# 如何计算神经网络梯度

## 1. 链式法则求偏导数

### 1.1 偏导数与梯度

$$\frac{dx^2}{dx} = 2x$$

$$\frac{\partial}{\partial x}(x^2+4y+7)=2x \\ \frac{\partial}{\partial y}(x^2+4y+7)=4$$

### 1.2 求向量梯度

$$\frac{\partial}{\partial x _0}{f(w,x)}=w _0 \\ \frac{\partial}{\partial x _1}{f(w,x)}=w _1 \\ … \\ \frac{\partial}{\partial x _{n-1}}{f(w,x)}=w _{n-1}$$

\begin{align} \nabla _x f(w,x) &= \frac{\partial}{\partial x}{f(w,x)} \\ &=\left[ \begin{array}{ccc} \frac{\partial f}{\partial x _0}& \frac{\partial f}{\partial x _1}& \frac{\partial f}{\partial x _2}& …& \frac{\partial f}{\partial x _{n-1}} \end{array} \right]^{\rm T} \\ &=\left[ \begin{array}{ccc} w _0& w _1& w _2& …& w _{n-1} \end{array} \right]^{\rm T} \\ &=w \\ &A^{\rm T}表示A的转置 \end{align}

$$\frac{\partial f(w,x)}{\partial w} = x, \quad \frac{\partial f(w,x)}{\partial b} = 1$$

### 1.3 常用的向量导数

$$\frac{\partial \left(u^{\rm T}v\right)}{\partial u} = v \\ \frac{\partial \left(u^{\rm T}v\right)}{\partial v} = u \\ \frac{\partial \left(x^{\rm T}Ax\right)}{\partial x} = 2Ax \\ \frac{\partial \left(x^Tx\right)}{\partial x} = 2x$$

### 1.4 导数的链式法则（Chain Rule）

$$y=f(g(x)) \\ y=f(u) \\ u=g(x) \\ \frac{\partial y}{\partial x} = \frac{\partial y}{\partial u} * \frac{\partial u}{\partial x}$$

$$y=f(u,v)=u^2+4v \\ u=g(x)=2x \\ v=h(x)=3x \\ \frac{\partial y}{\partial x} = \frac{\partial y}{\partial u} * \frac{\partial u}{\partial x} + \frac{\partial y}{\partial v} * \frac{\partial v}{\partial x}$$

$$x$$有两条路径到$$y$$，因此需要对两条路径求导并累加，这是神经网络求梯度的基础，神经网络的参数到最终的损失函数有多条路径，路径数是“指数级”（每一层的维度相乘），不可能手动一条路径一条路径求导再累加，需要找到一定的规律快速求解。

## 2. 计算神经网络的梯度

### 2.1 神经网络结构

$$W1$$表示输入层到隐藏层的参数，$$b1$$表示对应的偏置量，$$W2$$表示隐藏层到输出层的参数，$$b2$$表示对应的偏置量。根据神经网络的定义$$W1$$的“形状”shape为[6, 5]，$$b1$$为[6, 1]，$$W2$$的shape为[3, 6]，$$b2$$为[3, 1]用$$L0$$表示输入的维度，$$L1$$表示第一层的维度，$$L2$$表示第二层的维度。

### 2.2 神经网络前向传播计算方法

$$z1 = W1*x + b1 \\ a1 = {\rm sigmoid}(z1) \\ z2 = W2 * a1 + b2 \\ \hat{y} = {\rm softmax}(z2)$$
$${\rm softmax}$$定义为
$${\hat y} _i = \frac{\exp(z2 _i)}{\sum _{j=0}^{K-1} \exp(z2 _j)}$$
$${\rm softmax}$$函数的分母是归一化因子，为所有分子项之和。$${\rm softmax}$$函数输入是一个向量，输出是向量各维度的概率分布。

