第五章.与学习相关技巧

5.3 Batch Normalization

• Batch Norm以进行学习时的mini_batch为单位，按mini_batch进行正则化，具体而言，就是进行使数据分布的均值为0，方差为1的正则化。
• Batch Norm是调整各层激活值的分布使其拥有适当的广度。

1.优点

• 可以使学习快速进行
• 不那么依赖初始值
• 抑制过拟合(降低Dropout等的必要性)

2.数学式

1).均值：

2).方差：

3).正则化：

• 参数说明：
  ①.ε：微小值（1e-7），防止出现除数为0的情况

4).缩放和平移

• 参数说明：
  ①.α，β：一开始α=1，β=0，然后通过学习调整到合适的值。

3.通过MNIST数据集来对比使用Batch Norm层与不使用Batch Norm层的差异。

1).不同权重初始值的标准差：

2).图像分析：

• 从图像可知，几乎所有的情况下都是使用Batch Norm时学习进行的更快，同时也可以发现，在不使用Batch Norm的情况下，如果不赋予一个尺度好的初始值，学习将完全无法进行。

• 综上，通过使用Batch Norm，可以推动学习的进行，并且对权重初始值变得健壮，不那么依赖初始值。

3).代码实现：

import sys, ossys.path.append(os.pardir)
import numpy as np
import matplotlib.pyplot as plt
from dataset.mnist import load_mnist
from collections import OrderedDict# 加载数据
def get_data():(x_train, t_train), (x_test, t_test) = load_mnist(normalize=True)return (x_train, t_train), (x_test, t_test)def Sigmoid(x):return 1 / (1 + np.exp(-x))def Relu(x):return np.maximum(x, 0)def numerical_gradient(f, x):h = 1e-4  # 0.0001grad = np.zeros_like(x)it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])while not it.finished:idx = it.multi_indextmp_val = x[idx]x[idx] = float(tmp_val) + hfxh1 = f(x)  # f(x+h)x[idx] = tmp_val - hfxh2 = f(x)  # f(x-h)grad[idx] = (fxh1 - fxh2) / (2 * h)x[idx] = tmp_val  # 还原值it.iternext()return gradclass SoftmaxWithLoss:def __init__(self):self.loss = None  # 损失self.y = None  # softmax的输出self.t = None  # 监督数据(one_hot vector)# 输出层函数：softmaxdef softmax(self, x):if x.ndim == 2:x = x.Tx = x - np.max(x, axis=0)y = np.exp(x) / np.sum(np.exp(x), axis=0)return y.Tx = x - np.max(x)  # 溢出对策return np.exp(x) / np.sum(np.exp(x))# 交叉熵误差def cross_entropy_error(self, y, t):if y.ndim == 1:t = t.reshape(1, t.size)y = y.reshape(1, y.size)# 监督数据是one-hot-vector的情况下，转换为正确解标签的索引if t.size == y.size:t = t.argmax(axis=1)batch_size = y.shape[0]return -np.sum(np.log(y[np.arange(batch_size), t] + 1e-7)) / batch_size# 正向传播def forward(self, x, t):self.t = tself.y = self.softmax(x)self.loss = self.cross_entropy_error(self.y, self.t)return self.loss# 反向传播def backward(self, dout=1):batch_size = self.t.shape[0]if self.t.size == self.y.size:  # 监督数据是one-hot-vector的情况dx = (self.y - self.t) / batch_sizeelse:dx = self.y.copy()dx[np.arange(batch_size), self.t] -= 1dx = dx / batch_sizereturn dxclass Affine:def __init__(self, W, b):self.W = Wself.b = bself.x = Noneself.original_x_shape = None# 权重和偏置参数的导数self.dW = Noneself.db = None# 正向传播def forward(self, x):# 对应张量self.original_x_shape = x.shapex = x.reshape(x.shape[0], -1)self.x = xout = np.dot(self.x, self.w) + self.breturn out# 反向传播def backward(self, dout):dx = np.dot(dout, self.W.T)self.dW = np.dot(self.x.T, dout)self.db = np.sum(dout, axis=0)# 还原输入数据的形状dx = dx.reshape(*self.original_x_shape)return dxclass SGD:def __init__(self, lr):self.lr = lrdef update(self, params, grads):for key in params.keys():params[key] -= self.lr * grads[key]# 调整各层的激活值分布使其拥有适当的广度
class BatchNormlization:def __init__(self, gamma, beta, momentum, running_mean=None, running_var=None):self.gamma = gammaself.beta = betaself.momentum = momentumself.input_shape = None  # Conv层的情况下为4维，全连接层的情况下为2维# 测试时使用的平均值和方差self.running_mean = running_meanself.running_var = running_var# backward时使用的中间数据self.batch_size = Noneself.xc = Noneself.std = Noneself.dgamma = Noneself.dbeta = Nonedef __forward(self, x, train_flg):if self.running_mean is None:N, D = x.shapeself.running_mean = np.zeros(D)self.running_var = np.zeros(D)if train_flg:mu = x.mean(axis=0)xc = x - muvar = np.mean(xc ** 2, axis=0)std = np.sqrt(var + 1e-7)xn = xc / stdself.batch_size = x.shape[0]self.xc = xcself.xn = xnself.std = stdself.running_mean = self.momentum * self.running_mean + (1 - self.momentum) * muself.running_var = self.momentum * self.running_var + (1 - self.momentum) * varelse:xc = x - self.running_meanxn = xc / (np.sqrt(self.running_var + 1e-7))return self.gamma * xn + self.betadef forward(self, x, train_flg=True):self.input_shape = x.shapeif x.ndim != 2:N, C, H, W = x.shapex = x.reshape(N, -1)out = self.