寻找最优参数解：最速下降法，牛顿下降法，阻尼牛顿法，拟牛顿法

感谢于建民的投稿，转载请注明出处：数盟社区

机器学习的一个重要组成部分是如何寻找最优参数解。本文就常见寻优方法进行总结，并给出简单python2.7实现，可能文章有点长，大家耐心些。

寻找最优参数解，就是在一块参数区域上，去找到满足约束条件的那组参数。形象描述，比如代价函数是个碗状的，那我们就是去找最底部（代价最小）的那个地方的对应的参数值作为最优解。那么，如何找到那个底部的最优参数解呢，如何由一个初始值，一步一步地接近该最优解呢。寻优方法，提供了靠近最优解的方法，其中涉及到的核心点，无外乎两点：靠近最优解的方向和步幅（每步的长度）。

最优化，分为线性最优化理论和非线性最优化理论。其中线性最优化又称线性规划。目标函数和约束条件的表达是线性的，Y=αX；非线性最优化理论，是非线性的。其中包括梯度法，牛顿法，拟牛顿法（DFP/BFGS），约束变尺度（SQP），Lagrange乘子法，信赖域法等。

算法原理及简单推导

最速下降法（梯度下降法）

借助梯度，找到下降最快的方向，大小为最大变化率。

梯度：是方向导数中，变化最大的那个方向导数。
梯度方向：标量场中增长最快的方向。
梯度大小：最大变化率。
更新：沿着梯度的负向，更新参数（靠近最优解）。
*********************************************

*********************************************
梯度下降法
优点：方便直观，便于理解。
缺点：下降速度慢，有时参数会震荡在最优解附近无法终止。

牛顿下降法

牛顿下降法，是通过泰勒展开到二阶，推到出参数更新公式的。

调整了参数更新的方向和大小（牛顿方向）。
*********************************************

*********************************************
牛顿下降法
优点：对于正定二次函数，迭代一次，就可以得到极小值点。下降的目的性更强。
缺点：要求二阶可微分；收敛性对初始点的选取依赖性很大；每次迭代都要计算Hessian矩阵，计算量大；计算Dk时，方程组有时奇异或者病态，无法求解Dk或者Dk不是下降方向。

阻尼牛顿法

这是对牛顿法的改进，在求新的迭代点时，以Dk作为搜索方向，进行一维搜索，求步长控制量α，
*********************************************

*********************************************
阻尼牛顿法
优点：修改了下降方向，使得始终朝着下降的方向迭代。
缺点：与牛顿法一样。

一维搜索方法简介

一维无约束优化问题minF(α)，求解F(α)的极小值和极大值的数值迭代方法，即为一维搜索方法。常用的方法包括：试探法（黄金分割法，fibonacci方法，平分法，格点法）；插值法（牛顿法，抛物线法）。
（1）确定最优解所在区间[a,b] （进退法）
思想：从初始点α0开始，以步长h前进或者后退，试出三个点f(α0+h),f(α0),f(α0−h)，满足大，小，大规律。
*********************************************

*********************************************
（2）在[a, b]内，找到极小值（黄金分割法和平分法）
*********************************************

*********************************************
思考：如何在实际应用中，选择[a, b]，函数f是什么样子的？这些问题需要讨论。整个优化的目标是：找到最优θ，使得代价CostJ最小。故此，f=CostJ。

拟牛顿法 – DFP法

由于牛顿法计算二阶导数，计算量大，故此用其他方法（一阶导数）估计Hessian矩阵的逆。f(x)在Xk+1处，展开成二阶泰勒级数。

根据H的构造函数不同，分为不同的拟牛顿方法，下面为DFP方法：

*********************************************

*********************************************
拟牛顿法DFP：
优点：减少了二阶计算，运算量大大降低。

拟牛顿法 – BFGS法

若构造函数如下，则为BFGS法。

*********************************************

*********************************************
拟牛顿法是无约束最优化方法中最有效的一类算法。

算法的Python实现代码

Python2.7需要安装pandas, numpy, scipy, matplotlib。
下面给出Windows7下exe方式按照上面模块的简单方法。
numpy–http://sourceforge.net/projects/numpy/files/ –这里面也可以找到较新的scipy –
scipy–http://download.csdn.net/detail/caanyee/8241305
pandas-https://pypi.python.org/packages/2.7/p/pandas/pandas-0.12.0.win32-py2.7.exe#md5=80b0b9b891842ef4bdf451ac07b368e5
test.py

