Multilayer perceptron：多层感知器
本节实现两层网络（一个隐层）作为分类器实现手写数字分类。引入的内容：激活函数（双曲正切、L1和L2正则化）、Theano的共享变量、grad、floatX等。损失函数和错误率沿用了上一节的LogisticRegression类。本节没有使用反向传播来更新参数，用的依旧是损失函数对参数求导（梯度下降法）。网络隐层的激活函数为tanh，输出层即采用LogisticRegression。更新参数的机制：损失函数为LogisticRegression中的损失函数+两层网络的正则化的和，参数为两层分别的W和b。
要点如下：
1.初始化权重，众所周知在使用sigmoid激活函数时权重初始为零可能导致学习缓慢、隐层神经元的饱和。有许多方法初始化权重，文中给出：
当激活函数为双曲正切时 ：W取值为 
之间。
当激活函数为sigmoid时：W取值为：
之间。
2.网络中的超参数一般来说不能用梯度下降法优化，严格地讲，找到这些参数的最优解不可行。首先，我们不能简单地独立的优化它们中的每一个参数，其次不能应用之前的梯度下降法，最后优化问题非凸很难找到局部最小值。一个好的解决办法是反向传播，由Yann LeCun提出的。
3.学习速率：简单的解决办法是设为定值，随着时间减小学习速率有时也很好，一个简单的法则是μ0/(1+d*t)，μ0为初始设置的速率，d称为衰减常数控制衰减速率（10e-3或更小），t为迭代周期数。
总体代码如下：
# coding=UTF-8 # 两层网络、sgd优化（非bp）、early-stopping策略 import os import sys import timeit import numpy import theano import theano.tensor as T from Logistic_sgd import LogisticRegression, load_data #导入上一节的代码作为输出层 class HiddenLayer(object): #隐层类 def __init__(self, rng, input, n_in, n_out, W=None, b=None, activation=T.tanh): self.input = input if W is None: W_values = numpy.asarray( #W非初始化为零 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初始化为零 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)) self.params = [self.W, self.b] class MLP(object): #输出层 def __init__(self, rng, input, n_in, n_hidden, n_out): self.hiddenLayer = HiddenLayer(rng=rng, input=input, n_in=n_in, n_out=n_hidden, activation=T.tanh) self.logRegressionLayer = LogisticRegression(input=self.hiddenLayer.output, n_in=n_hidden, n_out=n_out) #引入输出层 self.L1 = (abs(self.hiddenLayer.W).sum()+ abs(self.logRegressionLayer.W).sum()) self.L2_sqr = ((self.hiddenLayer.W ** 2).sum() + (self.logRegressionLayer.W ** 2).sum()) #L1，L2正则化 self.negative_log_likelihood = (self.logRegressionLayer.negative_log_likelihood) self.errors = self.logRegressionLayer.errors self.params = self.hiddenLayer.params + self.logRegressionLayer.params #参数包括隐层和输出层 self.input = input def 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): 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] 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 print '...building the model' index = T.lscalar() x = T.matrix('x') y = T.ivector('y') rng = numpy.random.RandomState(1234) #随机数 classifier = MLP(rng=rng, input=x, n_in=28 * 28, n_hidden=n_hidden, n_out=10) #分类器 cost = (classifier.negative_log_likelihood(y) + L1_reg * classifier.L1 + L2_reg * classifier.L2_sqr) #损失函数 test_model = theano.function(inputs=[index], outputs=classifier.errors(y), #测试模型 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), #验证模型 givens={x: valid_set_x[index * batch_size:(index + 1) * batch_size], y: valid_set_y[index * batch_size:(index + 1) * batch_size]}) gparams = [T.grad(cost, param) for param in classifier.params] updates = [(param, param - learning_rate * gparam) for param, gparam in zip(classifier.params, gparams)] train_model = theano.function(inputs=[index], outputs=cost, updates=updates, #训练模型 givens={x: train_set_x[index * batch_size: (index + 1) * batch_size], y: train_set_y[index * batch_size: (index + 1) * batch_size]}) print '...training' patience = 10000 #early stopping策略 patience_increase = 2 improvement_threshold = 0.995 validation_frequency = min(n_train_batches, patience / 2) best_validation_loss = numpy.inf best_iter = 0 test_score = 0. start_time = timeit.default_timer() epoch = 0 done_looping = False while (epoch < n_epochs) and (not done_looping): #迭代优化过程（以下注释和上一节相同） epoch = epoch + 1 for minibatch_index in xrange(n_train_batches): minibatch_avg_cost = train_model(minibatch_index) iter = (epoch - 1) * n_train_batches + minibatch_index if (iter + 1) % validation_frequency == 0: 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 this_validation_loss < best_validation_loss: if (this_validation_loss < best_validation_loss * improvement_threshold): patience = max(patience, iter * patience_increase) best_validation_loss = this_validation_loss #最优解对应的验证损失值 best_iter = iter #最优解对应的迭代次数 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 = timeit.default_timer() print( ('Optimization compelete.Best validation scores of % %%''obtained at iteration %i,with test performance %f %%') % (best_validation_loss * 100., best_iter + 1, test_score * 100.)) if __name__ == '__main__': test_mlp()