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  • 梯度下降法基本原理

    2020-10-20 15:28:01
    求解线性回归模型–函数求极值 解析解 根据严格的推导和计算得到,是方程的精确...为了便于理解,我们先从最简单的一元凸函数开始介绍梯度下降法的求解过程。 一元凸函数求极值 这是一元函数fx=x平方+2的函数曲线,

    求解线性回归模型–函数求极值

    • 解析解
      • 根据严格的推导和计算得到,是方程的精确解
      • 能够在任意精度下满足方程

    但是在很多情况下,无法直接通过严格的公式推导,得到方程或者方程组的解析解,这时候只能够采用数值分析的方法得到近似解,这样的解也成为数值解

    • 数值解
      • 通过某种近似计算得到的解
      • 能够在给定的精度条件下满足方程

    我们就都介绍一种常用的求数值解的方法,梯度下降法。为了便于理解,我们先从最简单的一元凸函数开始介绍梯度下降法的求解过程。

    一元凸函数求极值

    在这里插入图片描述

    这是一元函数fx=x平方+2的函数曲线,这种形状的函数称为凸函数,他一定存在唯一的一个极小值点,这个点在一个斜率正好为0的位置。在求解一个问题的时候,如果我们能够把它转换为在凸函数上求极小值的问题,那么这个问题就算已经破解了。在上节课中我们介绍了求这个函数的解析解的方法,下面我们来看一下,如何采用数值计算的方法来得到它的数值解。这类问题可以采用迭代的方法来求解。

    在这里插入图片描述
    这类问题可以采用迭代的方法来求解,首先我们在函数曲线上取任意1点x0x_0作为初值,当然这个初值不会恰好在极值点上。下面,我们按照某个步长移动x找到x1x_1,使得函数在x1x_1上的值,小于在x0x_0上的值。不断重复这个步骤,直到无法找到更小的函数值,最后找到的这个x,就是使函数达到极小值时的位置。
    在这里插入图片描述

    例如取初始值x0=3x_0=3,那么这一点的函数值等于11。假设取步长是0.2,那么x可能分别向两个方向移动,到达3.2或者2.8的位置,究竟应该向哪个方向移动呢?对比它们的函数值分别是显然9.84小于12.24。因此,取x1=2.8x_1=2.8
    在这里插入图片描述

    下面再以2.8为起点,步长仍然是0.2,对比函数,在x=2.6x=2.6x=3x=3处的取值,显然,在x=2.6x=2.6时的函数值更小。因此,取x2=2.6x_2=2.6。下面再以2.6为起点,继续这个步骤不断移动x,直到函数达到最小值,这个图只是个示意图,这个步长的比例是夸张了,其实每一步0.2是要比这个小很多。

    为了更加严谨,我们用表格的形式来展示这个迭代的过程。
    在这里插入图片描述
    这是初值,x0=3,比较x=2.8x=2.8x=3.2x=3.2对应的函数值,因为9.84小于12.24,所以选择x1=2.8x_1=2.8,进入下一次迭代,这时初值是2.8,根据函数值要不断减小的原则,选择x1=2.6x_1=2.6,再进入下一次迭代。不断重复这个步骤,在第15次迭代时,到达x=0x=0x=0x=0时,函数值等于2。现在,无论往哪个方向移动,函数值都是2.04,比x=0x=0时更大。也就是说,到这里之后,函数值不能再继续减小了。因此,当x=0x=0时,f(x)f(x)就达到了最小值,迭代结束。

    采用这种方法,需要通过多次迭代,不断的接近极值点,因此要花费比较长的时间,如果想要速度更快一点,可以把步长的值加大,例如把步长改为0.5。
    在这里插入图片描述
    那么,只要经过6次迭代就可以达到极小值点。可见步长取值越大,找到极值点的速度就越快,那么是不是不长的取值越大越好呢?我们将步长设为0.7来试一下
    在这里插入图片描述
    可以看到在第4次迭代时x=0.2x=0.2,这时候可以取到的两个点分别是-0.5和0.9。取-0.5时函数值更小,因此下一步取-0.5,这时候比较x=1.2x=-1.2x=0.2x=0.2时函数的取值,取0.2时函数值更小,因此又回到了0.2。继续进行这个过程会一直在0.2和-0.5之间来回震荡,无法达到极小值点。
    在这里插入图片描述

