深度学习实践系列（2）- 搭建notMNIST的深度神经网络

如果你希望系统性的了解神经网络，请参考零基础入门深度学习系列 ，下面我会粗略的介绍一下本文中实现神经网络需要了解的知识。

什么是深度神经网络？

神经网络包含三层：输入层（X）、隐藏层和输出层：f(x)

每层之间每个节点都是完全连接的，其中包含权重（W）。每层都存在一个偏移值（b）。

每一层节点的计算方式如下：

其中g()代表激活函数，o()代表softmax输出函数。

使用Flow Graph的方式来表达如何正向推导神经网络，可以表达如下：

x: 输入值

a(x)：表示每个隐藏层的pre-activation的数据，由前一个隐藏层数据(h)、权重(w)和偏移值(b)计算而来

h(x)：表示每个隐藏层的数据

f(x)：表示输出层数据

激活函数ReLUs

激活函数有很多种类，例如sigmoid、tanh、ReLUs，对于深度神经网络而言，目前最流行的是ReLUs。

关于几种激活函数的对比可以参见：常用激活函数的总结与比较

ReLUs函数如下：

反向传播

现在，我们需要知道一个神经网络的每个连接上的权值是如何得到的。我们可以说神经网络是一个模型，那么这些权值就是模型的参数，也就是模型要学习的东西。然而，一个神经网络的连接方式、网络的层数、每层的节点数这些参数，则不是学习出来的，而是人为事先设置的。对于这些人为设置的参数，我们称之为超参数(Hyper-Parameters)。

接下来，我们将要介绍神经网络的训练算法：反向传播算法。

具体内容请参考：零基础入门深度学习(3) - 神经网络和反向传播算法

SGD

梯度下降算法是一种不断调整参数值从而达到减少Loss function的方法，通过不断迭代而获得最佳的权重值。梯度下降传统上是每次迭代都使用全部训练数据来进行参数调整，随机梯度下降则是使用少量训练数据来进行调整。

关于GD和SGD的区别可以参考：GD(梯度下降)和SGD(随机梯度下降)

正则化和Dropout

正则化和Dropout都是一些防止过度拟合的方法，详细介绍可以参考：正则化方法：L1和L2 regularization、数据集扩增、dropout

正则化：通过在Loss function中加入对权重(w)的惩罚，可以限制权重值变得非常大

Dropout: 通过随机抛弃一些节点，使得神经网络更加多样性，然后组合起来获得的结果更加通用。

好吧，基本的概念大概介绍了一遍，开始撸代码啦。

请先参考深度学习实践系列（1）- 从零搭建notMNIST逻辑回归模型，获得notMNIST.pickle的训练数据。

1. 引用第三方库

# These are all the modules we'll be using later. Make sure you can import them
# before proceeding further.
from __future__ import print_function
import numpy as np
import tensorflow as tf
from six.moves import cPickle as pickle
from six.moves import range

2. 读取数据

pickle_file = 'notMNIST.pickle'

with open(pickle_file, 'rb') as f:
  save = pickle.load(f)
  train_dataset = save['train_dataset']
  train_labels = save['train_labels']
  valid_dataset = save['valid_dataset']
  valid_labels = save['valid_labels']
  test_dataset = save['test_dataset']
  test_labels = save['test_labels']
  del save  # hint to help gc free up memory
  print('Training set', train_dataset.shape, train_labels.shape)
  print('Validation set', valid_dataset.shape, valid_labels.shape)
  print('Test set', test_dataset.shape, test_labels.shape)

Training set (200000, 28, 28) (200000,)
Validation set (10000, 28, 28) (10000,)
Test set (10000, 28, 28) (10000,)

3. 调整数据格式以便后续训练

image_size = 28
num_labels = 10

def reformat(dataset, labels):
  dataset = dataset.reshape((-1, image_size * image_size)).astype(np.float32)
  # Map 0 to [1.0, 0.0, 0.0 ...], 1 to [0.0, 1.0, 0.0 ...]
  labels = (np.arange(num_labels) == labels[:,None]).astype(np.float32)
  return dataset, labels
train_dataset, train_labels = reformat(train_dataset, train_labels)
valid_dataset, valid_labels = reformat(valid_dataset, valid_labels)
test_dataset, test_labels = reformat(test_dataset, test_labels)
print('Training set', train_dataset.shape, train_labels.shape)
print('Validation set', valid_dataset.shape, valid_labels.shape)
print('Test set', test_dataset.shape, test_labels.shape)