__forward(x, train_flg)return out.reshape(*self.input_shape)def __backward(self, dout):dbeta = dout.sum(axis=0)dgamma = np.sum(self.xn + dout, axis=0)dxn = self.gamma * doutdxc = dxn / self.stddstd = -np.sum((dxn * self.xc) / (self.std * self.std), axis=0)dvar = 0.5 * dstd / self.stddxc += (2.0 / self.batch_size) * self.xc * dvardmu = np.sum(dxc, axis=0)dx = dxc - dmu / self.batch_sizeself.dgamma = dgammaself.dbeta = dbetareturn dxdef backward(self, dout):if dout.ndim != 2:N, C, H, W = dout.shapedout = dout.reshape(N, -1)dx = self.__backward(dout)dx = dx.reshape(*self.input_shape)return dx# 在学习过程中随机删除神经元的方法:dropout_ratio神经元的删除比
class Dropout:def __int__(self, dropout_ratio=0.5):self.dropout_ratio = dropout_ratioself.mask = None# 正向传播def forward(self, x, train_flg=True):if train_flg:self.mask = np.random.rand(*x.shape) > self.dropout_ratioreturn x * self.maskelse:return x * (1 - self.dropout_ratio)# 反向传播def backward(self, dout):return dout * self.mask# 全连接的多层神经网络
class MultiLayerNet:def __init__(self, input_size, hidden_size_list, output_size, activation='relu', weight_init_std='relu',weight_decay_lambda=0, use_dropout=False, dropout_ratio=0.5, use_batchnorm=False):self.input_size = input_sizeself.hidden_size_list = hidden_size_listself.output_size = output_sizeself.hidden_layer_num = len(hidden_size_list)self.use_dropout = use_dropoutself.use_batchnorm = use_batchnormself.weight_decay_lambda = weight_decay_lambdaself.params = {}# 初始化权重self.__init_weight(weight_init_std)# 生成层activation_layer = {'sigmoid': Sigmoid, 'relu': Relu}self.layer = OrderedDict()for idx in range(1, self.hidden_layer_num + 1):self.layer['Affine' + str(idx)] = Affine(self.params['W' + str(idx)], self.params['b' + str(idx)])if self.use_batchnorm:self.params['gamma' + str(idx)] = np.ones(hidden_size_list[idx - 1])self.params['beta' + str(idx)] = np.zeros(hidden_size_list[idx - 1])self.layer['BatchNorm' + str(idx)] = BatchNormlization(self.params['gamma' + str(idx)],self.params['beta' + str(idx)])self.layer['Activation—_function' + str(idx)] = activation_layer[activation]if self.use_dropout:self.layer['Dropout' + str(idx)] = Dropout(dropout_ratio)idx = self.hidden_layer_num + 1self.layer['Affine' + str(idx)] = Affine(self.params['W' + str(idx)], self.params['b' + str(idx)])self.last_layer = SoftmaxWithLoss()def __init_weight(self, weight_init_std):"""设定权重的初始值Parameters----------weight_init_std : 指定权重的标准差（e.g. 0.01）指定'relu'或'he'的情况下设定“He的初始值”指定'sigmoid'或'xavier'的情况下设定“Xavier的初始值”"""all_size_list = [self.input_size] + self.hidden_size_list + [self.output_size]for idx in range(1, len(all_size_list)):scale = weight_init_stdif str(weight_init_std).lower() in ('relu', 'he'):scale = np.sqrt(2.0 / all_size_list[idx - 1])  # 使用ReLU的情况下推荐的初始值elif str(weight_init_std).lower() in ('sigmoid', 'xavier'):scale = np.sqrt(1.0 / all_size_list[idx - 1])  # 使用sigmoid的情况下推荐的初始值self.params['W' + str(idx)] = scale * np.random.randn(all_size_list[idx - 1], all_size_list[idx])self.params['b' + str(idx)] = np.zeros(all_size_list[idx])def predict(self, x, train_flg=False):for key, layer in self.layers.items():if "Dropout" in key or "BatchNorm" in key:x = layer.forward(x, train_flg)else:x = layer.forward(x)return xdef loss(self, x, t, train_flg=False):"""求损失函数参数x是输入数据，t是教师标签"""y = self.predict(x, train_flg)weight_decay = 0for idx in range(1, self.hidden_layer_num + 2):W = self.params['W' + str(idx)]weight_decay += 0.5 * self.weight_decay_lambda * np.sum(W ** 2)return self.last_layer.forward(y, t) + weight_decaydef accuracy(self, X, T):Y = self.predict(X, train_flg=False)Y = np.argmax(Y, axis=1)if T.ndim != 1: T = np.argmax(T, axis=1)accuracy = np.sum(Y == T) / float(X.shape[0])return accuracydef numerical_gradient(self, X, T):"""求梯度（数值微分）Parameters----------X : 输入数据T : 教师标签Returns-------具有各层的梯度的字典变量grads['W1']、grads['W2']、...