    产生震荡的原因是因为更新x的时候,步长太大,一下子跨过了最小值。在英文资料中把这种情况称为overshoot the minimum

    震荡现象也分为两种情况,一种是虽然来回震荡,但是震幅越来越小,最后还是可以收敛。另一种就是来回震荡无法收敛。

    通过这个例子我们发现,

    • 步长如果太小会增加迭代次数,收敛速度很慢。
    • 不长如果太大又会引起震荡,甚至导致无法收敛

    那么这个不长的职能不能自动地调节呢,在距离及时点比较远的地方,步长的取值可以大一些,使得算法尽快收敛。在距离极值点比较近的地方,可以使步长逐渐减小,避免跨过极小值点发生震荡。这个想法显然不错,但是怎样才能够实现这种不长的自动调节呢?
    在这里插入图片描述
    观察这个曲线可以发现,距离极值点比较远的地方,曲线比较陡峭,随着不断靠近极值点曲线逐渐变得平缓。曲线的陡峭程度可以用斜率表示,曲线越陡斜率越大,这时候我们希望步长也越大。曲线越平缓,斜率越小,我们希望步长也越小,因此只要用斜率去调节步长就可以了。
    在这里插入图片描述

    曲线在这一点的斜率就是函数在这个点的导数,让步长和斜率之间保持正比例关系。

    =ηdf(x)dx步长=\eta\frac{df(x)}{dx}η\eta是一个常数,称为学习率。

    就可以得到更新x的迭代公式x(k+1)=x(k)ηdf(x)dxx^{(k+1)}=x^{(k)}-\eta\frac{df(x)}{dx}

    这里我们采用加括号的上标来表示迭代的次数,这是为了和之前采用下标表示样本序号,采用上标表示属性区分开来。在第k+1次迭代中,x的取值是由第k次x的值减去步长得到。

    采用这种方法进行迭代,可以根据函数曲线的斜率实现,对步长的自适应调整,在这种迭代算法中,当x大于0时导数为正数。这时候,x(k+1)=x(k)ax^{(k+1)}=x^{(k)}-a,a为正数,变得更小,向原点的方向移动。
    在这里插入图片描述
    当x小于0时,导数为负数,x(k+1)=x(k)ax^{(k+1)}=x^{(k)}-a,a为负数,变得更大,x也是向原点的方向移动,
    在这里插入图片描述
    可见无论x是大于0还是小于0,最后都能够收敛于原点。采用这种方法导数的符号就直接决定了迭代更新的方向,而不需要像前面的例子那样,每次都去比较两个方向的函数值来确定x移动的方向。

    总结一下,采用这种迭代算法能够自动调节步长,自动确定下一次更新的方向,并且能够保证收敛性。

    这种一元凸函数求极值的方法可以推广到二元凸函数中,二元函数中有两个自变量z=f(x,y)z=f(x,y),x和y需要分别进行迭代计算。
    这是二元函数的迭代算法,
    x(k+1)=x(k)ηf(x,y)xx^{(k+1)}=x^{(k)}-\eta\frac{\partial{f(x, y)}}{\partial{x}}

    y(k+1)=y(k)ηf(x,y)yy^{(k+1)}=y^{(k)}-\eta\frac{\partial{f(x, y)}}{\partial{y}}

    x的更新通过函数对x的偏导数来调节,y的更新通过函数对y的偏导数来调节。
    在这里插入图片描述
    函数f对x的偏导数和对y的偏导数,所组成的向量是二元函数的梯度。

    • 导数:函数在某一点p的导数,其实就是函数在p点的变化率。

    • 偏导数:f(x,y)x\frac{\partial{f(x, y)}}{\partial{x}}二元函数对x的偏导数,就是函数在x方向上的变化率。
      f(x,y)y\frac{\partial{f(x, y)}}{\partial{y}}对y的偏导数就是函数在y方向的变化率

    • 方向导数:函数沿着某一个方向l的变化率,这个l可以是任何一个方向

    x偏导数和y偏导数都是方向导数的特例,那么函数在p点时沿着哪一个方向的变化最快呢?也就是说哪一个方向导数最大,这个最大的值又是多少呢?这个问题的答案就是梯度。

    • 梯度:gradf(x,y)=fxi+fyj\vec{grad}f(x, y)=\frac{\partial{f}}{\partial{x}}\vec{i}+\frac{\partial{f}}{\partial{y}}\vec{j}
      梯度是一个矢量,既有大小又有方向,也可以采用这样的向量形式来表示.
      =(f(x,y)xf(x,y)y)\nabla=(\begin{array}{c} \frac{\partial{f(x, y)}}{\partial{x}} \\ \frac{\partial{f(x, y)}}{\partial{y}} \end{array})
      • 它的大小就是所有方向导数中那个最大的值。
      • 它的方向就是取得最大方向导数的方向,或者说函数在某个点的梯度,就是指在这个点沿着这个方向的变化率最大。

