[深度学习]Python/Theano实现逻辑回归网络的代码分析

首先PO上主要Python代码(2.7), 这个代码在Deep Learning上可以找到.

 1 　　 # allocate symbolic variables for the data
 2     index = T.lscalar()  # index to a [mini]batch
 3     x = T.matrix('x')  # the data is presented as rasterized images
 4     y = T.ivector('y')  # the labels are presented as 1D vector of
 5                            # [int] labels
 6 
 7     # construct the logistic regression class
 8     # Each MNIST image has size 28*28
 9     classifier = LogisticRegression(input=x, n_in=24 * 48, n_out=10)
10 
11     # the cost we minimize during training is the negative log likelihood of
12     # the model in symbolic format
13     cost = classifier.negative_log_likelihood(y)
14 
15     # compiling a Theano function that computes the mistakes that are made by
16     # the model on a minibatch
17     test_model = theano.function(inputs=[index],
18             outputs=classifier.errors(y),
19             givens={
20                 x: test_set_x[index * batch_size: (index + 1) * batch_size],
21                 y: test_set_y[index * batch_size: (index + 1) * batch_size]})
22 
23     validate_model = theano.function(inputs=[index],
24             outputs=classifier.errors(y),
25             givens={
26                 x: valid_set_x[index * batch_size:(index + 1) * batch_size],
27                 y: valid_set_y[index * batch_size:(index + 1) * batch_size]})
28 
29     # compute the gradient of cost with respect to theta = (W,b)
30     g_W = T.grad(cost=cost, wrt=classifier.W)
31     g_b = T.grad(cost=cost, wrt=classifier.b)
32 
33     # specify how to update the parameters of the model as a list of
34     # (variable, update expression) pairs.
35     updates = [(classifier.W, classifier.W - learning_rate * g_W),
36                (classifier.b, classifier.b - learning_rate * g_b)]
37 
38     # compiling a Theano function `train_model` that returns the cost, but in
39     # the same time updates the parameter of the model based on the rules
40     # defined in `updates`
41     train_model = theano.function(inputs=[index],
42             outputs=cost,
43             updates=updates,
44             givens={
45                 x: train_set_x[index * batch_size:(index + 1) * batch_size],
46                 y: train_set_y[index * batch_size:(index + 1) * batch_size]})

代码长度不算太长, 只是逻辑关系需要厘清. 下面逐行分析这些代码. 

代码中的T是theano.tensor的代名词.

行1~行13:

# 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=24 * 48, 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)

声明index, x, y三个符号变量(类似Matlab的symbol), 分别用来指代训练样本批序号, 输入图像矩阵, 期望输出向量.

classifier是一个LR对象, 调用LR类的构造函数, 并将符号变量x作为输入, 我们就可以使用Theano.function方法在x和classifier中构造联系, 当x改变时, classifier也会改变.

cost指代classifier中的负对数相似度, 使用符号变量y作为输入, 此处的作用和classifier相同, 不再赘述.

行14~行28:

    # 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]})

这里的2个model是容易让人迷惑的地方, 关于theano.function, 需要一些基础知识:

比如声明2个符号变量a, b: a, b = T.iscalar(), T.iscalar() , 它们都是整形(i)标量(scalar), 再声明一个变量c:  c = a + b , 我们通过type(c)来查看其类型:

>>> type(c)
<class 'theano.tensor.var.TensorVariable'>
>>> type(a)
<class 'theano.tensor.var.TensorVariable'>

　　c的类型和a, b相同, 都是Tensor变量. 至此准备工作完成, 我们通过theano.function来构建关系:  add = theano.function(inputs = [a, b], output = c) . 这条语句就构造了一个函数add, 它接收a, b为输入, 输出为c. 我们在Python中这样使用它即可:

>>> add = theano.function(inputs = [a, b], outputs = c)
>>> test = add(100, 100)
>>> test
array(200)

好了, 有了基础知识, 就可以理解这2个model的含义:

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]})

输入是index, 输出则是classifier对象中的errors方法的返回值, 其中y作为errors方法的输入参数. 其中的classifier接收x作为输入参数.

givens关键字的作用是使用冒号后面的变量来替代冒号前面的变量, 本例中, 即使用测试数据中的第index批数据(一批有batch_size个)来替换x和y.

test_model用中文来解释就是: 接收第index批测试数据的图像数据x和期望输出y作为输入, 返回误差值的函数. 

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]})

这里同上, 只不过使用的是验证数据.

行29~行32:

    # 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)

计算的是梯度, 用于学习算法, T.grad(y, x) 计算的是相对于x的y的梯度.

行33~行37:

    # 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)]

updates是一个长度为2的list, 每个元素都是一组tuple, 在theano.function中, 每次调用对应函数, 使用tuple中的第二个元素来更新第一个元素.

行38~行46:

　　# 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]})

这里其余部分不再赘述. 需要注意的是增加了一个updates参数, 这个参数给定了每次调用train_model时对某些参数的修改(W, b). 另外输出也变成了cost函数(对数误差)而非test_model和valid-model中的errors函数(绝对误差).