### 2.4 梯度的反向传播

#### 2.4.1 求网络参数$$W2$$和$$b2$$的梯度

$$z2 _0 = W2[0] * a1 + b2 _0\\ z2 _1 = W2[1] * a1 + b2 _1\\ …\\ z2 _{L2-1} = W2[L2-1] * a1 + b2 _{L2-1}$$

\begin{align} \frac{\partial J(\theta)}{\partial b2 _l}&=\frac{\partial J(\theta)}{\partial z2 _l} * \frac{\partial z2 _l)}{\partial b2 _l}\\ &=\frac{\partial J(\theta)}{\partial z2 _l} \quad (\frac{\partial z2 _l)}{\partial b2 _l} = 1) \end{align}
$$\frac{\partial J(\theta)}{\partial b2}=\frac{\partial J(\theta)}{\partial z2}$$

$$W2[0] * a1 + b2 _0 = z2 _0\\ W2[1] * a1 + b2 _1 = z2 _1\\ …\\ W2[L2-1] * a1 + b2 _{L2-1} = z2 _{L2-1}$$

\begin{align} \frac{\partial J(\theta)}{\partial W2[t]} &= \frac{\partial J(\theta)}{\partial z2 _t} * \frac{\partial z2 _t}{\partial W2[t]} \\ &=\frac{\partial J(\theta)}{\partial z2 _t} * a1^{\rm T} \quad (\frac{\partial z2 _t)}{\partial W2[t]} = a1^{\rm T}, W2[t]为行向量)\\ &=(\frac{\partial J(\theta)}{\partial z2 _t}) * a1^{\rm T} \end{align}

$$\frac{\partial J(\theta)}{\partial W2}=\frac{\partial J(\theta)}{\partial z2} * a1^{\rm T}$$

#### 2.4.2 梯度的反向传播到前一层

\begin{align} \frac{\partial J(\theta)}{\partial a1 _t} &= \sum _{i = 0}^{L2 -1}\frac{\partial J(\theta)}{\partial z2 _i} * \frac{\partial z2 _i}{\partial a1 _t}\\ &=\sum _{i = 0}^{L2 – 1}\frac{\partial J(\theta)}{\partial z2 _i} * W2[i][t] \end{align}

\begin{align} \frac{\partial J(\theta)}{\partial a1} &= \pmatrix{\sum _{i = 0}^{L2 – 1} (\frac{\partial J(\theta)}{\partial z2 _i}) * W2[i][0] \cr \sum _{i = 0}^{L2 – 1} (\frac{\partial J(\theta)}{\partial z2 _i}) * W2[i][1] \cr … \cr … \cr … \cr \sum _{i = 0}^{L2 – 1} (\frac{\partial J(\theta)}{\partial z2 _i}) * W2[i][L1-1]} \\ &=W2^{\rm T} * (\frac{\partial J(\theta)}{\partial z2}) \end{align}

$$\frac{\partial J(\theta)}{\partial a1}=W2^{\rm T} * (\frac{\partial J(\theta)}{\partial z2})$$

$$\frac{\partial z}{\partial a} = \pmatrix{ \frac{\partial z _0}{\partial a _0} & \frac{\partial z _0}{\partial a _1} & … & \frac{\partial z _{0}}{\partial a _{n-1}} \cr \frac{\partial z _1}{\partial a _0} & \frac{\partial z _1}{\partial a _1} & … & \frac{\partial z _{1}}{\partial a _{n-1}} \cr …&…&…& \cr \frac{\partial z _{m-1}}{\partial a _0} & \frac{\partial z _{m-1}}{\partial a _1} & … & \frac{\partial z _{m-1}}{\partial a _{n-1}} \cr }$$

\begin{align} \frac{\partial J(\theta)}{\partial a1} = \frac{\partial J(\theta)}{\partial z2} * \frac{\partial z2}{\partial a1} \\ M _{(1, L1)} = M _{(1, L2)} * M _{(L2, L1)} \end{align}

$$(M _{(1, L1)})^{\rm T} = (M _{(1, L2)} * M _{(L2, L1)}) ^{\rm T}$$
$$(M _{(1, L1)})^{\rm T} = (M _{(L2, L1)}) ^{\rm T} * M _{(1, L2)}^{\rm T}$$