    这种二元函数求极值的迭代算法,其实就是每一步都是沿着梯度的方向移动,也就是沿着最陡的方向进行移动这样做计算的速度是最快的,这种方法被称为梯度下降法。
    在这里插入图片描述

    这就好像下山,我们在山腰上的某个位置环顾一圈之后,沿着当前最陡的方向,向下走一步。然后再环顾一圈,再沿着目前最陡的方向向下走一步。不断重复这个过程,保证每一步都是沿着当时最陡的方向向下走,那么一定可以以最少的步数到达山脚下,当然这要假设这个山的每个方向上都有路,都可以向下走。

    对于机器学习算法,只要能够把损失函数描述成凸函数,那么就一定可以采用梯度下降法,以最快的速度更新模型参数,找到使损失函数达到最小值的点的位置。

    展开全文
  • 科学计算中的无约束梯度算法的原理 梯度法尽管收敛速度较慢,但其迭代的几何概念比较...常常利用这个特点,将梯度法和其他方法配合使用构成更有效和实用的算法,在理论上梯度法仍不失为一种极为重要的基本优化方法。
  • 本文从一元函数的导数出发,不加详细证明的给出梯度下降背后的基本数据逻辑。 1. 从导数到梯度 (1) 一元函数的导数:自变量变化无穷小引起的因变量改变值的极限 f′(x)=lim⁡Δx→0f(x+Δx)−f(x)Δxf'(x)=\lim\...

    梯度下降法是机器学习算法中最常用的迭代优化数值算法,尤其是在深度学习神经网络的BP算法中扮演者重要作用。理解其基本原理是每个MLer的基础能力。

    本文从一元函数的导数出发,不加详细证明的给出梯度下降法背后的基本数据逻辑。

    1. 从导数到梯度

    (1) 一元函数的导数:自变量变化无穷小引起的因变量改变值的极限
    f(x)=limΔx0f(x+Δx)f(x)Δxf'(x)=\lim\limits_{\Delta x \rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
    (2) 多元函数的偏导:多元函数中,某个自变量(其它自变量固定不懂)变化无穷小引起的因变量改变值的极限
    f(xi)=limΔx0f(x+Δxei)f(x)Δxf'(\boldsymbol x_i)=\lim\limits_{\Delta x \rightarrow0}\frac{f(\boldsymbol x+\Delta x *\boldsymbol e_i)-f(\boldsymbol x)}{\Delta x}

    (3) 方向导数:多元函数中,自变量沿任意方向l\boldsymbol l变化无穷小引起的因变量改变值的极限,其值为一个标量
    f(x)l=limΔx0f(x+Δx)f(x)Δx\frac{\partial f(\boldsymbol x)}{\partial \boldsymbol l}=\lim\limits_{\Delta \boldsymbol x \rightarrow0}\frac{f(\boldsymbol x+\Delta \boldsymbol x )-f(\boldsymbol x)}{\Delta \boldsymbol x}式中Δx\Delta \boldsymbol x表示自变量沿任意方向的变化值

    (4) 梯度:多元函数中,各自变量偏导数组成的向量,其表示函数值变化最快的方向(后文会给出详细解释)
    f(x)=(f(x)x1,f(x)x2,...,f(x)xn)\nabla f(\boldsymbol x)=(\frac{\partial f(\boldsymbol x)}{\partial x_1},\frac{\partial f(\boldsymbol x)}{\partial x_2},...,\frac{\partial f(\boldsymbol x)}{\partial x_n})
    (5) 梯度和方向导数的关系

    假设多元函数的梯度方向与方向导数Δx\Delta \boldsymbol x所在方向l\boldsymbol l的夹角为θ\boldsymbol \theta,则该梯度和方向导数存在如下数学关系:
    f(x)l=f(x)cosθ\frac{\partial f(\boldsymbol x)}{\partial \boldsymbol l}=\nabla f(\boldsymbol x) \cdot cos \boldsymbol \theta式中\cdot表示点乘。

