梯度下降原理及数学推导

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从最小二乘法到梯度下降

前面我们学习了线性回归，采用最小二乘法来求回归方程参数的值，即通过对参数求偏导数，列出方程组，然后解方程组，得到参数的值。这种通过方程直接计算出来的解被称为解析解。
在数据量不多，且样本的特征不多的情况下，可以直接用最小二乘法求损失函数的解析解。但对于很多实际问题来说，数据量大、样本特征多，用最小二乘法求解，计算量会非常大，计算也很复杂，但是可以通过迭代的方式去求一个接近于实际最小值的点。
而梯度下降就是这种迭代式求最小值的方法中应用最为广泛的一种方法。

梯度下降的基本思想

梯度下降法的基本思想可以类比为一个下山的过程。

假设这样一个场景：一个人被困在山上，需要从山上下来(找到山的最低点)。但此时山上的浓雾很大，导致可视度很低；因此，下山的路径就无法确定，必须利用自己周围的信息一步一步地找到下山的路。这个时候，便可利用“梯度下降算法”来帮助自己下山。

怎么做呢？首先以他当前的所处的位置为基准，寻找这个位置最陡峭的方向，然后在这个位置朝最陡峭的方向下降的方向走一步，然后又继续以当前位置为基准，再找到这个位置最陡峭的方向，再在这个位置的地方朝最陡峭的方向下降的方向再走一步，按照这样的方式不断寻找，直到到达山的低处，或者说接近山的最低处。

在刚才提到的这个下山的过程中，可以做以下类比。

山可以看成是一个函数的图像，某个位置可以看作是函数上的一个点，某个位置最陡峭的方向可以看作是函数的梯度，在这个方向上，山的海拔增加最快，也就是函数值增加最快的方向，而朝着陡峭的方向下降的方向看作是梯度的反方向，也就是梯度下降的方向，梯度下降的方向，也就是函数值减少最快的方向；山的最低点看作是函数的最小值，到达山的最低点相当于找到了函数的最小值，但实际上，一般到达接近山的最低点就算完成下山了，也就是找到接近函数的最小值的点就可以了。
接下来，我们用数学语言来描述梯度及梯度下降。

梯度

什么是梯度？

设函数z=f(x,y)在区域D上具有连续的一阶偏导数，则有梯度的计算公式：

\mathbf{grad}f(x,y)=\frac{\partial f}{\partial x}i+\frac{\partial f}{\partial y}j

或

\nabla f=\frac{\partial f}{\partial x}i+\frac{\partial f}{\partial y}j

其中，i是沿x轴的单位向量(1,0)，j是沿y轴的单位向量(0,1)。
还可以将梯度写成如下形式：

\mathbf{grad}f(x,y)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})

所以，梯度本质上是一个向量。在数学中，向量表示方向，对于一个函数来说，梯度表示该函数增加（上升）最快的方向。沿着梯度的方向，可以找到这个函数的最大值。
在机器学习中，损失函数（或者代价函数）的值代表误差，误差肯定是越小越好，也就是要求损失函数的最小值，所以，我们在梯度前面加一个负号，变成梯度下降，而梯度下降表示函数减少（下降）最快的方向，沿着梯度下降的方向，就可以找到一个函数的最小值。

举个例子：二元函数的梯度

例如，给定一个二元函数

f(x,y)=x^2+y^2

函数图像如下图所示。

接着，求这个函数的梯度，就是下面这个样子。

grad f(x,y)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})=(2x,2y)

对于f(x,y)中的任意一点，梯度都可以由上式计算出来。

梯度下降

梯度下降基本步骤

  1. 首先，我们有一个可导函数（损失函数），现在的目标就是找到这个函数的最小值，也就是山底。
  2. 根据之前的场景假设，最快的下山的方式就是找到当前位置最陡峭的方向，然后沿着此方向向下走(梯度下降)，就能让函数值下降得最快！
  3. 所以，我们重复利用这个方法，反复求取梯度，最后就能到山底，即获得函数的最小值，这就类似于我们下山的过程。

二元函数的梯度下降

给定目标函数（损失函数）：

f(x,y)=x^2+y^2

这是一个二元函数，现在我们需要找到这个函数的最小值，当然，这个函数的最小值显然是0，在点(0,0)处取得。下面，我们演示如何通过梯度下降一步一步地接近这个最小值。

首先，求这个二元函数的梯度，如下。

grad f(x,y)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})=(2x,2y)

接着，给定一个初始点，及学习率（也叫步长）。
初始点：记为X_0=(2,4)，函数值f(2,4)=20
学习率：0.01，用来控制每一步下降的距离，每一步下降的距离为0.01\times grad f(x,y)，容易看出，学习率大，每一步下降的距离就大，学习率小，每一步下降的距离就小。

在初始点，函数值f(2,4)=20，梯度为：grad f(2,4)=(4,8)
接着，沿着梯度下降的方向走。
第1步： 到X_1，X_1=X_0-0.01\times(2,4)=(1.96,3.92)
说明：以上负号（减号）表示梯度下降。
此时，函数值f(1.96,3.92)=19.21

第2步： X_2=X_1-0.01\times grad f(1.96,3.92)=X_1-0.01\times(3.92,7.94)=(1.92,3.84)
此时，函数值f(1.92,3.84)=18.45

第3步： X_3=X_2-0.01\times grad f(1.92,3.84)=X_2-0.01\times(3.84,7.68)=(1.88,3.76)
此时，函数值f(1.88,3.76)=17.72
可以看到，随着步数的增加，函数值在不断减小，也就是不断靠近最小值0。