# coding = utf-8'''time: 2015.06.03author: yujianminobjection: BGD / SGD / mini-batch GD / QNGD / DFP / BFGS 实现了批量梯度下降、单个梯度下降； 最速下降法、牛顿下降法、阻尼牛顿法、拟牛顿DFP和BFGS'''import pandas as pdimport numpy as npimport scipy as spimport matplotlib.pyplot as pltdata = pd.read_csv("C:\\Users\\yujianmin\\Desktop\\python\\arraydataR.csv")print(data.ix[1:5, :])dataArray = np.array(data)'''x = dataArray[:, 0]y = dataArray[:, 1]plt.plot(x, y, 'o')plt.title('data is like this')plt.xlabel('x feature')plt.ylabel('y label')plt.show()'''def Myfunction_BGD(data, alpha, numIter, eplise):    ''' Batch Gradient Descent    :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    #eplise = 0.4    while i < numIter:        H = np.dot(x,theta)        J = (np.sum((y-H)**2))/(2*nRow)        print('Itering %d ;cost is:%f' %(i+1,J))        costJ.append(J)        Gradient = (np.dot(np.transpose(y-H),x))/nRow        Gradient = Gradient.reshape(nCol+1, 1)        if np.sum(np.fabs(Gradient))<= eplise:            return theta, costJ        else:            ## update            theta = theta + alpha * Gradient        i = i + 1    return theta, costJdef Myfunction_SGD(data, alpha, numIter, eplise):    ''' Stochastic Gradient Descent    :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    Loop = 0    costJ = []    while Loop <numIter:        H = np.dot(x,theta)        J = np.sum((y-H)**2)/(2*nRow)        print('Itering %d ;cost is:%f' %(Loop+1,J))        costJ.append(J)        i = 0        while i <nRow:            Gradient = (y[i] - np.dot(x[i], theta)) * x[i]            Gradient = Gradient.reshape(nCol+1, 1)            theta = theta + alpha * Gradient            i = i + 1        #eplise = 0.4        Gradient = (np.dot(np.transpose(y-H),x))/nRow        if np.sum(np.fabs(Gradient))<= eplise:            return theta, costJ        Loop = Loop + 1    return theta, costJdef Myfunction_NGD1(data, alpha, numIter, eplise):    ''' Newton Gradient Descent -- theta := theta - alpha*[f'']^(-1)*f'    :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    :reference:http://www.doc88.com/p-145660070193.html    :hessian = transpos(x) * x     '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    while i < numIter:        H = np.dot(x,theta)        J = (np.sum((y-H)**2))/(2*nRow)        ## update        print('Itering %d ;cost is:%f' %(i+1,J))        costJ.append(J)        Gradient = (np.dot(np.transpose(y-H),x))/nRow        Gradient = Gradient.reshape(nCol+1, 1)        #eplise = 0.4        if np.sum(np.fabs(Gradient))<=eplise:            return theta, costJ        Hessian = np.dot(np.transpose(x), x)/nRow        theta = theta + alpha * np.dot(np.linalg.inv(Hessian), Gradient)        #theta = theta + np.dot(np.linalg.inv(Hessian), Gradient)        i = i + 1    return theta, costJdef Myfunction_NGD2(data, alpha, numIter, eplise):    ''' Newton Gradient Descent -- theta := theta - [f'']^(-1)*f'    :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    :reference:http://www.doc88.com/p-145660070193.html    :hessian = transpos(x) * x     '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    while i < numIter:        H = np.dot(x,theta)        J = (np.sum((y-H)**2))/(2*nRow)        ## update        print('Itering %d ;cost is:%f' %(i+1,J))        costJ.append(J)        Gradient = (np.dot(np.transpose(y-H),x))/nRow        Gradient = Gradient.reshape(nCol+1, 1)        #eplise = 0.4        if np.sum(np.fabs(Gradient)) <= eplise:            return theta, costJ        Hessian = np.dot(np.transpose(x), x)/nRow        theta = theta + np.dot(np.linalg.inv(Hessian), Gradient)        i = i + 1    return theta, costJdef Myfunction_QNGD(data, alpha, numIter, eplise):    ''' Newton Gradient Descent -- theta := theta - alpha* [f'']^(-1)*f'--            alpha is search by ForwardAndBack method and huang jin fen ge     :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    :reference:http://www.doc88.com/p-145660070193.html    :hessian = transpos(x) * x     '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    #eplise = 0.4    while i < numIter:        