logistic回归的参数梯度更新方法的个人理解
logistic回归参数更新看了几篇博文，感觉理解不透彻，所以自己写一下，希望能有更深的理解。logistic回归输入是一个线性函数Wx+b\boldsymbol{W}\boldsymbol{x}+\boldsymbol{b}Wx+b，为了简单理解，考虑batchsize为1的情况。这时输入x\boldsymbol{x}x为一个n×1n\times1n×1的向量，标签y\boldsymbol{y}y我们采用oneHot编码为一个m×1m\times1m×1的向量，显然\boldsymbol{b}也是一个m×1m\times1m×1的向量，参数W\boldsymbol{W}W为一个m×nm\times nm×n的矩阵。若n=4n=4n=4、m=3m=3m=3，我们用图形表示logistic回归如下：
这里的标签y\boldsymbol{y}y采用onehot编码，长度为3，如果类别编号为1，则其编码为{1,0,0}T\{1,0,0\}^T{1,0,0}T，对应上图的话，就是y∗1=1y_*^1=1y∗1=1，y∗2=0y_*^2=0y∗2=0，y∗3=0y_*^3=0y∗3=0。损失函数LLL就是y1y^1y1和y∗1y_*^1y∗1的交叉熵损失+y2y^2y2和y∗2y_*^2y∗2的交叉熵损失+y3y^3y3和y∗3y_*^3y∗3的交叉熵损失。
L=∑i=13y∗ilogyi=y∗1logy1+y∗2logy2+y∗3logy3\begin{aligned} L&=\sum_{i=1}^3y^i_*\log{y^i}\\ &=y^1_*\log{y^1}+y^2_*\log{y^2}+y^3_*\log{y^3} \end{aligned} L=i=1∑3y∗ilogyi=y∗1logy1+y∗2logy2+y∗3logy3
上式中：
y1=ez1ez1+ez2+ez3y2=ez2ez1+ez2+ez3y3=ez3ez1+ez2+ez3\begin{aligned} y^1&=\frac{e^{z^1}}{e^{z^1}+e^{z^2}+e^{z^3}}\\ y^2&=\frac{e^{z^2}}{e^{z^1}+e^{z^2}+e^{z^3}}\\ y^3&=\frac{e^{z^3}}{e^{z^1}+e^{z^2}+e^{z^3}}\\ \end{aligned} y1y2y3=ez1+ez2+ez3ez1=ez1+ez2+ez3ez2=ez1+ez2+ez3ez3
而
z1=w1Tx+b1z2=w2Tx+b2z3=w3Tx+b3\begin{aligned} z^1=\boldsymbol{w_1}^T \boldsymbol{x}+b_1\\ z^2=\boldsymbol{w_2}^T \boldsymbol{x}+b_2\\ z^3=\boldsymbol{w_3}^T \boldsymbol{x}+b_3 \end{aligned} z1=w1Tx+b1z2=w2Tx+b2z3=w3Tx+b3
其中，w1={w11,w12,w13,w14}T\boldsymbol{w_1}=\{w_{11},w_{12},w_{13},w_{14}\}^Tw1={w11,w12,w13,w14}T，x={x1,x2,x3,x4}T\boldsymbol{x}=\{x_{1},x_{2},x_{3},x_{4}\}^Tx={x1,x2,x3,x4}T因此：
损失函数LLL对w1\boldsymbol{w_1}w1求导：
∂L∂w1=∂L∂y1∂y1∂z1∂z1∂w1+∂L∂y2∂y2∂z1∂z1∂w1+∂L∂y3∂y3∂z1∂z1∂w1=y1∗y1×y1(1−y1)×x−y2∗y2×y1y2×x−y3∗y3×y1y3×x=(y1∗(1−y1)−y2∗y1−y3∗y1)x=(y1∗−y1(y1∗+y2∗+y3∗))x=(y1∗−y1)x\begin{aligned} \frac{\partial L}{\partial \boldsymbol{w_1}}&=\frac{\partial L}{\partial y_1}\frac{\partial y_1}{\partial z^1}\frac{\partial z^1}{\partial \boldsymbol{w_1}}+\frac{\partial L}{\partial y_2}\frac{\partial y_2}{\partial z^1}\frac{\partial z^1}{\partial \boldsymbol{w_1}}+\frac{\partial