09.13 Theano 实例：更复杂的网络
Theano 实例：更复杂的网络
1import theano
2import theano.tensor as T
3import numpy as np
4from load import mnist
5from theano.sandbox.rng_mrg import MRG_RandomStreams as RandomStreams
6
7srng = RandomStreams()
8
9def floatX(X):
10 return np.asarray(X, dtype=theano.config.floatX)
Using gpu device 1: Tesla C2075 (CNMeM is disabled)
上一节我们用了一个简单的神经网络来训练 MNIST 数据，这次我们使用更复杂的网络来进行训练，同时加入 dropout 机制，防止过拟合。
这里采用比较简单的 dropout 机制，即将输入值按照一定的概率随机置零。
1def dropout(X, prob=0.):
2 if prob > 0:
3 X *= srng.binomial(X.shape, p=1-prob, dtype = theano.config.floatX)
4 X /= 1 - prob
5 return X
之前我们采用的的激活函数是 sigmoid，现在我们使用 rectify 激活函数。
这可以使用 T.nnet.relu(x, alpha=0) 来实现，它本质上相当于：T.switch(x > 0, x, alpha * x)，而 rectify 函数的定义为：
$$ \text{rectify}(x) = \left{ \begin{aligned} x, & \ x > 0 \ 0, & \ x < 0 \end{aligned}\right. $$
之前我们构造的是一个单隐层的神经网络结构，现在我们构造一个双隐层的结构即“输入-隐层1-隐层2-输出”的全连接结构。
$$ \begin{aligned} & h_1 = \text{rectify}(W_{h_1} \ x) \ & h_2 = \text{rectify}(W_{h_2} \ h_1) \ & o = \text{softmax}(W_o h_2) \end{aligned} $$
Theano 自带的 T.nnet.softmax() 的 GPU 实现目前似乎有 bug 会导致梯度溢出的问题，因此自定义了 softmax 函数：
1def softmax(X):
2 e_x = T.exp(X - X.max(axis=1).dimshuffle(0, 'x'))
3 return e_x / e_x.sum(axis=1).dimshuffle(0, 'x')
4
5def model(X, w_h1, w_h2, w_o, p_drop_input, p_drop_hidden):
6 """
7 input:
8 X: input data
9 w_h1: weights input layer to hidden layer 1
10 w_h2: weights hidden layer 1 to hidden layer 2
11 w_o: weights hidden layer 2 to output layer
12 p_drop_input: dropout rate for input layer
13 p_drop_hidden: dropout rate for hidden layer
14 output:
15 h1: hidden layer 1
16 h2: hidden layer 2
17 py_x: output layer
18 """
19 X = dropout(X, p_drop_input)
20 h1 = T.nnet.relu(T.dot(X, w_h1))
21
22 h1 = dropout(h1, p_drop_hidden)
23 h2 = T.nnet.relu(T.dot(h1, w_h2))
24
25 h2 = dropout(h2, p_drop_hidden)
26 py_x = softmax(T.dot(h2, w_o))
27 return h1, h2, py_x
随机初始化权重矩阵：
1def init_weights(shape):
2 return theano.shared(floatX(np.random.randn(*shape) * 0.01))
3
4w_h1 = init_weights((784, 625))
5w_h2 = init_weights((625, 625))
6w_o = init_weights((625, 10))
定义变量：
1X = T.matrix()
2Y = T.matrix()
定义更新的规则，之前我们使用的是简单的 SGD，这次我们使用 RMSprop 来更新，其规则为： $$ \begin{align} MS(w, t) & = \rho MS(w, t-1) + (1-\rho) \left(\left.\frac{\partial E}{\partial w}\right|{w(t-1)}\right)^2 \ w(t) & = w(t-1) - \alpha \left.\frac{\partial E}{\partial w}\right|{w(t-1)} / \sqrt{MS(w, t)} \end{align} $$
1def RMSprop(cost, params, accs, lr=0.001, rho=0.9, epsilon=1e-6):
2 grads = T.grad(cost=cost, wrt=params)
3 updates = []
4 for p, g, acc in zip(params, grads, accs):
5 acc_new = rho * acc + (1 - rho) * g ** 2
6 gradient_scaling = T.sqrt(acc_new + epsilon)
7 g = g / gradient_scaling
8 updates.append((acc, acc_new))
9 updates.append((p, p - lr * g))
10 return updates
训练函数：
1# 有 dropout，用来训练
2noise_h1, noise_h2, noise_py_x = model(X, w_h1, w_h2, w_o, 0.2, 0.5)
3cost = T.mean(T.nnet.categorical_crossentropy(noise_py_x, Y))
4params = [w_h1, w_h2, w_o]
5accs = [theano.shared(p.get_value() * 0.) for p in params]
6updates = RMSprop(cost, params, accs, lr=0.001)
7# 训练函数
8train = theano.function(inputs=[X, Y], outputs=cost, updates=updates, allow_input_downcast=True)
预测函数：
1# 没有 dropout，用来预测
2h1, h2, py_x = model(X, w_h1, w_h2, w_o, 0., 0.)
3# 预测的结果
4y_x = T.argmax(py_x, axis=1)
5predict = theano.function(inputs=[X], outputs=y_x, allow_input_downcast=True)
训练：
1trX, teX, trY, teY = mnist(onehot=True)
2
3for i in range(50):
4 for start, end in zip(range(0, len(trX), 128), range(128, len(trX), 128)):
5 cost = train(trX[start:end], trY[start:end])
6 print "iter {:03d} accuracy:".format(i + 1), np.mean(np.argmax(teY, axis=1) == predict(teX))
iter 001 accuracy: 0.943
iter 002 accuracy: 0.9665
iter 003 accuracy: 0.9732
iter 004 accuracy: 0.9763
iter 005 accuracy: 0.9767
iter 006 accuracy: 0.9802
iter 007 accuracy: 0.9795
iter 008 accuracy: 0.979
iter 009 accuracy: 0.9807
iter 010 accuracy: 0.9805
iter 011 accuracy: 0.9824
iter 012 accuracy: 0.9816
iter 013 accuracy: 0.9838
iter 014 accuracy: 0.9846
iter 015 accuracy: 0.983
iter 016 accuracy: 0.9837
iter 017 accuracy: 0.9841
iter 018 accuracy: 0.9837
iter 019 accuracy: 0.9835
iter 020 accuracy: 0.9844
iter 021 accuracy: 0.9837
iter 022 accuracy: 0.9839
iter 023 accuracy: 0.984
iter 024 accuracy: 0.9851
iter 025 accuracy: 0.985
iter 026 accuracy: 0.9847
iter 027 accuracy: 0.9851
iter 028 accuracy: 0.9846
iter 029 accuracy: 0.9846
iter 030 accuracy: 0.9853
iter 031 accuracy: 0.985
iter 032 accuracy: 0.9844
iter 033 accuracy: 0.9849
iter 034 accuracy: 0.9845
iter 035 accuracy: 0.9848
iter 036 accuracy: 0.9868
iter 037 accuracy: 0.9864
iter 038 accuracy: 0.9866
iter 039 accuracy: 0.9859
iter 040 accuracy: 0.9857
iter 041 accuracy: 0.9853
iter 042 accuracy: 0.9855
iter 043 accuracy: 0.9861
iter 044 accuracy: 0.9865
iter 045 accuracy: 0.9872
iter 046 accuracy: 0.9867
iter 047 accuracy: 0.9868
iter 048 accuracy: 0.9863
iter 049 accuracy: 0.9862
iter 050 accuracy: 0.9856