60行代码徒手实现深度神经网络

01

准备数据集

采用的数据集是sklearn中的breast cancer数据集，30维特征，569个样本。训练前进行MinMax标准化缩放至[0,1]区间。按照75/25比例划分成训练集和验证集。

# 获取数据集 
import numpy as np 
import pandas as pd 
from sklearn import datasets 
from sklearn import preprocessing 
from sklearn.model_selection import train_test_split 
breast = datasets.load_breast_cancer() 
scaler = preprocessing.MinMaxScaler() 
data = scaler.fit_transform(breast['data']) 
target = breast['target'] 
X_train,X_test,y_train,y_test = train_test_split(data,target)

02

模型结构图

03

正反传播公式

04

NN实现代码

import numpy as np 
import pandas as pd 
#定义激活函数 
ReLu = lambda z:np.maximum(0.0,z) 
d_ReLu = lambda z:np.where(z<0,0,1) 
LeakyReLu = lambda z:np.maximum(0.01*z,z) 
d_LeakyReLu = lambda z:np.where(z<0,0.01,1) 
Sigmoid = lambda z:1/(1+np.exp(-z)) 
d_Sigmoid = lambda z: Sigmoid(z)*(1-Sigmoid(z)) #d_Sigmoid = a(1-a) 
Tanh = np.tanh 
d_Tanh = lambda z:1 - Tanh(z)**2 #d_Tanh = 1 - a**2 
class NNClassifier(object): 
 def __init__(self,n = [np.nan,5,5,1],alpha = 0.1,ITERNUM = 50000, gfunc = 'ReLu'): 
 self.n = n #各层节点数 
 self.gfunc = gfunc #隐藏层激活函数 
 self.alpha,self.ITERNUM = alpha,ITERNUM 
 self.dfJ = pd.DataFrame(data = np.zeros((ITERNUM,1)),columns = ['J']) 
 self.W,self.b = np.nan,np.nan 
 # 确定各层激活函数 
 self.g = [eval(self.gfunc) for i in range(len(n))]; 
 self.g[-1] = Sigmoid;self.g[0] = np.nan 
 # 确定隐藏层激活函数的导数 
 self.d_gfunc = eval('d_' + self.gfunc) 
 def fit(self,X_train,y_train): 
 X,Y = X_train.T,y_train.reshape(1,-1) 
 m = X.shape[1] #样本个数 
 n = self.n; n[0] = X.shape[0] # 各层节点数量 
 # 节点值和参数初始化 
 A = [np.zeros((ni,m)) for ni in n];A[0] = X #各层节点输出值初始化 
 Z = [np.zeros((ni,m)) for ni in n];Z[0] = np.nan #各层节点中间值初始化 
 W = [np.nan] + [np.random.randn(n[i],n[i-1]) * 0.01 for i in range(1,len(n))] #各层系数参数 
 b = [np.zeros((ni,1)) for ni in n];b[0] = np.nan #n各层偏置参数 
 # 导数初始化 
 dA = [np.zeros(Ai.shape) for Ai in A] 
 dZ = [np.zeros(Ai.shape) for Ai in A] 
 dW = [np.zeros(Wi.shape) if isinstance(Wi,np.ndarray) else np.nan for Wi in W]  
 db = [np.zeros(bi.shape) if isinstance(bi,np.ndarray) else np.nan for bi in b]  
 for k in range(self.ITERNUM): 
 # ---------正向传播 ---------- 
 for i in range(1,len(n)): 
 Z[i] = np.dot(W[i],A[i-1]) + b[i] 
 A[i] = self.g[i](Z[i]) 
 J = (1/m) * np.sum(- Y*np.log(A[len(n)-1]) -(1-Y)*np.log(1-A[len(n)-1])) 
 self.dfJ.loc[k]['J']= J 
 # ----------反向传播 --------- 
 hmax = len(n) - 1 
 dA[hmax] = 1/m*(-Y/A[hmax] + (1-Y)/(1-A[hmax])) 
 dZ[hmax] = 1/m*(A[hmax]-Y) 
 dW[hmax] = np.dot(dZ[hmax],A[hmax-1].T) 
 db[hmax] = np.dot(dZ[hmax],np.ones((m,1))) 
 for i in range(len(n)-2,0,-1): 
 dA[i] = np.dot(W[i+1].T,dZ[i+1]) 
 dZ[i] = dA[i]* self.d_gfunc(Z[i]) 
 dW[i] = np.dot(dZ[i],A[i-1].T) 
 db[i] = np.dot(dZ[i],np.ones((m,1))) 
 #-----------梯度下降 --------- 
 for i in range(1,len(n)): 
 W[i] = W[i] - self.alpha*dW[i]  
 b[i] = b[i] - self.alpha*db[i] 
 # 显示进度 
 if (k+1)%1000 == 0: 
 print('progress rate：{}/{}'.format(k+1,self.ITERNUM),end = '\r') 
 self.W,self.b = W,b 
 def predict_prob(self,X_test): 
 # ---------正向传播 ---------- 
 W,b = self.W,self.b 
 Ai = X_test.T 
 for i in range(1,len(self.n)): 
 Zi = np.dot(W[i],Ai) + b[i] 
 Ai = self.g[i](Zi) 
 return(Ai.reshape(-1)) 
 def predict(self,X_test): 
 Y_prob = self.predict_prob(X_test) 
 Y_test = Y_prob.copy() 
 Y_test[Y_prob>=0.5] = 1 
 Y_test[Y_prob< 0.5] = 0 
 return(Y_test)