#### 2.4.3 张量求导的链式法则

$$x = g(u)\\y = f(u)$$

$$x$$是包含颜色深度的图像，$$u$$经过卷积之后“消掉”颜色深度的图像，函数$$y$$把图像经过一个变换生成向量。有点类似卷积神经网络，但是并不完全一样的结构，这里并没有指定$$g$$和$$f$$具体怎么变换。要计算$$\frac{\partial y}{\partial x}$$，扩展向量对向量导数的Jacobian矩阵，向量对向量的求导，两两求偏导之后构成Jacobian矩阵，张量对张量求导，两个张量元素分别求偏导组成张量，导数张量的shape为因变量张量和自变量张量shape的“拼接”（可想象为笛卡尔积）。例如，张量$$y$$对张量$$x$$求导，导数张量的shape为$$(n, d, w, h)$$，记为$$\frac{\partial y}{\partial x}=T _{(n, d, w, h)}^{y \rightarrow x}$$，下标为$$[i,j,k,l]$$的值$$T _{(n, d, w, h)}^{y \rightarrow x}[i,j,k,l]$$，表示$$y[i]$$对$$x[j,k,l]$$的偏导数。

$$\frac{\partial y}{\partial x}=T _{(n, d, m, n)}^{y \rightarrow x} \\ \frac{\partial y}{\partial u}=T _{(n, p, q)}^{y \rightarrow u}\\ \frac{\partial u}{\partial x}=T _{(p, q, d, w, h)}^{u \rightarrow x}$$

\begin{align} \frac{\partial y}{\partial x} &= \frac{\partial y}{\partial u} * \frac{\partial u}{\partial x} \\ &={\rm tensordot}(T _{(n, p, q)}^{y \rightarrow u}, \quad T _{(p, q, d, w, h)}^{u \rightarrow x}, \quad 2) \end{align}

$${\rm tensordot}(T1, T2, {\rm axes})$$表示两个张量的内积，axes参数指定所需累加的维度，本例中axes为2。axes参数指定了所需累加第一个张量的最后两个维度$$(p,q)$$和第二个张量的前面两个维度，同样也是$$(p,q)$$。这里第一个张量的最后两个维度和第二个张量的前两个维度必须相同，否则不能进行张量的内积。（axes参数的作用参考：numpy tensordot）。最后得到的张量维度为$$(n, d, w, h)$$，符合最开始定义的$$y$$对$$x$$导数张量的shape。张量内积累加中间变量的维度，和最开始定义的链式法则寻找所有路径然后累加有异曲同工之处，只是标量链式法则手动寻找所有路径，张量链式法则使用张量内积累加指定维度，表达更为简洁。

#### 2.4.4 激活层到非激活层的导数

$$\frac{\partial J(\theta)}{\partial z1} = {sigmoid^{\prime} (z1)} \circ \frac{\partial J(\theta)}{\partial a1}$$

$$\frac{\partial J(\theta)}{\partial b1}=\frac{\partial J(\theta)}{\partial z1}\\\frac{\partial J(\theta)}{\partial W1}=\frac{\partial J(\theta)}{\partial z1} * x^{\rm T}$$

#### 2.4.5 梯度反传计算方法总结

1、已知未使用激活函数的网络层$$z$$的梯度，计算该层神经网络的参数$$W$$和$$b$$的梯度

2、已知未使用激活函数的网络层$$z$$的梯度，计算前一网络层使用激活函数的网络层$$a$$的梯度

3、已知使用激活函数的网络层$$a$$的梯度，计算当前网络层未使用激活函数层$$z$$的梯度

### 2.5 求最后一层z的梯度

2.4节说明了梯度的反向传播算法，那么神经网络的最后一层的梯度如何计算呢？答案依然是链式法则求导。

\begin{align} J(\theta) &= -{y _k}*\log({\widehat y _k}) \\ &=-\log\frac{\exp(z2 _k)}{\sum _{j=0}^{K-1} \exp(z2 _j)} \end{align}