    2. 泰勒展开

    泰勒展开是用于进行函数逼近和估计的强有力工具。

    (1) 一元函数的一阶泰勒展开
    f(x)=f(x0)+f(x0)(xx0)+onf(x)=f(x_0)+f'(x_0)(x-x_0)+o^n(2) 一元函数的二阶泰勒展开
    f(x)=f(x0)+f(x0)(xx0)+12f(x0)(xx0)2+onf(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)( x-x_0)^2+o^n(3) 多元函数的一阶泰勒展开f(x)=f(x0)+[f(x)]T(xx0)+onf(\boldsymbol x)=f(\boldsymbol x_0)+[\nabla f(\boldsymbol x)]^T(\boldsymbol x-\boldsymbol x_0)+o^n(4) 多元函数的二阶泰勒展开f(x)=f(x0)+[f(x)]T(xx0)+12(xx0)TH(x0)(xx0)+onf(\boldsymbol x)=f(\boldsymbol x_0)+[\nabla f(\boldsymbol x)]^T(\boldsymbol x-\boldsymbol x_0)+ \frac{1}{2}(\boldsymbol x-\boldsymbol x_0)^TH(\boldsymbol x_0)(\boldsymbol x-\boldsymbol x_0)+o^n上式中H(x0)H(\boldsymbol x_0)表示Hessen矩阵H(x)H(\boldsymbol x)x0\boldsymbol x_0处的取值。
    H(x)=[fx11fx12...fx1nfx21fx22...fx2n............fxn1fxn2...fxnn]H(\boldsymbol x)=\left [\begin{matrix}\frac{\partial f}{\partial x_{11}}&\frac{\partial f}{\partial x_{12}}&...&\frac{\partial f}{\partial x_{1n}}\\\frac{\partial f}{\partial x_{21}}&\frac{\partial f}{\partial x_{22}}&...&\frac{\partial f}{\partial x_{2n}}\\...&...&...&...\\\frac{\partial f}{\partial x_{n1}}&\frac{\partial f}{\partial x_{n2}}&...&\frac{\partial f}{\partial x_{nn}}\end{matrix}\right]

    3. 梯度为0的数学意义

    对于一个凸函数,梯度为0处意味着全局的最小值点。
    在这里插入图片描述
    以一元函数为例(见上图)进行说明。
    对于位于右侧的点C而言,此时f(xC)>0f'(x_C)>0,因此有:
    f(xC)=f(xA)+f(xA)Δx+on>f(xA)f(x_C)=f(x_A)+f'(x_A)\Delta x+o^n>f(x_A)
    而对于位于左侧的点B而言,此时f(xB)<0f'(x_B)<0,因此有:
    f(xC)=f(xA)+f(xA)(Δx)+on>f(xA)f(x_C)=f(x_A)+f'(x_A)(-\Delta x)+o^n>f(x_A)易得点AA为局部最小值点,或对于凸函数而言即为全局最小值点。
    在这里插入图片描述

    将一元函数推广到二元或者多元的凸函数空间中,如下图所示。根据泰勒展开式f(x)=f(x0)+[f(x)]T(xx0)+onf(\boldsymbol x)=f(\boldsymbol x_0)+[\nabla f(\boldsymbol x)]^T(\boldsymbol x-\boldsymbol x_0)+o^n,其中xx0\boldsymbol x-\boldsymbol x_0表示空间点的变化向量,写成梯度的形式为:f(x)=f(x0)+[f(x)]Tf(x)cosθ+onf(\boldsymbol x)=f(\boldsymbol x_0)+[\nabla f(\boldsymbol x)]^T\nabla f(\boldsymbol x)*\cos\boldsymbol \theta+o^n上式中的*表示哈达玛积,对于凸函数而言,易知cosθ>0\cos \theta>0,因此:f(x)>f(x0)f(\boldsymbol x)>f(\boldsymbol x_0)这意味着x0\boldsymbol x_0处即为凸函数的最小值点。

    综上所述,对于凸函数,梯度为0处f(x)=0\nabla f(\boldsymbol x)=0就相当于是函数的全局最小值处。

    但注意其包含了两个前提假设:
    (1)目标函数为凸函数;
    (2)梯度存在,且梯度为0的等式f(x)=0\nabla f(\boldsymbol x)=0能够求解得到解析解。