是各层的权重grads['b1']、grads['b2']、...是各层的偏置"""loss_W = lambda W: self.loss(X, T, train_flg=True)grads = {}for idx in range(1, self.hidden_layer_num + 2):grads['W' + str(idx)] = numerical_gradient(loss_W, self.params['W' + str(idx)])grads['b' + str(idx)] = numerical_gradient(loss_W, self.params['b' + str(idx)])if self.use_batchnorm and idx != self.hidden_layer_num + 1:grads['gamma' + str(idx)] = numerical_gradient(loss_W, self.params['gamma' + str(idx)])grads['beta' + str(idx)] = numerical_gradient(loss_W, self.params['beta' + str(idx)])return gradsdef gradient(self, x, t):# forwardself.loss(x, t, train_flg=True)# backwarddout = 1dout = self.last_layer.backward(dout)layers = list(self.layers.values())layers.reverse()for layer in layers:dout = layer.backward(dout)# 设定grads = {}for idx in range(1, self.hidden_layer_num + 2):grads['W' + str(idx)] = self.layers['Affine' + str(idx)].dW + self.weight_decay_lambda * self.params['W' + str(idx)]grads['b' + str(idx)] = self.layers['Affine' + str(idx)].dbif self.use_batchnorm and idx != self.hidden_layer_num + 1:grads['gamma' + str(idx)] = self.layers['BatchNorm' + str(idx)].dgammagrads['beta' + str(idx)] = self.layers['BatchNorm' + str(idx)].dbetareturn grads# 加载数据
(x_train, t_train), (x_test, t_test) = get_data()# 抽取训练数据
x_train = x_train[:1000]
t_train = t_train[:1000]# 超参数
iter_num = 10000000
lr = 0.01
max_epochs = 201
train_size = x_train.shape[0]
batch_size = 100def __train(weight_init_std):bn_network = MultiLayerNet(input_size=784, hidden_size_list=[100, 100, 100, 100, 100], output_size=10,weight_init_std=weight_init_std, use_batchnorm=True)network = MultiLayerNet(input_size=784, hidden_size_list=[100, 100, 100, 100, 100], output_size=10,weight_init_std=weight_init_std)optimizer = SGD(lr)train_acc_list = []bn_train_acc_list = []iter_per_epoch = max(train_size / batch_size, 1)epoch_cnt = 0for i in range(iter_num):# 数据抽取batch_mask = np.random.choice(train_size, batch_size)x_batch = x_train[batch_mask]t_batch = t_train[batch_mask]for _network in (bn_network, network):grads = _network.gradient(x_batch, t_batch)optimizer.update(_network.params, grads)if i % iter_per_epoch == 0:train_acc = network.accuracy(x_train, t_train)train_acc_list.append(train_acc)bn_train_acc = _network.accuracy(x_train, t_train)bn_train_acc_list.append(bn_train_acc)print('train_acc,bn_train_acc|', str(train_acc) + str(bn_train_acc))epoch_cnt += 1if epoch_cnt >= max_epochs:breakreturn train_acc_list, bn_train_acc_list# 绘制图形
weight_scale_list = np.logspace(0, -4, num=16)
x = np.arange(max_epochs)for i, w in enumerate(weight_scale_list):print("============== " + str(i + 1) + "/16" + " ==============")train_acc_list, bn_train_acc_list = __train(w)plt.subplot(4, 4, i + 1)plt.title("W:" + str(w))if i == 15:plt.plot(x, bn_train_acc_list, label='Batch Normalization', markevery=2)plt.plot(x, train_acc_list, linestyle="--", label='Normal(without BatchNorm)', markevery=2)else:plt.plot(x, bn_train_acc_list, markevery=2)plt.plot(x, train_acc_list, linestyle="--", markevery=2)plt.ylim(0, 1.0)if i % 4:plt.yticks([])else:plt.ylabel("accuracy")if i < 12:plt.xticks([])else:plt.xlabel("epochs")plt.legend(loc='lower right')plt.show()