H = np.dot(x,theta)        J = (np.sum((y-H)**2))/(2*nRow)        ## update        print('Itering %d ;cost is:%f' %(i+1,J))        costJ.append(J)        Gradient = (np.dot(np.transpose(y-H),x))/nRow        Gradient = Gradient.reshape(nCol+1, 1)        if np.sum(np.fabs(Gradient))<= eplise:            return theta, costJ        else:            Hessian = np.dot(np.transpose(x), x)/nRow            Dk = - np.dot(np.linalg.inv(Hessian), Gradient)            ## find optimal [a,b] which contain optimal alpha            ## optimal alpha lead to min{f(theta + alpha*DK)}            alpha0 = 0            h = np.random.random(1)            alpha1 = alpha0            alpha2 = alpha0 + h            theta1 = theta + alpha1 * Dk            theta2 = theta + alpha2 * Dk            f1 = (np.sum((y-np.dot(x, theta1))**2))/(2*nRow)            f2 = (np.sum((y-np.dot(x, theta2))**2))/(2*nRow)            Loop = 1            a = 0            b = 0            while Loop >0:                print(' find [a,b] loop is %d' %Loop)                Loop = Loop + 1                if f1 > f2:                    h = 2*h                else:                    h = -h                    (alpha1, alpha2) = (alpha2, alpha1)                    (f1, f2) = (f2, f1)                alpha3 = alpha2 + h                theta3 = theta + alpha3 * Dk                f3 = (np.sum((y-np.dot(x, theta3))**2))/(2*nRow)                print('f3 - f2 is %f' %(f3-f2))                if f3 > f2:                    a = min(alpha1, alpha3)                    b = max(alpha1, alpha3)                    break                if f3 <= f2:                    alpha1 = alpha2                    alpha2 = alpha3                    f1 = f2                     f2 = f3            ## find optiaml alpha in [a,b] using huang jin fen ge fa             e = 0.01            while Loop >0:                alpha1 = a + 0.382 * (b - a)                alpha2 = a + 0.618 * (b - a)                theta1 = theta + alpha1* Dk                theta2 = theta + alpha2* Dk                f1 = (np.sum((y-np.dot(x, theta1))**2))/(2*nRow)                f2 = (np.sum((y-np.dot(x, theta2))**2))/(2*nRow)                if f1 > f2:                    a = alpha1                if f1< f2:                    b = alpha2                if np.fabs(a-b) <= e:                    alpha = (a+b)/2                    break            print('optimal alpha is %f' % alpha)            theta = theta + alpha * Dk        i = i + 1    return theta, costJdef Myfunction_DFP2(data, alpha, numIter, eplise):    ''' DFP -- theta := theta + alpha * Dk               --alpha is searched by huangjin method               --satisfied argmin{f(theta+alpha*Dk)}##    :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    :reference:http://blog.pfan.cn/miaowei/52925.html    :reference:http://max.book118.com/html/2012/1025/3119007.shtm ## important ##    :hessian is estimated by DFP method.    '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    Hessian = np.eye(nCol+1)    H = np.dot(x,theta)    J = (np.sum((y-H)**2))/(2*nRow)    #costJ.append(J)    Gradient = (np.dot(np.transpose(y-H),x))/nRow    Gradient = Gradient.reshape(nCol+1, 1)    Dk = - Gradient    #eplise = 0.4    while i < numIter:        if(np.sum(np.fabs(Dk)) <= eplise ): ## stop condition ##            return theta, costJ        else:            ## find alpha that min f(thetaK + alpha * Dk)            ## find optimal [a,b] which contain optimal alpha            ## optimal alpha lead to min{f(theta + alpha*DK)}            alpha0 = 0            h = np.random.random(1)            alpha1 = alpha0            alpha2 = alpha0 + h            theta1 = theta + alpha1 * Dk            theta2 = theta + alpha2 * Dk            f1 = (np.sum((y-np.dot(x, theta1))**2))/(2*nRow)            f2 = (np.sum((y-np.dot(x, theta2))**2))/(2*nRow)            Loop = 1            a = 0            b = 0            while Loop >0:                print(' find [a,b] loop is %d' %Loop)                Loop = Loop + 1                if f1 > f2:                    h = 2*h                