype=theano.config.floatX) b = theano.shared(value=b_values, name='b', borrow=True) self.W = W self.b = b lin_output = T.dot(input, self.W) + self.b self.output = (lin_output if activation is None else activation(lin_output)) # parameters of the model self.params = [self.W, self.b]class MLP(object): """Multi-Layer Perceptron Class A multilayer perceptron is a feedforward artificial neural network model that has one layer or more of hidden units and nonlinear activations. Intermediate layers usually have as activation function thanh or the sigmoid function (defined here by a ``SigmoidalLayer`` class) while the top layer is a softamx layer (defined here by a ``LogisticRegression`` class). """ def __init__(self, rng, input, n_in, n_hidden, n_out): """Initialize the parameters for the multilayer perceptron :type rng: numpy.random.RandomState :param rng: a random number generator used to initialize weights :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_hidden: int :param n_hidden: number of hidden units :type n_out: int :param n_out: number of output units, the dimension of the space in which the labels lie """ # Since we are dealing with a one hidden layer MLP, this will # translate into a TanhLayer connected to the LogisticRegression # layer; this can be replaced by a SigmoidalLayer, or a layer # implementing any other nonlinearity self.hiddenLayer = HiddenLayer(rng=rng, input=input, n_in=n_in, n_out=n_hidden, activation=T.tanh) # The logistic regression layer gets as input the hidden units # of the hidden layer self.logRegressionLayer = LogisticRegression( input=self.hiddenLayer.output, n_in=n_hidden, n_out=n_out) # L1 norm ; one regularization option is to enforce L1 norm to # be small self.L1 = abs(self.hiddenLayer.W).sum() \ + abs(self.logRegressionLayer.W).sum() # square of L2 norm ; one regularization option is to enforce # square of L2 norm to be small self.L2_sqr = (self.hiddenLayer.W ** 2).sum() \ + (self.logRegressionLayer.W ** 2).sum() # negative log likelihood of the MLP is given by the negative # log likelihood of the output of the model, computed in the # logistic regression layer self.negative_log_likelihood = self.logRegressionLayer.negative_log_likelihood # same holds for the function computing the number of errors self.errors = self.logRegressionLayer.errors # the parameters of the model are the parameters of the two layer it is # made out of self.params = self.hiddenLayer.params + self.logRegressionLayer.paramsdef test_mlp(learning_rate=0.01, L1_reg=0.00, L2_reg=0.0001, n_epochs=1000, dataset='../data/mnist.pkl.gz', batch_size=20, n_hidden=500): """ Demonstrate stochastic gradient descent optimization for a multilayer perceptron This is demonstrated on MNIST. :type learning_rate: float :param learning_rate: learning rate used (factor for the stochastic gradient :type L1_reg: float :param L1_reg: L1-norm's weight when added to the cost (see regularization) :type L2_reg: float :param L2_reg: L2-norm's weight when added to the cost (see regularization) :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 rng = numpy.random.RandomState(1234) # construct the MLP class classifier = MLP(rng=rng, input=x, n_in=28 * 28, n_hidden=n_hidden, n_out=10) # the cost we minimize during training is the negative log likelihood of # the model plus the regularization terms (L1 and L2); cost is expressed # here symbolically cost = classifier.negative_log_likelihood(y) \ + L1_reg * classifier.L1 \ + L2_reg * classifier.L2_sqr # 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 (sotred in params) # the resulting gradients will be stored in a list gparams gparams = [] for param in classifier.params: gparam = T.grad(cost, param) gparams.append(gparam) # specify how to update the parameters of the model as a list of # (variable, update expression) pairs updates = [] # given two list the zip A = [a1, a2, a3, a4] and B = [b1, b2, b3, b4] of # same length, zip generates a list C of same size, where each element # is a pair formed from the two lists : # C = [(a1, b1), (a2, b2), (a3, b3), (a4, b4)] for param, gparam in zip(classifier.params, gparams): updates.append((param, param - learning_rate * gparam)) # 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' # early-stopping parameters patience = 10000 # 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 best_iter = 0 test_score = 0. start_time = time.clock() epoch = 0 done_looping = False 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 best_iter = iter # 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. Best validation score of %f %% ' 'obtained at iteration %i, with test performance %f %%') % (best_validation_loss * 100., best_iter + 1, test_score * 100.)) print >> sys.stderr, ('The code for file ' + os.path.split(__file__)[1] + ' ran for %.2fm' % ((end_time - start_time) / 60.))if __name__ == '__main__': test_mlp()
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