『科学计算』通过代码理解SoftMax多分类

#计算Error，Cost，Grad
y_dash = self.softmax(X.dot(theta_n))   # 向前传播结果
        
Y = np.zeros((m,10))                    # one-hot编码label矩阵
for i in range(m):
    Y[i,y[i]]=1

error = np.sum(Y * np.log(y_dash), axis=1)   # 注意，这里是点乘
cost = -np.sum(error, axis=0)
grad = X.T.dot(y_dash-Y)
        
grad_n = grad.ravel()  

import numpy as np
import matplotlib.pyplot as plt
import math

def scale_n(x):
    return x
    #return (x-x.mean(axis=0))/(x.std(axis=0)+1e-10)


class SoftMaxModel(object):
    def __init__(self,alpha=0.06,threhold=0.0005):
        self.alpha = alpha         # 学习率
        self.threhold = threhold   # 循环终止阈值
        self.num_classes = 10      # 分类数
    
    def setup(self,X):
        # 初始化权重矩阵，注意，由于是多分类，所以权重由向量变化为矩阵
        # 而且这里面初始化的是flat为1维的矩阵
        m, n = X.shape  # 400,15   
        s = math.sqrt(6) / math.sqrt(n+self.num_classes)
        theta = np.random.rand(n*(self.num_classes))*2*s-s  #[15,1]
        return theta

    def softmax(self,x):
        # 先前传播softmax多分类
        # 注意输入的x是[batch数目n，类数目m]，输出是[batch数目n，类数目m]
        e = np.exp(x)
        temp = np.sum(e, axis=1,keepdims=True)
        return e/temp
        
    def get_cost_grad(self,theta,X,y):
        m, n = X.shape
        theta_n = theta.reshape(n, self.num_classes)

        #计算Error，Cost，Grad
        y_dash = self.softmax(X.dot(theta_n))   # 向前传播结果
        
        Y = np.zeros((m,10))                    # one-hot编码label矩阵
        for i in range(m):
            Y[i,y[i]]=1

        error = np.sum(Y * np.log(y_dash), axis=1)
        cost = -np.sum(error, axis=0)
        grad = X.T.dot(y_dash-Y)
        
        grad_n = grad.ravel()           
        return cost,grad_n
        
        
 
    def train(self,X,y,max_iter=50, batch_size=200):
        m, n = X.shape  # 400,15 
        theta = self.setup(X)

        #our intial prediction
        prev_cost = None
        loop_num = 0
        n_samples = y.shape[0]
        n_batches = n_samples // batch_size
        # Stochastic gradient descent with mini-batches
        while loop_num < max_iter:
            for b in range(n_batches):
                batch_begin = b*batch_size
                batch_end = batch_begin+batch_size
                X_batch = X[batch_begin:batch_end]
                Y_batch = y[batch_begin:batch_end]
                      

                #intial cost
                cost,grad = self.get_cost_grad(theta,X_batch,Y_batch)
                
                theta = theta- self.alpha * grad/float(batch_size)
                
            loop_num+=1
            if loop_num%10==0:
                print (cost,loop_num)
            if prev_cost:
                if prev_cost - cost <= self.threhold:
                    break

            prev_cost = cost
                       
            
        self.theta = theta
        print (theta,loop_num)

    def train_scipy(self,X,y):
        m,n = X.shape
        import scipy.optimize
        options = {‘maxiter‘: 50, ‘disp‘: True}
        J = lambda x: self.get_cost_grad(x, X, y)
        theta = self.setup(X)

        result = scipy.optimize.minimize(J, theta, method=‘L-BFGS-B‘, jac=True, options=options)
        self.theta = result.x

    def predict(self,X):
        m,n = X.shape
        theta_n = self.theta.reshape(n, self.num_classes)
        a = np.argmax(self.softmax(X.dot(theta_n)),axis=1)
        return a


    def grad_check(self,X,y):
        epsilon = 10**-4
        m, n = X.shape   
        
        sum_error=0
        N=300
        
        for i in range(N):
            theta = self.setup(X)
            j = np.random.randint(1,len(theta))
            theta1=theta.copy()
            theta2=theta.copy()
            theta1[j]+=epsilon
            theta2[j]-=epsilon

            cost1,grad1 = self.get_cost_grad(theta1,X,y)
            cost2,grad2 = self.get_cost_grad(theta2,X,y)
            cost3,grad3 = self.get_cost_grad(theta,X,y)

            sum_error += np.abs(grad3[j]-(cost1-cost2)/float(2*epsilon))
        print ("grad check error is %e\n"%(sum_error/float(N)))
        

if __name__=="__main__":

    
    import cPickle, gzip
    # Load the dataset
    f = gzip.open(‘mnist.pkl.gz‘, ‘rb‘)
    train_set, valid_set, test_set = cPickle.load(f)
    f.close()    
    train_X = scale_n(train_set[0])
    train_y = train_set[1]
    test_X = scale_n(test_set[0])
    test_y = test_set[1]



    l_model = SoftMaxModel()

    l_model.grad_check(test_X[0:200,:],test_y[0:200])

    l_model.train_scipy(train_X,train_y)


    predict_train_y = l_model.predict(train_X)
    b = predict_train_y!=train_y

    error_train = np.sum(b, axis=0)/float(b.size)    

    predict_test_y = l_model.predict(test_X)
    b = predict_test_y!=test_y

    error_test = np.sum(b, axis=0)/float(b.size)

    print ("Train Error rate = %.4f, \nTest Error rate = %.4f\n"%(error_train,error_test))

def train_scipy(self,X,y):
        m,n = X.shape
        import scipy.optimize
        options = {‘maxiter‘: 50, ‘disp‘: True}
        J = lambda x: self.get_cost_grad(x, X, y)
        theta = self.setup(X)

        result = scipy.optimize.minimize(J, theta, method=‘L-BFGS-B‘, jac=True, options=options)
        self.theta = result.x