05

单隐层神经网络

设置1个隐藏层，隐藏层节点数为5，隐藏层使用Sigmoid激活函数。

# 采用Sigmoid激活函数 
NN = NNClassifier(n = [np.nan,5,1],alpha = 0.02, 
 ITERNUM = 200000, gfunc = 'Sigmoid') 
NN.fit(X_train,y_train) 
# 绘制目标函数迭代曲线 
%matplotlib inline 
NN.dfJ.plot(figsize = (12,8)) 
# 测试在验证集的auc得分 
from sklearn.metrics import roc_auc_score 
Y_prob = NN.predict_prob(X_test) 
roc_auc_score(list(y_test),list(Y_prob))

隐藏层使用Tanh激活函数。

# 采用 Tanh激活函数 
NN = NNClassifier(n = [np.nan,5,1],alpha = 0.02, 
 ITERNUM = 200000, gfunc = 'Tanh') 
NN.fit(X_train,y_train) 
# 绘制目标函数迭代曲线 
%matplotlib inline 
NN.dfJ.plot(figsize = (12,8)) 
# 测试在验证集的auc得分 
from sklearn.metrics import roc_auc_score 
Y_prob = NN.predict_prob(X_test) 
roc_auc_score(list(y_test),list(Y_prob))

隐藏层使用ReLu激活函数。

# 采用 ReLu激活函数 
NN = NNClassifier(n = [np.nan,5,1],alpha = 0.02, 
 ITERNUM = 200000, gfunc = 'ReLu') 
NN.fit(X_train,y_train) 
# 绘制目标函数迭代曲线 
%matplotlib inline 
NN.dfJ.plot(figsize = (12,8)) 
# 测试在验证集的auc得分 
from sklearn.metrics import roc_auc_score 
Y_prob = NN.predict_prob(X_test) 
roc_auc_score(list(y_test),list(Y_prob))

隐藏层使用LeakyReLu激活函数。

# 采用 LeakyReLu激活函数 
NN = NNClassifier(n = [np.nan,5,1],alpha = 0.02, 
 ITERNUM = 200000, gfunc = 'LeakyReLu') 
NN.fit(X_train,y_train) 
# 绘制目标函数迭代曲线 
%matplotlib inline 
NN.dfJ.plot(figsize = (12,8)) 
# 测试在验证集的auc得分 
from sklearn.metrics import roc_auc_score 
Y_prob = NN.predict_prob(X_test) 
roc_auc_score(list(y_test),list(Y_prob))

以上试验似乎表明，在当前的数据集上，隐藏层采用ReLu激活函数是一个最好的选择，AUC最高得分为0.99958。

06

双隐层神经网络

设置2个隐藏层，隐藏层节点数都为5，隐藏层都使用ReLu激活函数。

# 设置两个隐藏层，采用ReLu激活函数 
NN = NNClassifier(n = [np.nan,5,5,1],alpha = 0.02, 
 ITERNUM = 200000, gfunc = 'ReLu') 
NN.fit(X_train,y_train) 
# 绘制目标函数迭代曲线 
%matplotlib inline 
NN.dfJ.plot(figsize = (12,8)) 
# 测试在验证集的auc得分 
from sklearn.metrics import roc_auc_score 
Y_prob = NN.predict_prob(X_test) 
roc_auc_score(list(y_test),list(Y_prob))

AUC得分0.99874比采用单隐藏层的最优得分0.99958有所降低，可能是模型复杂度过高，我们尝试减少隐藏层节点的个数至3以降低模型复杂度。

# 双隐藏层，隐藏层节点数为3 
NN = NNClassifier(n = [np.nan,3,3,1],alpha = 0.02, 
 ITERNUM = 200000, gfunc = 'ReLu') 
NN.fit(X_train,y_train) 
# 绘制目标函数迭代曲线 
%matplotlib inline 
NN.dfJ.plot(figsize = (12,8)) 
# 测试在验证集的auc得分 
from sklearn.metrics import roc_auc_score 
Y_prob = NN.predict_prob(X_test) 
roc_auc_score(list(y_test),list(Y_prob))

AUC得分0.99979，又有所提高。

和sklearn中自带的神经网络分类器进行对比。

# 和sklearn中的模型对比 
from sklearn.neural_network import MLPClassifier 
# 第一隐藏层神经元个数为3，第二隐藏层神经元个数为3 
MLPClf = MLPClassifier(hidden_layer_sizes=(3,3),max_iter=200000,activation='relu') 
MLPClf.fit(X_train,y_train) 
# 绘制目标函数迭代曲线 
dfJ = pd.DataFrame(data = np.array(MLPClf.loss_curve_),columns = ['J']) 
dfJ.plot(figsize = (12,8)) 
# 测试在验证集的auc得分 
from sklearn.metrics import roc_auc_score 
Y_prob = MLPClf.predict_proba(X_test)[:,1] 
roc_auc_score(list(y_test),list(Y_prob))

以上试验表明，针对当前数据数据集，选择ReLu激活函数，采用双隐藏层，每个隐藏层节点数设置为3是一个不错的选择，AUC得分为0.99979。该得分高于采用CV交叉验证优化超参数后的逻辑回归模型的0.99897的AUC得分。