else:                    h = -h                    (alpha1, alpha2) = (alpha2, alpha1)                    (f1, f2) = (f2, f1)                alpha3 = alpha2 + h                theta3 = theta + alpha3 * Dk                f3 = (np.sum((y-np.dot(x, theta3))**2))/(2*nRow)                print('f3 - f2 is %f' %(f3-f2))                if f3 > f2:                    a = min(alpha1, alpha3)                    b = max(alpha1, alpha3)                    break                if f3 <= f2:                    alpha1 = alpha2                    alpha2 = alpha3                    f1 = f2                     f2 = f3            ## find optiaml alpha in [a,b] using huang jin fen ge fa             e = 0.01            while Loop >0:                alpha1 = a + 0.382 * (b - a)                alpha2 = a + 0.618 * (b - a)                theta1 = theta + alpha1* Dk                theta2 = theta + alpha2* Dk                f1 = (np.sum((y-np.dot(x, theta1))**2))/(2*nRow)                f2 = (np.sum((y-np.dot(x, theta2))**2))/(2*nRow)                if f1 > f2:                    a = alpha1                if f1< f2:                    b = alpha2                if np.fabs(a-b) <= e:                    alpha = (a+b)/2                    break            print('optimal alpha is %f' % alpha)            theta_old = theta            theta = theta + alpha * Dk            ## update the Hessian matrix ##            H = np.dot(x,theta)            J = (np.sum((y-H)**2))/(2*nRow)            ## update             print('Itering %d ;cost is:%f' %(i+1,J))            costJ.append(J)            # here to estimate Hessian'inv #            # sk = ThetaNew - ThetaOld = alpha * inv(H) * Gradient            sk = theta - theta_old            #yk = DelX(k+1) - DelX(k)            DelXK = - (np.dot(np.transpose(y-np.dot(x, theta)),x))/nRow            DelXk = - (np.dot(np.transpose(y-np.dot(x, theta_old)),x))/nRow            yk = (DelXK - DelXk).reshape(nCol+1, 1)            #z1 = (sk * sk') # a matrix            #z2 = (sk' * yk) # a value            z1 = sk * np.transpose(sk)            z2 = np.dot(np.transpose(sk),yk)            #z3 = (H * yk * yk' * H) # a matrix            #z4 = (yk' * H * yk) # a value            z3 = np.dot(np.dot(np.dot(Hessian, yk), np.transpose(yk)), Hessian)            z4 = np.dot(np.dot(np.transpose(yk), Hessian),yk)            DHessian = z1/z2 - z3/z4            Hessian = Hessian + DHessian            Dk = - np.dot(Hessian, DelXK.reshape(nCol+1,1))        i = i + 1    return theta, costJdef Myfunction_DFP1(data, alpha, numIter, eplise):    ''' DFP -- theta := theta + alpha * Dk               alpha is fixed ##    :type data: array     :param data: contain x and y(label)     :type step: int/float numeric    :param step: length of step when update the theta    :reference:http://blog.pfan.cn/miaowei/52925.html    :reference:http://max.book118.com/html/2012/1025/3119007.shtm ## important ##    :hessian is estimated by DFP method.    '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    Hessian = np.eye(nCol+1)    H = np.dot(x,theta)    J = (np.sum((y-H)**2))/(2*nRow)    #costJ.append(J)    Gradient = (np.dot(np.transpose(y-H),x))/nRow    Gradient = Gradient.reshape(nCol+1, 1)    Dk = - Gradient    #eplise = 0.4    while i < numIter:        if(np.sum(np.fabs(Dk)) <= eplise ): ## stop condition ##            return theta, costJ        else:            ## find alpha that min f(thetaK + alpha * Dk)            ## here for simple alpha is parameter 'alpha'            alpha = alpha            theta_old = theta            theta = theta + alpha * Dk            ## update the Hessian matrix ##            H = np.dot(x,theta)            J = (np.sum((y-H)**2))/(2*nRow)            ## update             print('Itering %d ;cost is:%f' %(i+1,J))            costJ.append(J)            # here to estimate Hessian'inv #            # sk = ThetaNew - ThetaOld = alpha * inv(H) * Gradient            sk = theta - theta_old            #yk = DelX(k+1) - DelX(k)            DelXK = - (np.dot(np.transpose(y-np.dot(x, theta)),x))/nRow            DelXk = - (np.dot(np.transpose(y-np.dot(x, theta_old)),x))/nRow            yk = (DelXK - DelXk).reshape(nCol+1, 1)            #z1 = (sk * sk') # a matrix            #z2 = (sk' * yk) # a value            z1 = sk * np.transpose(sk)            z2 = np.dot(np.transpose(sk),yk)            #z3 = (H * yk * yk' * H) # a matrix            #z4 = (yk' * H * yk) # a value            z3 = np.dot(np.dot(np.dot(Hessian, yk), np.transpose(yk)), Hessian)            z4 = np.dot(np.dot(np.transpose(yk), Hessian),yk)            DHessian = z1/z2 - z3/z4            Hessian = Hessian + DHessian            Dk = - np.dot(Hessian, DelXK.reshape(nCol+1,1))            i = i + 1    return theta, costJdef Myfunction_BFGS1(data, alpha, numIter, eplise):    ''' BFGS     :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    :reference:http://blog.pfan.cn/miaowei/52925.html    :reference:http://max.book118.com/html/2012/1025/3119007.shtm ## important ##    :hessian is estimated by BFGS method.    '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    Hessian = np.eye(nCol+1)    H = np.dot(x,theta)    J = (np.sum((y-H)**2))/(2*nRow)    #costJ.append(J)    Gradient = (np.dot(np.transpose(y-H),x))/nRow    Gradient = Gradient.reshape(nCol+1, 1)    Dk = - Gradient    #eplise = 0.4    while i < numIter:        if(np.sum(np.fabs(Dk)) <= eplise ): ## stop condition ##            return theta, costJ        else:            ## find alpha that min J(thetaK + alpha * Dk)            ## here for simple alpha is parameter 'alpha'            alpha = alpha            theta_old = theta            theta = theta + alpha * Dk            ## update the Hessian matrix ##            H = np.dot(x,theta)            J = (np.sum((y-H)**2))/(2*nRow)            ## update             print('Itering %d ;cost is:%f' %(i+1,J))            costJ.append(J)            # here to estimate Hessian #            # sk = ThetaNew - ThetaOld = alpha * inv(H) * Gradient            sk = theta - theta_old            #yk = DelX(k+1) - DelX(k)            DelXK = - (np.dot(np.transpose(y-np.dot(x, theta)),x))/nRow            DelXk = - (np.dot(np.transpose(y-np.dot(x, theta_old)),x))/nRow            yk = (DelXK - DelXk).reshape(nCol+1, 1)            #z1 = yk' * H * yk # a value            #z2 = (sk' * yk) # a value            z1 = np.dot(np.dot(np.transpose(yk), Hessian), yk)            z2 = np.dot(np.transpose(sk),yk)            #z3 = sk * sk' # a matrix            #z4 = sk * yk' * H # a matrix            z3 = np.dot(sk, np.transpose(sk))            z4 = np.dot(np.dot(sk, np.transpose(yk)), Hessian)            DHessian = (1+z1/z2) * (z3/z2) - z4/z2            Hessian = Hessian + DHessian            Dk = - np.dot(Hessian, DelXK.reshape(nCol+1,1))            i = i + 1    return theta, costJdef Myfunction_BFGS2(data, alpha, numIter, eplise):    ''' BFGS     :type data: array      :param data: contain x and y(label)    :type step: int/float numeric    :param step: length of step when update the theta    :reference:http://blog.pfan.cn/miaowei/52925.html    :reference:http://max.book118.com/html/2012/1025/3119007.shtm ## important ##    :hessian is estimated by BFGS method.    '''    nCol = data.shape[1]-1    nRow = data.shape[0]    print nCol    print nRow    x = data[:, :nCol]    print x[1:5, :]    z = np.ones(nRow).reshape(nRow, 1)    x = np.hstack((z, x))  ## vstack merge like rbind in R; hstack like cbind in R;    y = data[:, (nCol)].reshape(nRow, 1)    #theta = np.random.random(nCol+1).reshape(nCol+1, 1)    theta = np.ones(nCol+1).reshape(nCol+1, 1)    i = 0    costJ = []    Hessian = np.eye(nCol+1)    H = np.dot(x,theta)    J = (np.sum((y-H)**2))/(2*nRow)    #costJ.append(J)    Gradient = (np.dot(np.transpose(y-H),x))/nRow    Gradient = Gradient.reshape(nCol+1, 1)    Dk = - Gradient    #eplise = 0.4    while i < numIter:        if(np.sum(np.fabs(Dk)) <= eplise ): ## stop condition ##            return theta, costJ        else:            ## find alpha that min J(thetaK + alpha * Dk)            alpha = alpha            ## find optimal [a,b] which contain optimal alpha            ## optimal alpha lead to min{f(theta + alpha*DK)}            '''            alpha0 = 0            h = np.random.random(1)            alpha1 = alpha0            alpha2 = alpha0 + h            theta1 = theta + alpha1 * Dk            theta2 = theta + alpha2 * Dk            f1 = (np.sum((y-np.dot(x, theta1))**2))/(2*nRow)            f2 = (np.sum((y-np.dot(x, theta2))**2))/(2*nRow)            Loop = 1            a = 0            b = 0            while Loop >0:                print(' find [a,b] loop is %d' %Loop)                Loop = Loop + 1                if f1 > f2:                    h = 2*h                else:                    h = -h                    (alpha1, alpha2) = (alpha2, alpha1)                    (f1, f2) = (f2, f1)                alpha3 = alpha2 + h                theta3 = theta + alpha3 * Dk                f3 = (np.sum((y-np.dot(x, theta3))**2))/(2*nRow)                print('f3 - f2 is %f' %(f3-f2))                if f3 > f2:                    a = min(alpha1, alpha3)                    b = max(alpha1, alpha3)                    break                if f3 <= f2:                    alpha1 = alpha2                    alpha2 = alpha3                    f1 = f2                     f2 = f3            ## find optiaml alpha in [a,b] using huang jin fen ge fa             e = 0.01            while Loop >0:                alpha1 = a + 0.382 * (b - a)                alpha2 = a + 0.618 * (b - a)                theta1 = theta + alpha1* Dk                theta2 = theta + alpha2* Dk                f1 = (np.sum((y-np.dot(x, theta1))**2))/(2*nRow)                f2 = (np.sum((y-np.dot(x, theta2))**2))/(2*nRow)                if f1 > f2:                    a = alpha1                if f1< f2:                    b = alpha2                if np.fabs(a-b) <= e:                    alpha = (a+b)/2                    break            print('optimal alpha is %f' % alpha)            '''            ## Get Dk and update Hessian            theta_old = theta            theta = theta + alpha * Dk            ## update the Hessian matrix ##            H = np.dot(x,theta)            J = (np.sum((y-H)**2))/(2*nRow)            ## update             print('Itering %d ;cost is:%f' %(i+1,J))            costJ.append(J)            # here to estimate Hessian #            # sk = ThetaNew - ThetaOld = alpha * inv(H) * Gradient            sk = theta - theta_old            #yk = DelX(k+1) - DelX(k)            DelXK = - (np.dot(np.transpose(y-np.dot(x, theta)),x))/nRow            DelXk = - (np.dot(np.transpose(y-np.dot(x, theta_old)),x))/nRow            yk = (DelXK - DelXk).reshape(nCol+1, 1)            #z1 = yk' * H * yk # a value            #z2 = (sk' * yk) # a value            z1 = np.dot(np.dot(np.transpose(yk), Hessian), yk)            z2 = np.dot(np.transpose(sk),yk)            #z3 = sk * sk' # a matrix            #z4 = sk * yk' * H # a matrix            z3 = np.dot(sk, np.transpose(sk))            z4 = np.dot(np.dot(sk, np.transpose(yk)), Hessian)            DHessian = (1+z1/z2) * (z3/z2) - z4/z2            Hessian = Hessian + DHessian            Dk = - np.dot(Hessian, DelXK.reshape(nCol+1,1))            i = i + 1    return theta, costJ## test ##num = 10000#theta, costJ = Myfunction_BGD(dataArray, alpha=0.0005, numIter=num, eplise=0.4) ###theta, costJ = Myfunction_SGD(dataArray, alpha=0.00005, numIter=num, eplise=0.4)#theta, costJ = Myfunction_NGD1(dataArray, alpha=0.0005, numIter=num, eplise=0.4) ## alpha is fixed ###theta, costJ = Myfunction_NGD2(dataArray, alpha=0.0005, numIter=num, eplise=0.4) ## alpha is 1 ###theta, costJ = Myfunction_QNGD(dataArray, alpha=0.0005, numIter=num, eplise=0.4) ## alpha is searched ###theta, costJ = Myfunction_DFP1(dataArray, alpha=0.0005, numIter=num, eplise=0.4) ## alpha is fixed ###theta, costJ = Myfunction_DFP2(dataArray, alpha=0.0005, numIter=num, eplise=0.4) ## alpha is searched ##theta, costJ = Myfunction_BFGS1(dataArray, alpha=0.0005, numIter=num, eplise=0.4) ## alpha is fxied ##print thetaklen = len(costJ)leng = np.linspace(1, klen, klen)plt.plot(leng, costJ)plt.show()