    但在实际问题中,这两点往往是难以完全满足的,此时可以通过梯度下降法进行迭代数值计算。

    4. 梯度下降法的数学意义

    根据上面的泰勒展开式:f(x)=f(x0)+[f(x)]T(xx0)+on=f(x0)+[f(x)]Tf(x)cosθ+on=f(x0)+i(fxi)2cosθi+on\begin{aligned}f(\boldsymbol x)&=f(\boldsymbol x_0)+[\nabla f(\boldsymbol x)]^T(\boldsymbol x-\boldsymbol x_0)+o^n\\&=f(\boldsymbol x_0)+[\nabla f(\boldsymbol x)]^T\nabla f(\boldsymbol x)*\cos\boldsymbol \theta+o^n\\&=f(\boldsymbol x_0)+\sum_i(\frac{\partial f}{\partial x_i})^2\cos\theta_i+o^n\end{aligned}cosθi=1\cos\theta_i=1,即xx0\boldsymbol x-\boldsymbol x_0的方向与f(x)\nabla f(\boldsymbol x)方向一致,此时函数值增加最快,这对应梯度上升法。

    cosθi=1\cos\theta_i=-1,即xx0\boldsymbol x-\boldsymbol x_0的方向与f(x)\nabla f(\boldsymbol x)负方向一致,此时函数值下降最快,这对应梯度下降法。

    如下图中的图a所示,梯度下降法的意义在于每步迭代中,通过沿着当前点的梯度下降方向(即函数值下降最快的方向),找到下一步的迭代点。通过这种方法,不断逼近梯度值为0的点。
    在这里插入图片描述
    梯度下降法的迭代公式可写作:x:=xαf(x)\boldsymbol x:=\boldsymbol x- \alpha \nabla f(\boldsymbol x)其中决定了迭代方向,那究竟要迭代多少呢,这个由超参数学习率α\alpha决定。

    α\alpha值的设定很大程度上决定了梯度下降法的执行效率和最终结果的准确性。很遗憾,对于不同的问题其α\alpha值需要通过反复验证进行选取。若α\alpha取值小(如上图a),可稳步迭代到梯度为0处,但迭代步数往往过大;而若α\alpha取值大,则会出现如上图b所示的震荡现象,导致结果的不收敛。

    5. 梯度下降法的缺陷及改进方向

    上文大致从一元函数的导数出发,大致介绍了梯度下降法的来龙去脉,显然梯度下降法的适用性存在先决条件、使用效果等方面的缺陷:
    (1)梯度不存在。梯度下降法必然用到了函数的梯度,但对于L1正则、Relu激活函数等诸多问题,目标优化函数并不存在梯度,这该如何处理?
    (2)非凸函数的最优化。凸函数的梯度下降法可以稳定找到全局最优解,而对于非凸函数确只能保证局部最优解,甚至只是鞍点。又该如何避免或改善此类问题?
    (3)函数的信息利用率不高。梯度信息法仅利用了当前每个迭代点的一阶梯度信息,是否能够通过使用历史迭代点的信息以及该点的高阶梯度信息,得到更快更准确的迭代方法?

    基于上述缺陷,数学家们或对梯度下降法进行改进,或提出新的迭代算法来解决问题。本系列的后文会分别进行介绍。

    展开全文
  • 梯度下降是求解无约束最优化问题一种最常用方法,实现简单,每一步需要求解目标函数的梯度向量。 牛顿和拟牛顿 牛顿是求解无约束最优化问题常用方法,收敛速度快,每一步迭代需要求解目标函数海森...

    梯度下降法

    梯度下降法是求解无约束最优化问题的一种最常用方法,实现简单,每一步需要求解目标函数的梯度向量。
    以二元函数 z=f(x,y) 等值线俯视图为例:
    这里写图片描述

    牛顿法和拟牛顿法

    牛顿法是求解无约束最优化问题的常用方法,收敛速度快,每一步迭代需要求解目标函数的海森矩阵的逆矩阵,计算较为复杂。

    拟牛顿法则用正定矩阵近似海森矩阵的逆矩阵,简化了牛顿法。

    海森矩阵(Hessian Matrix)是一个多元函数的二阶偏导数构成的方阵,描述了函数的局部曲率。

    梯度下降法 牛顿法/拟牛顿法 拉格朗日对偶性
    解决问题 无约束最优化问题 约束最优化问题
    目标函数要求 函数一阶偏导数存在 函数二阶偏导数存在 拉格朗日对偶性
    思想 通过取负梯度来迭代更新,寻找极值点 利用极小值的必要条件——偏导数为0——来迭代更新,寻找极值点 拉格朗日对偶性