Training set (200000, 784) (200000, 10)
Validation set (10000, 784) (10000, 10)
Test set (10000, 784) (10000, 10)

4. 定义神经网络

# With gradient descent training, even this much data is prohibitive.
# Subset the training data for faster turnaround.
train_subset = 10000

graph = tf.Graph()
with graph.as_default():

  # Input data.
  # Load the training, validation and test data into constants that are
  # attached to the graph.
  tf_train_dataset = tf.constant(train_dataset[:train_subset, :])
  tf_train_labels = tf.constant(train_labels[:train_subset])
  tf_valid_dataset = tf.constant(valid_dataset)
  tf_test_dataset = tf.constant(test_dataset)
  
  # Variables.
  # These are the parameters that we are going to be training. The weight
  # matrix will be initialized using random values following a (truncated)
  # normal distribution. The biases get initialized to zero.
  weights = tf.Variable(
    tf.truncated_normal([image_size * image_size, num_labels]))
  biases = tf.Variable(tf.zeros([num_labels]))
  
  # Training computation.
  # We multiply the inputs with the weight matrix, and add biases. We compute
  # the softmax and cross-entropy (it's one operation in TensorFlow, because
  # it's very common, and it can be optimized). We take the average of this
  # cross-entropy across all training examples: that's our loss.
  logits = tf.matmul(tf_train_dataset, weights) + biases
  loss = tf.reduce_mean(
    tf.nn.softmax_cross_entropy_with_logits(labels=tf_train_labels, logits=logits))
  
  # Optimizer.
  # We are going to find the minimum of this loss using gradient descent.
  optimizer = tf.train.GradientDescentOptimizer(0.5).minimize(loss)
  
  # Predictions for the training, validation, and test data.
  # These are not part of training, but merely here so that we can report
  # accuracy figures as we train.
  train_prediction = tf.nn.softmax(logits)
  valid_prediction = tf.nn.softmax(
    tf.matmul(tf_valid_dataset, weights) + biases)
  test_prediction = tf.nn.softmax(tf.matmul(tf_test_dataset, weights) + biases)

5. 使用梯度下降（GD）训练神经网络

num_steps = 801

def accuracy(predictions, labels):
  return (100.0 * np.sum(np.argmax(predictions, 1) == np.argmax(labels, 1))
          / predictions.shape[0])

with tf.Session(graph=graph) as session:
  # This is a one-time operation which ensures the parameters get initialized as
  # we described in the graph: random weights for the matrix, zeros for the
  # biases. 
  tf.global_variables_initializer().run()
  print('Initialized')
  for step in range(num_steps):
    # Run the computations. We tell .run() that we want to run the optimizer,
    # and get the loss value and the training predictions returned as numpy
    # arrays.
    _, l, predictions = session.run([optimizer, loss, train_prediction])
    if (step % 100 == 0):
      print('Loss at step %d: %f' % (step, l))
      print('Training accuracy: %.1f%%' % accuracy(
        predictions, train_labels[:train_subset, :]))
      # Calling .eval() on valid_prediction is basically like calling run(), but
      # just to get that one numpy array. Note that it recomputes all its graph
      # dependencies.
      print('Validation accuracy: %.1f%%' % accuracy(
        valid_prediction.eval(), valid_labels))
  print('Test accuracy: %.1f%%' % accuracy(test_prediction.eval(), test_labels))

Initialized
Loss at step 0: 16.516306
Training accuracy: 11.4%
Validation accuracy: 11.7%
Loss at step 100: 2.269041
Training accuracy: 71.8%
Validation accuracy: 70.2%
Loss at step 200: 1.816886
Training accuracy: 74.8%
Validation accuracy: 72.6%
Loss at step 300: 1.574824
Training accuracy: 76.0%
Validation accuracy: 73.6%
Loss at step 400: 1.415523
Training accuracy: 77.1%
Validation accuracy: 73.9%
Loss at step 500: 1.299691
Training accuracy: 78.0%
Validation accuracy: 74.4%
Loss at step 600: 1.209450
Training accuracy: 78.6%
Validation accuracy: 74.6%
Loss at step 700: 1.135888
Training accuracy: 79.0%
Validation accuracy: 74.9%
Loss at step 800: 1.074228
Training accuracy: 79.5%
Validation accuracy: 75.0%
Test accuracy: 82.3%

