2.5BatchNormalzation

BatchNormalzation是一种解决深度神经网络层数太多，而没有办法有效前向传递的问题，因为每层的输出值都会有不同的均值和方差，所以输出数据的分布也不一样。

如果对于输入的X*W本身得到的值通过tanh激活函数已经输出为1，在通过下一层的神经元之后结果并没有什么改变，因为该值已经很大（神经网络过敏感问题）

对于这种情况我们可以首先做一些批处理正则化

　　可以看出没有BN的时候，每层的值迅速全部都变为0，也可以说，所有的神经元迅速都已经死去了，而有BN，relu之后，每层的值都能有一个比较好的分布效果

大部分神经元都还活着

relu下的误差对比，no BN的曲线是没有的，很快消失

如果改为tanh，误差变化如下

"""
Build two networks.
1. Without batch normalization
2. With batch normalization

Run tests on these two networks.
"""

#  Batch Normalization

import numpy as np
import tensorflow as tf
import matplotlib.pyplot as plt

ACTIVATION = tf.nn.relu #tf.nn.tanh
N_LAYERS = 7
N_HIDDEN_UNITS = 30

def fix_seed(seed=1):
    # reproducible
    np.random.seed(seed)
    tf.set_random_seed(seed)

def plot_his(inputs, inputs_norm):
    # plot histogram for the inputs of every layer
    for j, all_inputs in enumerate([inputs, inputs_norm]):
        for i, input in enumerate(all_inputs):
            plt.subplot(2, len(all_inputs), j*len(all_inputs)+(i+1))
            plt.cla()
            if i == 0:
                the_range = (-7, 10)
            else:
                the_range = (-1, 1)
            plt.hist(input.ravel(), bins=15, range=the_range, color='#FF5733')
            plt.yticks(())
            if j == 1:
                plt.xticks(the_range)
            else:
                plt.xticks(())
            ax = plt.gca()
            ax.spines['right'].set_color('none')
            ax.spines['top'].set_color('none')
        plt.title("%s normalizing" % ("Without" if j == 0 else "With"))
    plt.draw()
    plt.pause(0.01)

def built_net(xs, ys, norm):
    def add_layer(inputs, in_size, out_size, activation_function=None, norm=False):
        # weights and biases (bad initialization for this case)
        Weights = tf.Variable(tf.random_normal([in_size, out_size], mean=0., stddev=1.))
        biases = tf.Variable(tf.zeros([1, out_size]) + 0.1)

        # fully connected product
        Wx_plus_b = tf.matmul(inputs, Weights) + biases

        # normalize fully connected product
        if norm:
            # Batch Normalize
            fc_mean, fc_var = tf.nn.moments(
                Wx_plus_b,
                axes=[0],   # the dimension you wanna normalize, here [0] for batch
                            # for image, you wanna do [0, 1, 2] for [batch, height, width] but not channel
            )
            scale = tf.Variable(tf.ones([out_size]))
            shift = tf.Variable(tf.zeros([out_size]))
            epsilon = 0.001

            # apply moving average for mean and var when train on batch
            ema = tf.train.ExponentialMovingAverage(decay=0.5)
            def mean_var_with_update():
                ema_apply_op = ema.apply([fc_mean, fc_var])
                with tf.control_dependencies([ema_apply_op]):
                    return tf.identity(fc_mean), tf.identity(fc_var)
            mean, var = mean_var_with_update()
            #all parmeter batch normalization
            Wx_plus_b = tf.nn.batch_normalization(Wx_plus_b, mean, var, shift, scale, epsilon)
            # similar with this two steps:
            # Wx_plus_b = (Wx_plus_b - fc_mean) / tf.sqrt(fc_var + 0.001)
            # Wx_plus_b = Wx_plus_b * scale + shift

        # activation
        if activation_function is None:
            outputs = Wx_plus_b
        else:
            outputs = activation_function(Wx_plus_b)

        return outputs

    fix_seed(1)

    if norm:
        # BN for the first input
        fc_mean, fc_var = tf.nn.moments(
            xs,
            axes=[0],
        )
        scale = tf.Variable(tf.ones([1]))
        shift = tf.Variable(tf.zeros([1]))
        epsilon = 0.001
        # apply moving average for mean and var when train on batch
        ema = tf.train.ExponentialMovingAverage(decay=0.5)
        def mean_var_with_update():
            ema_apply_op = ema.apply([fc_mean, fc_var])
            with tf.control_dependencies([ema_apply_op]):
                return tf.identity(fc_mean), tf.identity(fc_var)
        mean, var = mean_var_with_update()
        xs = tf.nn.batch_normalization(xs, mean, var, shift, scale, epsilon)

    # record inputs for every layer
    layers_inputs = [xs]

    # build hidden layers
    for l_n in range(N_LAYERS):
        layer_input = layers_inputs[l_n]
        in_size = layers_inputs[l_n].get_shape()[1].value

        output = add_layer(
            layer_input,    # input
            in_size,        # input size
            N_HIDDEN_UNITS, # output size
            ACTIVATION,     # activation function
            norm,           # normalize before activation
        )
        layers_inputs.append(output)    # add output for next run

    # build output layer
    prediction = add_layer(layers_inputs[-1], 30, 1, activation_function=None)

    cost = tf.reduce_mean(tf.reduce_sum(tf.square(ys - prediction), reduction_indices=[1]))
    train_op = tf.train.GradientDescentOptimizer(0.001).minimize(cost)
    return [train_op, cost, layers_inputs]

# make up data
fix_seed(1)
x_data = np.linspace(-7, 10, 2500)[:, np.newaxis]
np.random.shuffle(x_data)
noise = np.random.normal(0, 8, x_data.shape)
y_data = np.square(x_data) - 5 + noise

# plot input data
plt.scatter(x_data, y_data)
plt.show()

xs = tf.placeholder(tf.float32, [None, 1])  # [num_samples, num_features]
ys = tf.placeholder(tf.float32, [None, 1])
#create 2 neuralnetwork
train_op, cost, layers_inputs = built_net(xs, ys, norm=False)   # without BN
train_op_norm, cost_norm, layers_inputs_norm = built_net(xs, ys, norm=True) # with BN

sess = tf.Session()
if int((tf.__version__).split('.')[1]) < 12 and int((tf.__version__).split('.')[0]) < 1:
    init = tf.initialize_all_variables()
else:
    init = tf.global_variables_initializer()
sess.run(init)

# record cost
cost_his = []
cost_his_norm = []
record_step = 5

plt.ion()
plt.figure(figsize=(7, 3))
for i in range(250):
    if i % 50 == 0:
        # plot histogram
        all_inputs, all_inputs_norm = sess.run([layers_inputs, layers_inputs_norm], feed_dict={xs: x_data, ys: y_data})
        plot_his(all_inputs, all_inputs_norm)

    # train on batch
    sess.run([train_op, train_op_norm], feed_dict={xs: x_data[i*10:i*10+10], ys: y_data[i*10:i*10+10]})

    if i % record_step == 0:
        # record cost
        cost_his.append(sess.run(cost, feed_dict={xs: x_data, ys: y_data}))
        cost_his_norm.append(sess.run(cost_norm, feed_dict={xs: x_data, ys: y_data}))

plt.ioff()
plt.figure()
plt.plot(np.arange(len(cost_his))*record_step, np.array(cost_his), label='no BN')     # no norm
plt.plot(np.arange(len(cost_his))*record_step, np.array(cost_his_norm), label='BN')   # norm
plt.legend()
plt.show()

数据集图示