2023.1.25

现在已经学习过了，如果我们设置了合适的权重初始值，则各层的激活值分布会呈现适当的广度，从而可以时神经网络模型顺利的进行学习。

而 batch normalization算法 的思想就是为了使得各层有适当的广度，“强制性”地调整激活值的分布。

 Batch Normalization算法：

这是一个提出自2015年的方法，但是已经广泛的应用于机器学习......

他的作用是调整各层激活函数进行正规化的层，即Bacth Normalization 层，所以将他插入在Affine层与ReLU层之间；

具体，在神经网络进行学习时，以mini-batch为单位，按mini-batch进行正规化（正规化：就是时数据分布均值为0、反差为1的正规化）

正规化数学公式：

；；；其中  是个很小的值，防止除以0的情况 。

之后Batch Normalization层会对正规划的数据进行缩放和平移： 

 =1和  = 0 是参数，然后经过学习调整到合适的值；

优点：

一、可以使学习快速进行；

二、不那么以来初始值；

三、可以抑制过拟合；

观察Batch Normalization的计算图：

 其反向操作比较复杂，并没有推导（教材上也没有推导）

运用MNIST数据集对Bacth Normalization的评估：

观察使用Batch Normalization层和不适用Batch Normalizaton层，会出现什么现象；

 进行了16次的对比，可以说使用了Bacth Normalization后，学习变得更快了。在不同的权重初始值的标准差为各种不同的值的时学习过程也做了实验；

事实是，如果没有一个好的初始值，神经网络的学习将难以进行；

通过使用Batch Norlization层 推动神经网络学习的进行。并且，对权重初始值变得 健壮（使得神经网络模型不那么依赖初始值）

实验代码：

import sys, os
from collections import OrderedDict
import numpy as np
import matplotlib.pyplot as plt
from dataset.mnist import load_mnistsys.path.append(os.pardir)def softmax(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)return np.exp(x) / np.sum(np.exp(x))def cross_entropy_error(y, t):if y.ndim == 1:t = t.reshape(1, t.size)y = y.reshape(1, y.size)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_sizedef sigmoid(x):return 1 / (1 + np.exp(-x))class Relu:def __init__(self):self.mask = Nonedef forward(self, x):self.mask = (x <= 0)out = x.copy()out[self.mask] = 0return outdef backward(self, dout):dout[self.mask] = 0dx = doutreturn dxclass Sigmoid:def __init__(self):self.out = Nonedef forward(self, x):out = sigmoid(x)self.out = outreturn outdef backward(self, dout):dx = dout * (1.0 - self.out) * self.outreturn 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 = Nonedef 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 outdef 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)dx = dx.reshape(*self.original_x_shape)  # 还原输入数据的形状（对应张量）return dxclass SoftmaxWithLoss:def __init__(self):self.loss = Noneself.y = None  # softmax的输出self.t = None  # 监督数据def forward(self, x, t):self.t = tself.y = softmax(x)self.loss = cross_entropy_error(self.y, self.t)return self.lossdef 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 Dropout:def __init__(self, dropout_ratio=0.5):self.dropout_ratio = dropout_ratioself.mask = Nonedef forward(self, x, train_flg=True):if train_flg:self.mask = np.random.rand(*x.shape) > self.dropout_ratioreturn x * self.maskelse:return x * (1.0 - self.dropout_ratio)def backward(self, dout):return dout * self.maskclass BatchNormalization:def __init__(self, gamma, beta, momentum=0.9, 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=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 __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 + 10e-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 + 10e-7)))out = self.gamma * xn + self.betareturn outdef 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 dxdef __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 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 MultiLayerNetExtend:def __init__(self, input_size, hidden_size_list, output_size,activation='relu', weight_init_std='relu', weight_decay_lambda=0,use_dropout=False, dropout_ration=0.5, use_batchnorm=False):self.input_size = input_sizeself.output_size = output_sizeself.hidden_size_list = hidden_size_listself.hidden_layer_num = len(hidden_size_list)self.use_dropout = use_dropoutself.weight_decay_lambda = weight_decay_lambdaself.use_batchnorm = use_batchnormself.params = {}# 初始化权重self.__init_weight(weight_init_std)# 生成层activation_layer = {'sigmoid': Sigmoid, 'relu': Relu}self.layers = OrderedDict()for idx in range(1, self.hidden_layer_num + 1):self.layers['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.layers['BatchNorm' + str(idx)] = BatchNormalization(self.params['gamma' + str(idx)],self.params['beta' + str(idx)])self.layers['Activation_function' + str(idx)] = activation_layer[activation]()if self.use_dropout:self.layers['Dropout' + str(idx)] = Dropout(dropout_ration)idx = self.hidden_layer_num + 1self.layers['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):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):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):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 gradsclass Adam:def __init__(self, lr=0.001, beta1=0.9, beta2=0.999):self.lr = lrself.beta1 = beta1self.beta2 = beta2self.iter = 0self.m = Noneself.v = Nonedef update(self, params, grads):if self.m is None:self.m, self.v = {}, {}for key, val in params.items():self.m[key] = np.zeros_like(val)self.v[key] = np.zeros_like(val)self.iter += 1lr_t = self.lr * np.sqrt(1.0 - self.beta2 ** self.iter) / (1.0 - self.beta1 ** self.iter)for key in params.keys():# self.m[key] = self.beta1*self.m[key] + (1-self.beta1)*grads[key]# self.v[key] = self.beta2*self.v[key] + (1-self.beta2)*(grads[key]**2)self.m[key] += (1 - self.beta1) * (grads[key] - self.m[key])self.v[key] += (1 - self.beta2) * (grads[key] ** 2 - self.v[key])params[key] -= lr_t * self.m[key] / (np.sqrt(self.v[key]) + 1e-7)class SGD:def __init__(self, lr=0.01):self.lr = lrdef update(self, params, grads):for key in params.keys():params[key] -= self.lr * grads[key](x_train, t_train), (x_test, t_test) = load_mnist(normalize=True)# 减少学习数据
x_train = x_train[:1000]
t_train = t_train[:1000]max_epochs = 20
train_size = x_train.shape[0]
batch_size = 100
learning_rate = 0.01def __train(weight_init_std):bn_network = MultiLayerNetExtend(input_size=784, hidden_size_list=[100, 100, 100, 100, 100], output_size=10,weight_init_std=weight_init_std, use_batchnorm=True)network = MultiLayerNetExtend(input_size=784, hidden_size_list=[100, 100, 100, 100, 100], output_size=10,weight_init_std=weight_init_std)optimizer = SGD(lr=learning_rate)train_acc_list = []bn_train_acc_list = []iter_per_epoch = max(train_size / batch_size, 1)epoch_cnt = 0for i in range(1000000000):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)bn_train_acc = bn_network.accuracy(x_train, t_train)train_acc_list.append(train_acc)bn_train_acc_list.append(bn_train_acc)print("epoch:" + str(epoch_cnt) + " | " + str(train_acc) + " - " + str(bn_train_acc))epoch_cnt += 1if epoch_cnt >= max_epochs:breakreturn train_acc_list, bn_train_acc_list# 3.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()