6. 使用随机梯度下降（SGD）算法

batch_size = 128

graph = tf.Graph()
with graph.as_default():

  # Input data. For the training data, we use a placeholder that will be fed
  # at run time with a training minibatch.
  tf_train_dataset = tf.placeholder(tf.float32,
                                    shape=(batch_size, image_size * image_size))
  tf_train_labels = tf.placeholder(tf.float32, shape=(batch_size, num_labels))
  tf_valid_dataset = tf.constant(valid_dataset)
  tf_test_dataset = tf.constant(test_dataset)
  
  # Variables.
  weights = tf.Variable(
    tf.truncated_normal([image_size * image_size, num_labels]))
  biases = tf.Variable(tf.zeros([num_labels]))
  
  # Training computation.
  logits = tf.matmul(tf_train_dataset, weights) + biases
  loss = tf.reduce_mean(
    tf.nn.softmax_cross_entropy_with_logits(labels=tf_train_labels, logits=logits))
  
  # Optimizer.
  optimizer = tf.train.GradientDescentOptimizer(0.5).minimize(loss)
  
  # Predictions for the training, validation, and test data.
  train_prediction = tf.nn.softmax(logits)
  valid_prediction = tf.nn.softmax(
    tf.matmul(tf_valid_dataset, weights) + biases)
  test_prediction = tf.nn.softmax(tf.matmul(tf_test_dataset, weights) + biases)

num_steps = 3001

with tf.Session(graph=graph) as session:
  tf.global_variables_initializer().run()
  print("Initialized")
  for step in range(num_steps):
    # Pick an offset within the training data, which has been randomized.
    # Note: we could use better randomization across epochs.
    offset = (step * batch_size) % (train_labels.shape[0] - batch_size)
    # Generate a minibatch.
    batch_data = train_dataset[offset:(offset + batch_size), :]
    batch_labels = train_labels[offset:(offset + batch_size), :]
    # Prepare a dictionary telling the session where to feed the minibatch.
    # The key of the dictionary is the placeholder node of the graph to be fed,
    # and the value is the numpy array to feed to it.
    feed_dict = {tf_train_dataset : batch_data, tf_train_labels : batch_labels}
    _, l, predictions = session.run(
      [optimizer, loss, train_prediction], feed_dict=feed_dict)
    if (step % 500 == 0):
      print("Minibatch loss at step %d: %f" % (step, l))
      print("Minibatch accuracy: %.1f%%" % accuracy(predictions, batch_labels))
      print("Validation accuracy: %.1f%%" % accuracy(
        valid_prediction.eval(), valid_labels))
  print("Test accuracy: %.1f%%" % accuracy(test_prediction.eval(), test_labels))

Initialized
Minibatch loss at step 0: 18.121506
Minibatch accuracy: 11.7%
Validation accuracy: 15.0%
Minibatch loss at step 500: 1.192153
Minibatch accuracy: 80.5%
Validation accuracy: 76.1%
Minibatch loss at step 1000: 1.309419
Minibatch accuracy: 75.8%
Validation accuracy: 76.8%
Minibatch loss at step 1500: 0.739157
Minibatch accuracy: 83.6%
Validation accuracy: 77.3%
Minibatch loss at step 2000: 0.854160
Minibatch accuracy: 85.2%
Validation accuracy: 77.5%
Minibatch loss at step 2500: 1.045702
Minibatch accuracy: 76.6%
Validation accuracy: 78.8%
Minibatch loss at step 3000: 0.940078
Minibatch accuracy: 79.7%
Validation accuracy: 78.8%
Test accuracy: 85.8%

7. 使用ReLUs激活函数

batch_size = 128
hidden_layer_size = 1024

graph = tf.Graph()
with graph.as_default():

    # Input data. For the training data, we use a placeholder that will be fed
    # at run time with a training minibatch.
    tf_train_dataset = tf.placeholder(tf.float32,
                                    shape=(batch_size, image_size * image_size))
    tf_train_labels = tf.placeholder(tf.float32, shape=(batch_size, num_labels))
    tf_valid_dataset = tf.constant(valid_dataset)
    tf_test_dataset = tf.constant(test_dataset)