L}{\partial y_3}\frac{\partial y_3}{\partial z^1}\frac{\partial z^1}{\partial \boldsymbol{w_1}}\\ &=\frac{y_1^*}{y_1}\times y_1(1-y_1)\times \boldsymbol{x}-\frac{y_2^*}{y_2}\times y_1y_2\times \boldsymbol{x}-\frac{y_3^*}{y_3}\times y_1y_3\times \boldsymbol{x}\\ &=(y_1^*(1-y_1)-y_2^*y_1-y_3^*y_1)\boldsymbol{x}\\ &=(y_1^*-y_1(y_1^*+y_2^*+y_3^*))\boldsymbol{x}\\ &=(y_1^*-y_1)\boldsymbol{x}\\ \end{aligned} ∂w1∂L=∂y1∂L∂z1∂y1∂w1∂z1+∂y2∂L∂z1∂y2∂w1∂z1+∂y3∂L∂z1∂y3∂w1∂z1=y1y1∗×y1(1−y1)×x−y2y2∗×y1y2×x−y3y3∗×y1y3×x=(y1∗(1−y1)−y2∗y1−y3∗y1)x=(y1∗−y1(y1∗+y2∗+y3∗))x=(y1∗−y1)x
注意(y1∗+y2∗+y3∗)(y_1^*+y_2^*+y_3^*)(y1∗+y2∗+y3∗)是标签onehot编码的三个值，和正好为1。同理可得到剩下的两个导数：
∂L∂w2=(y2∗−y2)x∂L∂w3=(y3∗−y3)x\frac{\partial L}{\partial \boldsymbol{w_2}} = (y_2^*-y_2)\boldsymbol{x}\\ \frac{\partial L}{\partial \boldsymbol{w_3}} = (y_3^*-y_3)\boldsymbol{x} ∂w2∂L=(y2∗−y2)x∂w3∂L=(y3∗−y3)x
交叉熵损失函数LLL关于w\boldsymbol{w}w的梯度为：
[(y1∗−y1)x1(y2∗−y2)x1(y3∗−y3)x1(y1∗−y1)x2(y2∗−y2)x2(y3∗−y3)x2(y1∗−y1)x3(y2∗−y2)x3(y3∗−y3)x3(y1∗−y1)x4(y2∗−y2)x4(y3∗−y3)x4(y1∗−y1)x5(y2∗−y2)x5(y3∗−y3)x5]T\left[ \begin{aligned} &(y_1^*-y_1)x1&(y_2^*-y_2)x1\space\space\space\space&(y_3^*-y_3)x1\\ &(y_1^*-y_1)x2&(y_2^*-y_2)x2\space\space\space\space&(y_3^*-y_3)x2\\ &(y_1^*-y_1)x3&(y_2^*-y_2)x3\space\space\space\space&(y_3^*-y_3)x3\\ &(y_1^*-y_1)x4&(y_2^*-y_2)x4\space\space\space\space&(y_3^*-y_3)x4\\ &(y_1^*-y_1)x5&(y_2^*-y_2)x5\space\space\space\space&(y_3^*-y_3)x5\\ \end{aligned} \right]^T (y1∗−y1)x1(y1∗−y1)x2(y1∗−y1)x3(y1∗−y1)x4(y1∗−y1)x5(y2∗−y2)x1 (y2∗−y2)x2 (y2∗−y2)x3 (y2∗−y2)x4 (y2∗−y2)x5 (y3∗−y3)x1(y3∗−y3)x2(y3∗−y3)x3(y3∗−y3)x4(y3∗−y3)x5T
这样交叉熵损失函数LLL关于w\boldsymbol{w}w的梯度用numpy的外积计算表示为：
∂L∂w=numpy.outer(x,y∗−y)\frac{\partial L}{\partial \boldsymbol{w}}=numpy.outer(\boldsymbol{x},\boldsymbol{y^*}-\boldsymbol{y}) ∂w∂L=numpy.outer(x,y∗−y)
用同样的方法可以推导出：
∂L∂b=y∗−y\frac{\partial L}{\partial \boldsymbol{b}}=\boldsymbol{y^*}-\boldsymbol{y} ∂b∂L=y∗−y
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