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。

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