    # Variables.
    
    # Hidden layer (RELU magic)
    
    weights_hidden = tf.Variable(
        tf.truncated_normal([image_size * image_size, hidden_layer_size]))
    biases_hidden = tf.Variable(tf.zeros([hidden_layer_size]))
    hidden = tf.nn.relu(tf.matmul(tf_train_dataset, weights_hidden) + biases_hidden)

    # Output layer

    weights_output = tf.Variable(
        tf.truncated_normal([hidden_layer_size, num_labels]))
    biases_output = tf.Variable(tf.zeros([num_labels]))
    
    # Training computation.
    
    logits = tf.matmul(hidden, weights_output) + biases_output
    
    loss = tf.reduce_mean(
        tf.nn.softmax_cross_entropy_with_logits(labels=tf_train_labels, logits=logits))
  
    # Optimizer.

    optimizer = tf.train.GradientDescentOptimizer(0.5).minimize(loss)
  
    # Predictions for the training, validation, and test data.

    # Creation of hidden layer of RELU for the validation and testing process
    
    
    train_prediction = tf.nn.softmax(logits)
    
    hidden_validation = tf.nn.relu(tf.matmul(tf_valid_dataset, weights_hidden) + biases_hidden)
    valid_prediction = tf.nn.softmax(
    tf.matmul(hidden_validation, weights_output) + biases_output)
    
    hidden_prediction = tf.nn.relu(tf.matmul(tf_test_dataset, weights_hidden) + biases_hidden)
    test_prediction = tf.nn.softmax(tf.matmul(hidden_prediction, weights_output) + biases_output)

num_steps = 3001

def accuracy(predictions, labels):
  return (100.0 * np.sum(np.argmax(predictions, 1) == np.argmax(labels, 1))
          / predictions.shape[0])

with tf.Session(graph=graph) as session:
  tf.initialize_all_variables().run()
  print("Initialized")
  for step in range(num_steps):
    # Pick an offset within the training data, which has been randomized.
    # Note: we could use better randomization across epochs.
    offset = (step * batch_size) % (train_labels.shape[0] - batch_size)
    # Generate a minibatch.
    batch_data = train_dataset[offset:(offset + batch_size), :]
    batch_labels = train_labels[offset:(offset + batch_size), :]
    # Prepare a dictionary telling the session where to feed the minibatch.
    # The key of the dictionary is the placeholder node of the graph to be fed,
    # and the value is the numpy array to feed to it.
    feed_dict = {tf_train_dataset : batch_data, tf_train_labels : batch_labels}
    _, l, predictions = session.run(
      [optimizer, loss, train_prediction], feed_dict=feed_dict)
    if (step % 500 == 0):
      print("Minibatch loss at step %d: %f" % (step, l))
      print("Minibatch accuracy: %.1f%%" % accuracy(predictions, batch_labels))
      print("Validation accuracy: %.1f%%" % accuracy(
        valid_prediction.eval(), valid_labels))
  print("Test accuracy: %.1f%%" % accuracy(test_prediction.eval(), test_labels))

Initialized
Minibatch loss at step 0: 282.291931
Minibatch accuracy: 14.1%
Validation accuracy: 32.1%
Minibatch loss at step 500: 18.090569
Minibatch accuracy: 82.0%
Validation accuracy: 79.7%
Minibatch loss at step 1000: 15.504422
Minibatch accuracy: 75.0%
Validation accuracy: 80.8%
Minibatch loss at step 1500: 5.314545
Minibatch accuracy: 87.5%
Validation accuracy: 80.6%
Minibatch loss at step 2000: 3.442260
Minibatch accuracy: 86.7%
Validation accuracy: 81.5%
Minibatch loss at step 2500: 2.226066
Minibatch accuracy: 83.6%
Validation accuracy: 82.6%
Minibatch loss at step 3000: 2.228517
Minibatch accuracy: 83.6%
Validation accuracy: 82.5%
Test accuracy: 89.6%

8. 正则化

import math
batch_size = 128
hidden_layer_size = 1024 # Doubled because half of the results are discarded
regularization_beta = 5e-4

graph = tf.Graph()
with graph.as_default():