MNIST数据集的导入代码：

代码需要在一个命名为命名为dataset的文件夹下命名为mnist，并且与上个代码在同一个文件夹；

# coding: utf-8
try:import urllib.request
except ImportError:raise ImportError('You should use Python 3.x')
import os.path
import gzip
import pickle
import os
import numpy as npurl_base = 'http://yann.lecun.com/exdb/mnist/'
key_file = {'train_img':'train-images-idx3-ubyte.gz','train_label':'train-labels-idx1-ubyte.gz','test_img':'t10k-images-idx3-ubyte.gz','test_label':'t10k-labels-idx1-ubyte.gz'
}dataset_dir = os.path.dirname(os.path.abspath(__file__))
save_file = dataset_dir + "/mnist.pkl"train_num = 60000
test_num = 10000
img_dim = (1, 28, 28)
img_size = 784def _download(file_name):file_path = dataset_dir + "/" + file_nameif os.path.exists(file_path):returnprint("Downloading " + file_name + " ... ")urllib.request.urlretrieve(url_base + file_name, file_path)print("Done")def download_mnist():for v in key_file.values():_download(v)def _load_label(file_name):file_path = dataset_dir + "/" + file_nameprint("Converting " + file_name + " to NumPy Array ...")with gzip.open(file_path, 'rb') as f:labels = np.frombuffer(f.read(), np.uint8, offset=8)print("Done")return labelsdef _load_img(file_name):file_path = dataset_dir + "/" + file_nameprint("Converting " + file_name + " to NumPy Array ...")    with gzip.open(file_path, 'rb') as f:data = np.frombuffer(f.read(), np.uint8, offset=16)data = data.reshape(-1, img_size)print("Done")return datadef _convert_numpy():dataset = {}dataset['train_img'] =  _load_img(key_file['train_img'])dataset['train_label'] = _load_label(key_file['train_label'])    dataset['test_img'] = _load_img(key_file['test_img'])dataset['test_label'] = _load_label(key_file['test_label'])return datasetdef init_mnist():download_mnist()dataset = _convert_numpy()print("Creating pickle file ...")with open(save_file, 'wb') as f:pickle.dump(dataset, f, -1)print("Done!")def _change_one_hot_label(X):T = np.zeros((X.size, 10))for idx, row in enumerate(T):row[X[idx]] = 1return Tdef load_mnist(normalize=True, flatten=True, one_hot_label=False):"""读入MNIST数据集Parameters----------normalize : 将图像的像素值正规化为0.0~1.0one_hot_label : one_hot_label为True的情况下，标签作为one-hot数组返回one-hot数组是指[0,0,1,0,0,0,0,0,0,0]这样的数组flatten : 是否将图像展开为一维数组Returns-------(训练图像, 训练标签), (测试图像, 测试标签)"""if not os.path.exists(save_file):init_mnist()with open(save_file, 'rb') as f:dataset = pickle.load(f)if normalize:for key in ('train_img', 'test_img'):dataset[key] = dataset[key].astype(np.float32)dataset[key] /= 255.0if one_hot_label:dataset['train_label'] = _change_one_hot_label(dataset['train_label'])dataset['test_label'] = _change_one_hot_label(dataset['test_label'])if not flatten:for key in ('train_img', 'test_img'):dataset[key] = dataset[key].reshape(-1, 1, 28, 28)return (dataset['train_img'], dataset['train_label']), (dataset['test_img'], dataset['test_label']) if __name__ == '__main__':init_mnist()