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226 lines
23 KiB
Markdown
226 lines
23 KiB
Markdown
# Theano 实例：线性回归
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## 基本模型
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在用 `theano` 进行线性回归之前，先回顾一下 `theano` 的运行模式。
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`theano` 是一个符号计算的数学库，一个基本的 `theano` 结构大致如下：
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* 定义符号变量
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* 编译用符号变量定义的函数，使它能够用这些符号进行数值计算。
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* 将函数应用到数据上去
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In [1]:
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```py
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%matplotlib inline
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from matplotlib import pyplot as plt
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import numpy as np
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import theano
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from theano import tensor as T
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```
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```py
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Using gpu device 0: GeForce GTX 850M
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```
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简单的例子：$y = a \times b, a, b \in \mathbb{R}$
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定义 $a, b, y$：
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In [2]:
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```py
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a = T.scalar()
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b = T.scalar()
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y = a * b
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```
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编译函数：
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In [3]:
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```py
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multiply = theano.function(inputs=[a, b], outputs=y)
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```
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将函数运用到数据上：
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In [4]:
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```py
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print multiply(3, 2) # 6
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print multiply(4, 5) # 20
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```
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```py
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6.0
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20.0
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```
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## 线性回归
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回到线性回归的模型，假设我们有这样的一组数据：
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In [5]:
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```py
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train_X = np.linspace(-1, 1, 101)
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train_Y = 2 * train_X + 1 + np.random.randn(train_X.size) * 0.33
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```
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分布如图：
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In [6]:
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```py
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plt.scatter(train_X, train_Y)
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plt.show()
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```
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### 定义符号变量
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我们使用线性回归的模型对其进行模拟： $$\bar{y} = wx + b$$
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首先我们定义 $x, y$：
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In [7]:
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```py
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X = T.scalar()
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Y = T.scalar()
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```
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可以在定义时候直接给变量命名，也可以之后修改变量的名字：
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In [8]:
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```py
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X.name = 'x'
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Y.name = 'y'
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```
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我们的模型为：
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In [9]:
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```py
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def model(X, w, b):
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return X * w + b
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```
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在这里我们希望模型得到 $\bar{y}$ 与真实的 $y$ 越接近越好，常用的平方损失函数如下： $$C = |\bar{y}-y|^2$$
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有了损失函数，我们就可以使用梯度下降法来迭代参数 $w, b$ 的值，为此，我们将 $w$ 和 $b$ 设成共享变量：
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In [10]:
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```py
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w = theano.shared(np.asarray(0., dtype=theano.config.floatX))
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w.name = 'w'
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b = theano.shared(np.asarray(0., dtype=theano.config.floatX))
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b.name = 'b'
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```
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定义 $\bar y$：
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In [11]:
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```py
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Y_bar = model(X, w, b)
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theano.pp(Y_bar)
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```
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Out[11]:
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```py
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'((x * HostFromGpu(w)) + HostFromGpu(b))'
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```
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损失函数及其梯度：
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In [12]:
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```py
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cost = T.mean(T.sqr(Y_bar - Y))
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grads = T.grad(cost=cost, wrt=[w, b])
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```
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定义梯度下降规则：
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In [13]:
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```py
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lr = 0.01
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updates = [[w, w - grads[0] * lr],
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[b, b - grads[1] * lr]]
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```
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### 编译训练模型
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每运行一次，参数 $w, b$ 的值就更新一次：
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In [14]:
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```py
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train_model = theano.function(inputs=[X,Y],
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outputs=cost,
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updates=updates,
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allow_input_downcast=True)
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```
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### 将训练函数应用到数据上
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训练模型，迭代 100 次：
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In [15]:
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```py
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for i in xrange(100):
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for x, y in zip(train_X, train_Y):
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train_model(x, y)
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```
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显示结果：
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In [16]:
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```py
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print w.get_value() # 接近 2
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print b.get_value() # 接近 1
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plt.scatter(train_X, train_Y)
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plt.plot(train_X, w.get_value() * train_X + b.get_value(), 'r')
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plt.show()
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```
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```py
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1.94257426262
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1.00938093662
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```
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