    # Input data. For the training data, we use a placeholder that will be fed
    # at run time with a training minibatch.
    tf_train_dataset = tf.placeholder(tf.float32,
                                    shape=(batch_size, image_size * image_size))
    tf_train_labels = tf.placeholder(tf.float32, shape=(batch_size, num_labels))
    tf_valid_dataset = tf.constant(valid_dataset)
    tf_test_dataset = tf.constant(test_dataset)

    # Variables.
    
    # Hidden layer (RELU magic)
    
    weights_hidden_1 = tf.Variable(
        tf.truncated_normal([image_size * image_size, hidden_layer_size], 
                           stddev=1 / math.sqrt(float(image_size * image_size))))
    biases_hidden_1 = tf.Variable(tf.zeros([hidden_layer_size]))
    hidden_1 = tf.nn.relu(tf.matmul(tf_train_dataset, weights_hidden_1) + biases_hidden_1)

    weights_hidden_2 = tf.Variable(tf.truncated_normal([hidden_layer_size, hidden_layer_size],
                                                      stddev=1 / math.sqrt(float(image_size * image_size))))
    biases_hidden_2 = tf.Variable(tf.zeros([hidden_layer_size]))
    hidden_2 = tf.nn.relu(tf.matmul(hidden_1, weights_hidden_2) + biases_hidden_2)
    
    # Output layer

    weights_output = tf.Variable(
        tf.truncated_normal([hidden_layer_size, num_labels],
                           stddev=1 / math.sqrt(float(image_size * image_size))))
    biases_output = tf.Variable(tf.zeros([num_labels]))
    
    # Training computation.
    
    logits = tf.matmul(hidden_2, weights_output) + biases_output
    
    loss = tf.reduce_mean(
        tf.nn.softmax_cross_entropy_with_logits(labels=tf_train_labels, logits=logits))
  
    # L2 regularization on hidden and output weights and biases
    
    regularizers = (tf.nn.l2_loss(weights_hidden_1) + tf.nn.l2_loss(biases_hidden_1) + 
                    tf.nn.l2_loss(weights_hidden_2) + tf.nn.l2_loss(biases_hidden_2) +
                    tf.nn.l2_loss(weights_output) + tf.nn.l2_loss(biases_output))

    loss = loss + regularization_beta * regularizers
    
    # Optimizer.

    optimizer = tf.train.GradientDescentOptimizer(0.5).minimize(loss)
  
    # Predictions for the training, validation, and test data.

    # Creation of hidden layer of RELU for the validation and testing process
    
    
    train_prediction = tf.nn.softmax(logits)
    
    hidden_validation_1 = tf.nn.relu(tf.matmul(tf_valid_dataset, weights_hidden_1) + biases_hidden_1)
    hidden_validation_2 = tf.nn.relu(tf.matmul(hidden_validation_1, weights_hidden_2) + biases_hidden_2)
    valid_prediction = tf.nn.softmax(
    tf.matmul(hidden_validation_2, weights_output) + biases_output)
    
    hidden_prediction_1 = tf.nn.relu(tf.matmul(tf_test_dataset, weights_hidden_1) + biases_hidden_1)
    hidden_prediction_2 = tf.nn.relu(tf.matmul(hidden_prediction_1, weights_hidden_2) + biases_hidden_2)
    test_prediction = tf.nn.softmax(tf.matmul(hidden_prediction_2, weights_output) + biases_output)

num_steps = 3001

def accuracy(predictions, labels):
  return (100.0 * np.sum(np.argmax(predictions, 1) == np.argmax(labels, 1))
          / predictions.shape[0])