\begin{align} \frac {\partial J(\theta)} {\partial z2 _k} &= \frac {\partial} {\partial z2 _k} ({-\log\frac{\exp(z2 _k)}{\sum _{j=0}^{K-1} \exp(z2 _j)}}) \\ &=-\frac{\sum _{j=0}^{K-1} \exp(z2 _j)}{\exp(z2 _k)} * \frac {\partial} {\partial z2 _k} (\frac{\exp(z2 _k)}{\sum _{j=0}^{K-1} \exp(z2 _j)})\\ &=-\frac{\sum _{j=0}^{K-1} \exp(z2 _j)}{\exp(z2 _k)} * \frac{exp(z2) * (\sum _{j=0}^{K-1} \exp(z2 _j)) – \exp(z2 _k) * \exp(z2 _k)}{(\sum _{j=0}^{K-1} \exp(z2 _j))^2} \\ &=-(1-\frac{\exp(z2 _k)}{\sum _{j=0}^{K-1} \exp(z2 _j)}) \\ &=\widehat y _k – 1 \\ &=\widehat y _k – y _k \end{align}

\begin{align} \frac {\partial J(\theta)} {\partial z2 _l} &= \frac {\partial} {\partial z2 _l} ({-\log\frac{\exp(z2 _l)}{\sum _{j=0}^{K-1} \exp(z2 _j)}}) \\ &=-\frac{\sum _{j=0}^{K-1} \exp(z2 _j)}{\exp(z2 _k)} * \frac {\partial} {\partial z2 _l} (\frac{\exp(z2 _k)}{\sum _{j=0}^{K-1} \exp(z2 _j)})\\ &=-\frac{\sum _{j=0}^{K-1} \exp(z2 _j)}{\exp(z2 _k)} * \frac{0 * (\sum _{j=0}^{K-1} \exp(z2 _j)) – \exp(z2 _k) * \exp(z2 _l)}{(\sum _{j=0}^{K-1} \exp(z2 _j))^2} \\ &=\widehat y _l\\ &=\widehat y _l – y _l \qquad (y _l = 0) \end{align}

\begin{align} \frac {\partial J(\theta)} {\partial z2} &= [\frac {\partial J(\theta)} {\partial z2 _0}, \frac {\partial J(\theta)} {\partial z2 _1}, ……,\frac {\partial J(\theta)} {\partial z2 _{K-1}}]^{\rm T}\\ &=[\widehat y _0 – y _0, \widehat y _1 – y _1, ……, \widehat y _{K-1} – y _{K-1}]^{\rm T} \\ &=\widehat y – y \end{align}

### 2.6 思考

1、经典神经网络（只有相邻层全连接，非激活层到激活层一一对应连接，除此之外无其他方式的连接）计算参数梯度的时间复杂度是多少？
3、1.1节中，为什么说梯度方向（反方向）是函数当前点前往极大值（极小值）点的“最优”方向？（“最优”是指学习率确定的情况下，仅改变前进方向，函数值向极值变化最大的方向。这里“最优”方向，并不是真正意义的最优方向，对于凸函数，牛顿法比基于梯度的方法迭代方向更优，但是牛顿法并不适用于神经网络的优化，有两方面原因，1）牛顿法在凸优化能够得到理论上最优解，但是在非凸函数上效果并不理想，神经网络几乎所有情况都不是凸优化；2）牛顿法需要计算Hessian Matrix的逆，矩阵逆算法是三次方的时间复杂度，尽管有近似的方法计算牛顿法的逆矩阵，但是时间复杂度，收敛速度等因素综合起来并不一定比基于梯度的方法更好，关于神经网络和深度模型优化方面的问题建议深入阅读Deep Learning第八章

## “如何计算神经网络梯度”的583个回复

1. layor说道：

1.2那，对x1求偏导，结果写成w0了。

2. Pires说道：

1. 假设一层包括一个非激活层和一个激活层，假设每层有N个神经元，总共有L层，那么时间复杂度是O(L*N*N)
2. 前面层次的梯度是从该层到末层梯度的累乘，如果用sigmoid函数作为激活层，其梯度在0-0.25之间，所以多次乘积下来前面层次的梯度会趋近于0.（所以越来越多人放弃sigmoid函数作为激活层，而选择ReLU激活函数）
3. 梯度的定义就是：在该位置，自变量单位变化的情况下，函数值变化最大的方向。基于这个定义，梯度指出了在每个位置下，最快逼近函数极值的方向，因为学习率（步长）确定，所以它是“最优”方向。

1. 其实ReLU也有梯度激活的问题，会出现大梯度之后突然就为0再也不激活了，所以也有leaked ReLU之类的变种

3. 2.4.2的第二个独立公式中W1应该是W2吧

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