实验数据和结果展示

数据csv格式

0   28.224016691   33.249216932   35.820842773   36.870968784   30.984885315   38.782212966   38.467533247   41.960658458   36.826564139   35.508112110  35.7464718111  36.1711098712  37.5116599913  41.2710925714  44.0384267715  48.0300170516  45.5040184317  45.0263560818  51.7057403419  46.7635988120  52.648759521  48.8138359322  50.6945125423  55.5420040324  54.5563958625  53.1903622326  58.8926909127  54.7888425128  57.903395129  62.2111496730  64.5102546831  62.2071053732  62.9473630433  60.3044793334  65.3204440635  65.8290345236  66.3787221637  69.7564055338  66.0211259439  65.8711903940  74.2720975141  67.5766162842  73.1944408843  69.453311744  74.9112981745  71.2118760946  77.096254547  81.9506683748  78.0463683849  83.4284252650  80.4021756351  78.6865020652  82.9139521553  85.0966311554  88.7154090755  87.7395556  89.1865477657  91.0933744158  83.9561442259  93.3068317960  93.2761859661  88.0785923862  89.1066785663  95.6144366664  93.3989910665  94.3825875866  96.8764180267  96.8789694668  97.009441269  100.07611570  104.761990571  100.791709372  99.8552336273  106.901849474  103.606106375  103.410505876  106.430457677  110.735724978  107.042045579  107.283422180  113.929949681  111.218762782  116.410059683  108.023725684  112.777359285  117.346495786  117.197680787  120.053852188  114.458496489  122.2860022

结果展示

横轴是迭代次数，纵轴是代价

 Batch Gradient Descent- 批量梯度下降法 
 Stochastic Gradient Descent- 随机梯度下降法 
 Newton下降法，固定alpha=1 
 Newton下降法，固定alpha=0.0005 
 DFP，alpha是一维搜索得到的 
 阻尼牛顿法，alpha是一维搜索得到的

总结

不管什么最优化方法，都是试图去寻找代价下降最快的方向和合适的步幅。

作者简介：于建民，关注领域数据挖掘，模式识别。我的博客。

from: http://dataunion.org/20714.html