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Softmax Regressions

$\text{evidence}_i = \sum_j W_{i,~ j} x_j + b_i \tag{1}$ $y = \text{softmax}(\text{evidence}) \tag{2}$ $\text{softmax}(x) = \text{normalize}(\exp(x)) = \frac{\exp(x_i)}{\sum_j \exp(x_j)} \tag{3}$

完整的代码如下:

from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("MNIST_data/", one_hot=True)
import tensorflow as tf
x = tf.placeholder(tf.float32, [None, 784])
W = tf.Variable(tf.zeros([784, 10]))
b = tf.Variable(tf.zeros([10]))
y = tf.nn.softmax(tf.matmul(x, W) + b)
y_ = tf.placeholder(tf.float32, [None, 10])
# cross_entropy = tf.reduce_mean(-tf.reduce_sum(y_ * tf.log(y), reduction_indices=[1]))
cross_entropy = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(labels=y_, logits=y))
train_step = tf.train.GradientDescentOptimizer(0.5).minimize(cross_entropy)  # learning rate = 0.5
sess = tf.InteractiveSession()
tf.global_variables_initializer().run()
for _ in xrange(1000):
    batch_xs, batch_ys = mnist.train.next_batch(100)
    sess.run(train_step, feed_dict={x: batch_xs, y_:batch_ys})
correct_prediction = tf.equal(tf.argmax(y, 1), tf.argmax(y_, 1))
accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))
print sess.run(accuracy, feed_dict={x: mnist.test.images, y_: mnist.test.labels})
# Output: 0.9192

以上代码仅仅只有一层隐层神经元，然后就连上softmax，十分简单的结构。

tensorflow的后台底层(backend)是用C++写的，连接backend的叫做Session。所以从上面的代码也可以看出整个代码分为两个部分，前一部分构建graph，后一部分就是创建一个Session来连接这个graph，并运行它。也就是说，tensorflow在计算反向传播等一系列深度学习的操作的时候，都是用C++在跑，而python的作用是构建外壳，也就是graph，来调用这个C++核心，实现交互。

文档里建议使用InteractiveSession，而不是之前的Session，说InteractiveSession的好处是更加灵活，可以交错的构建computation graph，有点像IPython，如果使用Session，则需要构建完整的graph。这段话我不太理解，仅从上面这段代码来看，InteractiveSession也是构建了完整的graph，跟Session好像并无太大区别，唯一我发现的区别就是使用feed_dict来传参。

假如使用Session来操作，上面的部分代码改为如下:

sess = tf.Session()
init = tf.global_variables_initializer()
sess.run(init)
for _ in xrange(1000):
    batch_xs, batch_ys = mnist.train.next_batch(100)
    sess.run(train_step, {x: batch_xs, y_:batch_ys})

更深层次的区别有待我继续探索。

还有一个小点，就是文档建议使用官方给出的函数tf.nn.softmax_cross_entropy_with_logits来计算cross-entropy，而不是自己写（就是被注释掉的那一句).

Build a Multilayer Convolutional Network

简单的一层隐层网络能达到近似92%的精确度，现在使用多层卷积网络可以达到99.2%的精确度。代码如下：

# Weight Initialization
def weight_variable(shape):
    initial = tf.truncated_normal(shape, stddev=0.1)
    return tf.Variable(initial)

def bias_variable(shape):
    initial = tf.constant(0.1, shape=shape)
    return tf.Variable(initial)

# Convolution and Pooling
def conv2d(x, W):
    return tf.nn.conv2d(x, W, strides=[1, 1, 1, 1], padding='SAME')

def max_pool_2x2(x):
    return tf.nn.max_pool(x, ksize=[1, 2, 2, 1], strides=[1, 2, 2, 1], padding='SAME')

# First Convolutional Layer
W_conv1 = weight_variable([5, 5, 1, 32])
b_conv1 = bias_variable([32])

x_image = tf.reshape(x, [-1, 28, 28, 1])

h_conv1 = tf.nn.relu(conv2d(x_image, W_conv1) + b_conv1)
h_pool1 = max_pool_2x2(h_conv1)

# Second Convolutional Layer
W_conv2 = weight_variable([5, 5, 32, 64])
b_conv2 = bias_variable([64])

h_conv2 = tf.nn.relu(conv2d(h_pool1, W_conv2) + b_conv2)
h_pool2 = max_pool_2x2(h_conv2)

# Densely Connected Layer
W_fc1 = weight_variable([7*7*64, 1024])
b_fc1 = bias_variable([1024])

h_pool2_flat = tf.reshape(h_pool2, [-1, 7*7*64])
h_fc1 = tf.nn.relu(tf.matmul(h_pool2_flat, W_fc1) + b_fc1)

# Dropout
keep_prob = tf.placeholder(tf.float32)
h_fc1_drop = tf.nn.dropout(h_fc1, keep_prob)

# Readout Layer
W_fc2 = weight_variable([1024, 10])
b_fc2 = bias_variable([10])

y_conv = tf.matmul(h_fc1_drop, W_fc2) + b_fc2

# Train and Evaluate the Model
cross_entropy = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(labels=y_, logits=y_conv))
train_step = tf.train.AdamOptimizer(1e-4).minimize(cross_entropy)
correct_prediction = tf.equal(tf.argmax(y_conv,1), tf.argmax(y_,1))
accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))
sess.run(tf.global_variables_initializer())
for i in range(5000):
  batch = mnist.train.next_batch(50)
  if i%100 == 0:
    train_accuracy = accuracy.eval(feed_dict={
        x:batch[0], y_: batch[1], keep_prob: 1.0})
    print("step %d, training accuracy %g"%(i, train_accuracy))
  train_step.run(feed_dict={x: batch[0], y_: batch[1], keep_prob: 0.5})

print("test accuracy %g"%accuracy.eval(feed_dict={
    x: mnist.test.images, y_: mnist.test.labels, keep_prob: 1.0}))

上面代码的训练循环次数，改为20000次，可以达到99.2%，5000次大概只有98.6%的准确率。

tf.nn.conv2d(input, filter, strides, padding, use_cudnn_on_gpu=None, data_format=None, name=None) Given an input tensor of shape [batch, in_height, in_width, in_channels] and a filter / kernel tensor of shape [filter_height, filter_width, in_channels, out_channels]。

padding='SAME':

out_height = ceil(float(in_height) / float(strides[1]))
out_width  = ceil(float(in_width) / float(strides[2]))

padding='VALID':

out_height = ceil(float(in_height - filter_height + 1) / float(strides[1]))
out_width  = ceil(float(in_width - filter_width + 1) / float(strides[2]))

Reference

Tensorflo文档
StackOverflow关于tf.nn.conv2d的理解的提问

MNIST example

Softmax Regressions

Build a Multilayer Convolutional Network

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