    拉格朗日对偶性

    在约束最优化问题中,利用拉格朗日对偶性将原始问题转换为对偶问题。

    1. 原始问题

    最优化问题:

    minxRnf(x)

    s.t.c(x)0,i=1,2,...,k

    hj(x)=0,j=1,2,...,l

    引入拉格朗日函数得到:

    L(x,α,β)=f(x)+i=1kαici(x)+j=1lβjhj(x)

    定义问题:
    Θp(x)=maxα,β;αi0L(x,α,β)

    p=minxΘp(x)

    2. 对偶问题

    定义对偶问题与对偶问题的最优值

    ΘD(α,β)=minxL(x,α,β)

    d=maxα,β;αi0ΘD(α,β)

    3.原始问题和对偶问题的关系

    a. 若原始问题与对偶问题都有最优解,则:
    d*=p*

    ΘD(α,β)=minxL(x,α,β)L(x,α,β)maxα,β;αi0L(x,α,β)=p

    b. 假设f(x)和c(x)是凸函数,h(x)是仿射函数;并假设不等式约束是严格可行的,即存在x,对所有c(x)<0,则存在x*是原始问题最优解,alpha和beta是对偶问题的解,并且有:

    p=d=L(x,α,β)

    c. KKT条件

    假设f(x)和c(x)是凸函数,h(x)是仿射函数;并假设不等式约束是严格可行的,则x,alpha,beta分别是原始问题和对偶问题的解的充要条件可以归结为——KKT条件。

    展开全文
  • 牛顿梯度下降、最小二乘法的原理以及利用它们解决实际问题python编程 一、牛顿法原理 1、产生背景 2、牛顿迭代公式 二、梯度下降法原理 根据计算梯度时所用数据量不同,可以分为三种基本方法:批量梯度...

    牛顿法、梯度下降法、最小二乘法的原理以及利用它们解决实际问题的python编程

    一、牛顿法原理

    1、产生背景

    2、牛顿迭代公式

    二、梯度下降法原理

    根据计算梯度时所用数据量不同,可以分为三种基本方法:批量梯度下降法(Batch Gradient Descent, BGD)、小批量梯度下降法(Mini-batch Gradient Descent, MBGD)以及随机梯度下降法(Stochastic Gradient Descent, SGD)。
    梯度下降法的一般求解框架

    三、最小二乘法原理

    最小二乘法(又称最小平方法)是一种数学优化技术。它通过最小化误差的平方和寻找数据的最佳函数匹配。利用最小二乘法可以简便地求得未知的数据,并使得这些求得的数据与实际数据之间误差的平方和为最小。最小二乘法还可用于曲线拟合。其他一些优化问题也可通过最小化能量或最大化熵用最小二乘法来表达。
    详细原理请参考:https://baike.baidu.com/item/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95/2522346#4

    四、运用梯度下降法原理解决实际问题的python编程举例

    1、问题如下

    2、导入所需要的包

    import numpy as np
    import matplotlib.pyplot as plt
    import matplotlib as mpl
    import math
    from mpl_toolkits.mplot3d import Axes3D
    import warnings
    

    3、画出函数图像

    def f2(x1,x2):
        return x1 ** 2 + 2 * x2 ** 2 -4*x1- 2 * x1*x2
    X1 = np.arange(-4,4,0.2)
    X2 = np.arange(-4,4,0.2)
    X1, X2 = np.meshgrid(X1, X2) # 生成xv、yv,将X1、X2变成n*m的矩阵,方便后面绘图
    Y = np.array(list(map(lambda t : f2(t[0],t[1]),zip(X1.flatten(),X2.flatten()))))
    Y.shape = X1.shape # 1600的Y图还原成原来的(40,40)
    %matplotlib inline
    #作图
    fig = plt.figure(facecolor='w')
    ax = Axes3D(fig)
    ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
    ax.set_title(u'$ x1 ** 2 + 2 * x2 ** 2 -4*x1- 2 * x1*x2 $')
    plt.show()
    