\cdots

继续，到第100步的时候，已经下降到了点(0.27, 0.53)
此时，函数值f(0.27, 0.53)=0.35
可以看到，当下降100步的时候，函数值已经很小了。

当下降到200步的时候，函数值为0.006，已经非常接近真实最小值0了，此时误差小于0.01。
这就是用梯度下降求解这个二元函数的一个基本过程。

从上面可以看到，学习率对于下降的速度有着很大的影响，有人可能会说，把学习率设置的大一点，不就能很快下降到最小值了吗？俗话说，“步子太大，容易扯着蛋”，如果学习率过大，可能会错过最低点，见下图。

上图表示，如果设置一个很小的学习率，一步一步地会到达最低点，尽管会花很长的时间，如果设置一个过大的学习率，则会错过最低点。
所以，在实际问题中，要先设置一个小的学习率，然后观察下降速度，再根据情况调整学习率。

梯度下降数学推导

对于多元线性回归，假设函数为

h_w(x_1,x_2,…,x_n)=w_1x_1+w_2x_2+…+w_nx_n+b

假设有m个样本，(x_1^{(0)}, x_2^{(0)}, …x_n^{(0)}, y_0), (x_1^{(1)}, x_2^{(1)}, …x_n^{(1)},y_1), … (x_1^{(m)}, x_2^{(m)}, …x_n^{(m)}, y_m)

对于上方的假设函数，损失函数为：

l(w_0,w_1,…,w_n)=\frac{1}{2m}\sum_{j=1}^m(h_w(x_0^{(j)},x_1^{(j)},…,x_n^{(j)})-y_i)^2

说明：这里除以m表示平均损失，又除了一个2是为了方便后面的计算，并不影响对于目标函数的优化。

接下来要求这个损失函数的最小值。

梯度下降算法过程：

初始化w_1,w_2,…,w_n,b，步长\alpha。

  1. 求损失函数的梯度，对于参数w_i，其梯度表达式为：

    \frac{\partial}{\partial w_i} l(w_1,w_2,…,w_n,b)=\frac{1}{m}\sum_{j=0}^m(h_w(x_0^{(j)},x_1^{(j)},…,x_n^{(j)})-y_i)x_i^{(j)}

  2. 用步长乘以损失函数的梯度，得到当前位置下降的距离

    \alpha\frac{\partial}{\partial w_i}l(w_0,w_1,…,w_n)

  3. 更新参数，对于参数w_i，其更新表达式为：

    w_i=w_i-\alpha\frac{\partial}{\partial w_i}l(w_1,w_2,…,w_n,b)=w_i-\alpha \frac{1}{m}\sum_{j=0}^m(h_w(x_0^{(j)},x_1^{(j)},…,x_n^{(j)})-y_i)x_i^{(j)}

    说明：因为是梯度下降，所以这里是减号，如果是梯度上升，那就是加号了！

  4. 确定是否所有的参数w_i梯度下降的距离都小于\epsilon，如果小于\epsilon则算法终止，否则进入步骤3

梯度下降算法家族成员

批量梯度下降

批量梯度下降法是梯度下降法最常用的形式，具体做法也就是在更新参数时使用所有的样本来进行更新，这个方法对应于前面说的线性回归的梯度下降算法。

w_i=w_i-\alpha \frac{1}{m}\sum_{j=0}^m(h_w(x_0^{(j)},x_1^{(j)},…,x_n^{(j)})-y_i)x_i^{(j)}

随机梯度下降

随机梯度下降法和批量梯度下降法原理类似，区别在于求梯度时没有用所有的m个样本的数据，而是选取一个样本来求梯度，对应的参数更新公式为：

w_i=w_i-\alpha(h_w(x_0^{(j)},x_1^{(j)},…,x_n^{(j)})-y_i)x_i^{(j)}

随机梯度下降法和批量梯度下降法是两个极端，一个采用所有数据来梯度下降，一个用一个样本来梯度下降。自然各自的优缺点都非常突出。
对于训练速度来说，随机梯度下降法由于每次仅仅采用一个样本来迭代，训练速度很快，而批量梯度下降法在样本量很大的时候，训练速度不能让人满意。
对于准确度来说，随机梯度下降法用于仅仅用一个样本决定梯度方向，导致解很有可能不是最优。
对于收敛速度来说，由于随机梯度下降法一次迭代一个样本，导致迭代方向变化很大，不能很快的收敛到局部最优解。

小批量梯度下降

小批量梯度下降法是批量梯度下降法和随机梯度下降法的折衷，也就是对于m个样本，采用x个样子来迭代，1<x<m。一般可以取x=10，当然根据样本的数据，可以调整这个x的值。对应的参数更新公式是：

w_i=w_i-\alpha\sum_{j=t}^{t+x-1}(h_w(x_0^{(j)},x_1^{(j)},…,x_n^{(j)})-y_i)x_i^{(j)}

梯度下降法和最小二乘法相比，梯度下降法需要选择步长，而最小二乘法不需要。梯度下降法是迭代求解，最小二乘法是计算解析解。

如果样本量不算很大，且存在解析解，最小二乘法比起梯度下降法要有优势，计算速度很快。但是如果样本量很大，用最小二乘法由于需要求一个超级大的逆矩阵，这时就很难或者很慢才能求解解析解了，使用迭代的梯度下降法比较有优势。

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