with tf.Session(graph=graph) as session:
  tf.initialize_all_variables().run()
  print("Initialized")
  for step in range(num_steps):
    # Pick an offset within the training data, which has been randomized.
    # Note: we could use better randomization across epochs.
    offset = (step * batch_size) % (train_labels.shape[0] - batch_size)
    # Generate a minibatch.
    batch_data = train_dataset[offset:(offset + batch_size), :]
    batch_labels = train_labels[offset:(offset + batch_size), :]
    # Prepare a dictionary telling the session where to feed the minibatch.
    # The key of the dictionary is the placeholder node of the graph to be fed,
    # and the value is the numpy array to feed to it.
    feed_dict = {tf_train_dataset : batch_data, tf_train_labels : batch_labels}
    _, l, predictions = session.run(
      [optimizer, loss, train_prediction], feed_dict=feed_dict)
    if (step % 500 == 0):
      print("Minibatch loss at step %d: %f" % (step, l))
      print("Minibatch accuracy: %.1f%%" % accuracy(predictions, batch_labels))
      print("Validation accuracy: %.1f%%" % accuracy(
        valid_prediction.eval(), valid_labels))
  print("Test accuracy: %.1f%%" % accuracy(test_prediction.eval(), test_labels))

Initialized
Minibatch loss at step 0: 2.769384
Minibatch accuracy: 8.6%
Validation accuracy: 34.8%
Minibatch loss at step 500: 0.735681
Minibatch accuracy: 89.1%
Validation accuracy: 86.2%
Minibatch loss at step 1000: 0.791112
Minibatch accuracy: 85.9%
Validation accuracy: 86.9%
Minibatch loss at step 1500: 0.523572
Minibatch accuracy: 93.0%
Validation accuracy: 88.1%
Minibatch loss at step 2000: 0.487140
Minibatch accuracy: 95.3%
Validation accuracy: 88.5%
Minibatch loss at step 2500: 0.529468
Minibatch accuracy: 89.8%
Validation accuracy: 88.4%
Minibatch loss at step 3000: 0.531258
Minibatch accuracy: 86.7%
Validation accuracy: 88.9%
Test accuracy: 94.7%

9. Dropout

import math
batch_size = 128
hidden_layer_size = 2048 # Doubled because half of the results are discarded
regularization_beta = 5e-4

graph = tf.Graph()
with graph.as_default():

    # Input data. For the training data, we use a placeholder that will be fed
    # at run time with a training minibatch.
    tf_train_dataset = tf.placeholder(tf.float32,
                                    shape=(batch_size, image_size * image_size))
    tf_train_labels = tf.placeholder(tf.float32, shape=(batch_size, num_labels))
    tf_valid_dataset = tf.constant(valid_dataset)
    tf_test_dataset = tf.constant(test_dataset)

    # Variables.
    
    # Hidden layer (RELU magic)
    
    weights_hidden_1 = tf.Variable(
        tf.truncated_normal([image_size * image_size, hidden_layer_size], 
                           stddev=1 / math.sqrt(float(image_size * image_size))))
    biases_hidden_1 = tf.Variable(tf.zeros([hidden_layer_size]))
    hidden_1 = tf.nn.relu(tf.matmul(tf_train_dataset, weights_hidden_1) + biases_hidden_1)

    weights_hidden_2 = tf.Variable(tf.truncated_normal([hidden_layer_size, hidden_layer_size],
                                                      stddev=1 / math.sqrt(float(image_size * image_size))))
    biases_hidden_2 = tf.Variable(tf.zeros([hidden_layer_size]))
    hidden_2 = tf.nn.relu(tf.matmul(tf.nn.dropout(hidden_1, 0.5), weights_hidden_2) + biases_hidden_2)
    
    # Output layer

    weights_output = tf.Variable(
        tf.truncated_normal([hidden_layer_size, num_labels],
                           stddev=1 / math.sqrt(float(image_size * image_size))))
    biases_output = tf.Variable(tf.zeros([num_labels]))
    
    # Training computation.
    
    logits = tf.matmul(tf.nn.dropout(hidden_2, 0.5), weights_output) + biases_output
    
    loss = tf.reduce_mean(
        tf.nn.softmax_cross_entropy_with_logits(labels=tf_train_labels, logits=logits))
  
    # L2 regularization on hidden and output weights and biases
    
    regularizers = (tf.nn.l2_loss(weights_hidden_1) + tf.nn.l2_loss(biases_hidden_1) + 
                    tf.nn.l2_loss(weights_hidden_2) + tf.nn.l2_loss(biases_hidden_2) +
                    tf.nn.l2_loss(weights_output) + tf.nn.l2_loss(biases_output))

    loss = loss + regularization_beta * regularizers
    
    # Optimizer.

    optimizer = tf.train.GradientDescentOptimizer(0.5).minimize(loss)
  