    4、求极小点和极值点

    # 解决中文显示问题
    mpl.rcParams['font.sans-serif'] = [u'SimHei']
    mpl.rcParams['axes.unicode_minus'] = False
    %matplotlib inline
    # 二维原始图像
    def f2(x, y):
        return x ** 2 + 2 * y ** 2 -4*x- 2 * x*y 
    ## 偏函数
    def hx1(x, y):
        return 2*x-4-2*y
    def hx2(x, y):
        return 4*y-2*x
    x1 = 4
    x2 = 4
    alpha = 0.001
    #保存梯度下降经过的点
    GD_X1 = [x1]
    GD_X2 = [x2]
    GD_Y = [f2(x1,x2)]
    # 定义y的变化量和迭代次数
    y_change = f2(x1,x2)
    iter_num = 0
    while(iter_num < 10000) :
        tmp_x1 = x1 - alpha * hx1(x1,x2)
        tmp_x2 = x2 - alpha * hx2(x1,x2)
        tmp_y = f2(tmp_x1,tmp_x2)    
        f_change = np.absolute(tmp_y - f2(x1,x2))
        x1 = tmp_x1
        x2 = tmp_x2
        GD_X1.append(x1)
        GD_X2.append(x2)
        GD_Y.append(tmp_y)
        iter_num += 1
    print(u"最终结果为:(%.5f, %.5f, %.5f)" % (x1, x2, f2(x1,x2)))
    print(u"迭代过程中X的取值,迭代次数:%d" % iter_num)
    print(GD_X1)
    # 作图
    fig = plt.figure(facecolor='w',figsize=(20,18))
    ax = Axes3D(fig)
    ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
    ax.plot(GD_X1,GD_X2,GD_Y,'ko-')
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    ax.set_zlabel('z')
    ax.set_title(u'函数;\n学习率:%.3f; 最终解:(%.3f, %.3f, %.3f);迭代次数:%d' % (alpha, x1, x2, f2(x1,x2), iter_num))
    plt.show()
    
    最终结果为:(4.00043, 2.00027, -8.00000)
    迭代过程中X的取值,迭代次数:10000
    [4, 4.004, 4.007975999999999, 4.011928127999999, 4.015856511327999, 4.019761276643519, 4.0236425499411, 4.02750045655379, 4.031335121156604, 4.0351466677699745, 4.0389352197631805, 4.042700899857756, 4.046443830130885, 4.050164132018776, 4.053861926320018, 4.057537333198925, 4.061190472188854, 4.0648214621955105, 4.068430421500238, 4.072017467763288, 4.075582718027074, 4.079126288719406, 4.082648295656712, 4.08614885404724, 4.089628078494242, 4.093086082999145, 4.096522980964705, 4.0999388851981395, 4.10333390791425, 4.106708160738522, 4.110061754710218, 4.11339480028544, 4.116707407340193, 4.119999685173415, 4.123271742510009, 4.126523687503843, 4.129755627740743, 4.132967670241472, 4.136159921464686, 4.139332487309879, 4.142485473120316, 4.1456189836859405, 4.148733123246279, 4.151827995493321, 4.154903703574387, 4.157960350094985, 4.1609980371216455, 4.164016866184745, 4.167016938281318, 4.169998353877847, 4.172961212913043, 4.175905614800612, 4.178831658432005, 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    在这里插入图片描述

    从上面求出的结果可以得到极小点为(4,2),极小值为-8

    5、用Excel求上个函数的极小点和极小值

    实验结果如下图:

    依旧可以清晰地看出该函数的极小点为(4,2),极值点为-8

    五、使用梯度下降法和最小二乘法求解多元函数python编程举例

    1、问题如下

    用梯度下降法和最小二乘法根据以下图片的数据进行多元线性回归并求解相关系数

    将数据写入Excel中

    在这里插入图片描述

    2、使用梯度下降法求解多元函数

    代码如下:
    import matplotlib.pyplot as plt
    import numpy as np
    from numpy import array
    import pandas as pd
    # 读取数据文件
    df=pd.read_csv("C:/Users/LOL/Desktop/店铺多元回归.csv")
    %matplotlib notebook
    df=df.values
    x_data=df[:,:2]
    y_data=df[:,2]
    #定义学习率、斜率、截据
    #设方程为y=theta1*x1+theta2*x2+theta0
    lr=0.00001
    theta0=0
    theta1=0
    theta2=0
    #定义最大迭代次数
    epochs=10000
    #定义最小二乘法函数-损失函数(代价函数)
    def compute_error(theta0,theta1,theta2,x_data,y_data):
        totalerror=0
        for i in range(0,len(x_data)):#定义一共有多少样本点
            totalerror=totalerror+(y_data[i]-(theta1*x_data[i,0]+theta2*x_data[i,1]+theta0))**2
        return totalerror/float(len(x_data))/2
    #梯度下降算法求解参数
    def gradient_descent_runner(x_data,y_data,theta0,theta1,theta2,lr,epochs):
        m=len(x_data)
        for i in range(epochs):
            theta0_grad=0
            theta1_grad=0
            theta2_grad=0
            for j in range(0,m):
                theta0_grad-=(1/m)*(-(theta1*x_data[j,0]+theta2*x_data[j,1]+theta2)+y_data[j])
                theta1_grad-=(1/m)*x_data[j,0]*(-(theta1*x_data[j,0]+theta2*x_data[j,1]+theta0)+y_data[j])
                theta2_grad-=(1/m)*x_data[j,1]*(-(theta1*x_data[j,0]+theta2*x_data[j,1]+theta0)+y_data[j])
            theta0=theta0-lr*theta0_grad
            theta1=theta1-lr*theta1_grad
            theta2=theta2-lr*theta2_grad
        return theta0,theta1,theta2
    #进行迭代求解
    theta0,theta1,theta2=gradient_descent_runner(x_data,y_data,theta0,theta1,theta2,lr,epochs)
    print('迭代次数:{0} 学习率:{1}之后 a0={2},a1={3},a2={4},代价函数为{5}'.format(epochs,lr,theta0,theta1,theta2,compute_error(theta0,theta1,theta2,x_data,y_data)))
    print("多元线性回归方程为:y=",theta1,"X1+",theta2,"X2+",theta0)
    
    
    迭代次数:10000 学习率:1e-05之后 a0=5.3774162274868,a1=45.0533119768975,a2=-0.19626929358281256,代价函数为366.7314528822914
    多元线性回归方程为:y= 45.0533119768975 X1+ -0.19626929358281256 X2+ 5.3774162274868
    

    3、用最小二乘法求解

    代码如下:
    import numpy as np
    import pandas as pd
    #变量初始化
    X=[]
    Y=[]
    B=[]
    Q_e=0
    Q_E=0
    #从csv文件中读取数据
    def get_data(file_name):
        data=pd.read_csv(file_name,header=0)
        data=np.array(data)
        Y=data[:,data.shape[1]-1]#预测对象位于最后一列
        X=data[:,0:data.shape[1]-1]
        print(X.shape)
        return X,Y
        return X,Y
    X,Y=get_data('C:/Users/LOL/Desktop/店铺多元回归.csv')
    X=np.mat(np.c_[np.ones(X.shape[0]),X])#为系数矩阵增加常数项系数
    Y=np.mat(Y)#数组转化为矩阵
    B=np.linalg.inv(X.T*X)*(X.T)*(Y.T)
    print("第一项为常数项,其他为回归系数",B)#输出系数,第一项为常数项,其他为回归系数
    print("输入店铺面积,距离最近的车站距离,预测营业额:",np.mat([1,10,80])*B ,"万日元")#预测结果
    #相关系数
    Y_mean=np.mean(Y)
    for i in range(Y.size):
        Q_e+=pow(np.array((Y.T)[i]-X[i]*B),2)
        Q_E+=pow(np.array(X[i]*B)-Y_mean,2)
    R2=Q_E/(Q_e+Q_E)
    print("R2的值:",R2)
    
    (10, 2)
    第一项为常数项,其他为回归系数 [[65.32391639]
     [41.51347826]
     [-0.34088269]]
    输入店铺面积,距离最近的车站距离,预测营业额: [[453.1880841]] 万日元
    R2的值: [[0.94523585]]
    

    4、使用Excel求解

    如图

    由上面三种方法求到的结果对比,可以清晰地看出用最小二乘法和Excel求出的结果一致,而用梯度下降法求解有一定的误差

    六、总结

    最小二乘法:能通过最小化误差的平方和寻找数据的最佳函数匹配,但是使用有一定局限性,在回归过程中,回归的关联式不可能全部通过每个回归数据点。
    梯度下降法:是通过梯度方向和步长,直接求解目标函数的最小值时的参数,越接近最优值时,步长应该不断减小,否则会在最优值附近来回震荡。

    七、参考文献

    1、https://blog.csdn.net/qq_35946969/article/details/84446000
    2、https://www.jianshu.com/p/424b7b70df7b
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