    # Predictions for the training, validation, and test data.

    # Creation of hidden layer of RELU for the validation and testing process
    
    
    train_prediction = tf.nn.softmax(logits)
    
    hidden_validation_1 = tf.nn.relu(tf.matmul(tf_valid_dataset, weights_hidden_1) + biases_hidden_1)
    hidden_validation_2 = tf.nn.relu(tf.matmul(hidden_validation_1, weights_hidden_2) + biases_hidden_2)
    valid_prediction = tf.nn.softmax(
    tf.matmul(hidden_validation_2, weights_output) + biases_output)
    
    hidden_prediction_1 = tf.nn.relu(tf.matmul(tf_test_dataset, weights_hidden_1) + biases_hidden_1)
    hidden_prediction_2 = tf.nn.relu(tf.matmul(hidden_prediction_1, weights_hidden_2) + biases_hidden_2)
    test_prediction = tf.nn.softmax(tf.matmul(hidden_prediction_2, weights_output) + biases_output)

num_steps = 5001

def accuracy(predictions, labels):
  return (100.0 * np.sum(np.argmax(predictions, 1) == np.argmax(labels, 1))
          / predictions.shape[0])

with tf.Session(graph=graph) as session:
  tf.initialize_all_variables().run()
  print("Initialized")
  for step in range(num_steps):
    # Pick an offset within the training data, which has been randomized.
    # Note: we could use better randomization across epochs.
    offset = (step * batch_size) % (train_labels.shape[0] - batch_size)
    # Generate a minibatch.
    batch_data = train_dataset[offset:(offset + batch_size), :]
    batch_labels = train_labels[offset:(offset + batch_size), :]
    # Prepare a dictionary telling the session where to feed the minibatch.
    # The key of the dictionary is the placeholder node of the graph to be fed,
    # and the value is the numpy array to feed to it.
    feed_dict = {tf_train_dataset : batch_data, tf_train_labels : batch_labels}
    _, l, predictions = session.run(
      [optimizer, loss, train_prediction], feed_dict=feed_dict)
    if (step % 500 == 0):
      print("Minibatch loss at step %d: %f" % (step, l))
      print("Minibatch accuracy: %.1f%%" % accuracy(predictions, batch_labels))
      print("Validation accuracy: %.1f%%" % accuracy(
        valid_prediction.eval(), valid_labels))
  print("Test accuracy: %.1f%%" % accuracy(test_prediction.eval(), test_labels))

WARNING:tensorflow:From <ipython-input-12-3684c7218154>:8: initialize_all_variables (from tensorflow.python.ops.variables) is deprecated and will be removed after 2017-03-02.
Instructions for updating:
Use `tf.global_variables_initializer` instead.
Initialized
Minibatch loss at step 0: 4.059163
Minibatch accuracy: 7.8%
Validation accuracy: 31.5%
Minibatch loss at step 500: 1.626858
Minibatch accuracy: 86.7%
Validation accuracy: 84.8%
Minibatch loss at step 1000: 1.492026
Minibatch accuracy: 82.0%
Validation accuracy: 85.8%
Minibatch loss at step 1500: 1.139689
Minibatch accuracy: 92.2%
Validation accuracy: 87.1%
Minibatch loss at step 2000: 0.970064
Minibatch accuracy: 93.0%
Validation accuracy: 87.1%
Minibatch loss at step 2500: 0.963178
Minibatch accuracy: 87.5%
Validation accuracy: 87.6%
Minibatch loss at step 3000: 0.870884
Minibatch accuracy: 87.5%
Validation accuracy: 87.6%
Minibatch loss at step 3500: 0.898399
Minibatch accuracy: 85.2%
Validation accuracy: 87.7%
Minibatch loss at step 4000: 0.737084
Minibatch accuracy: 91.4%
Validation accuracy: 88.0%
Minibatch loss at step 4500: 0.646125
Minibatch accuracy: 88.3%
Validation accuracy: 87.7%
Minibatch loss at step 5000: 0.685591
Minibatch accuracy: 88.3%
Validation accuracy: 88.6%
Test accuracy: 94.4%

最后总结一下各种算法的训练表现，可以看出使用正则化和Dropout后训练效果明显变好，最后趋近于95%的准确率了。