title: 关于吴恩达机器学习中反向传播的理解
date: 2022-04-12
categories: 数学
mathjax: true
tags:
- 机器学习
- 线性代数
原文
在机器学习视频反向传播章节[1]中:
我们用 \(\delta\) 来表示误差,则: \(\boldsymbol\delta^{\left(4\right)}=\boldsymbol a^{\left(4\right)}−\boldsymbol y\) 。我们利用这个误差值来计算前一层的误差:
\(\boldsymbol\delta^{\left(3\right)}=\left(\boldsymbol\Theta^{\left(3\right)}\right)^T\boldsymbol\delta^{\left(4\right)}\cdot g^\prime\left(\boldsymbol z^{\left(3\right)}\right)\) 。其中 \(g^\prime\left(\boldsymbol{z}^{\left(3\right)}\right)\) 是 \(S\) 形函数的导数,
\(g^\prime\left(\boldsymbol z^{\left(3\right)}\right)=\boldsymbol a^{\left(3\right)}\cdot\left(1−\boldsymbol a^{\left(3\right)}\right)\) 。而 \(\left(\boldsymbol\Theta^{\left(3\right)}\right)^T\boldsymbol\delta^{\left(4\right)}\) 则是权重导致的误差的和。
问题
\[\boldsymbol\delta^{\left(3\right)}=\left(\boldsymbol\Theta^{\left(3\right)}\right)^T\boldsymbol\delta^{\left(4\right)}\cdot g^\prime\left(\boldsymbol z^{\left(3\right)}\right) \]看到这道算式时我百思不得其解。为什么凭空会有转置?
在我自己推一遍之后,发现原公式中可能有些不严谨的地方,所以在此阐述我的理解,欢迎大家指正:
前提
对数似然代价函数: \(J\left(\Theta\right)=y\ln h_\Theta\left(x\right)+\left(1-y\right)\ln\left(1-h_\Theta\left(x\right)\right)\)
估计函数: \(h_\Theta\left(x\right)=\sum_i\Theta_ix_i= \begin{bmatrix}\Theta_1&\Theta_2&\cdots&\Theta_n\end{bmatrix} \begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix}\)
Logistic
激活函数: \(g\left(x\right)=\frac1{1+{\rm e}^{-x}}\)
此外激活函数导数为: \(g^\prime\left(x\right)=g\left(x\right)\left[1-g\left(x\right)\right]\)
我的理解
flowchart LR x1--"(Θ<sub>1</sub><sup>(1)</sup>)<sub>1</sub>"-->z12 x1--"(Θ<sub>1</sub><sup>(1)</sup>)<sub>2</sub>"-->z22 x2--"(Θ<sub>2</sub><sup>(1)</sup>)<sub>1</sub>"-->z22 x2--"(Θ<sub>2</sub><sup>(1)</sup>)<sub>2</sub>"-->z12 a12--"(Θ<sub>1</sub><sup>(2)</sup>)<sub>1</sub>"-->z13 a12--"(Θ<sub>1</sub><sup>(2)</sup>)<sub>2</sub>"-->z23 a22--"(Θ<sub>2</sub><sup>(2)</sup>)<sub>1</sub>"-->z23 a22--"(Θ<sub>2</sub><sup>(2)</sup>)<sub>2</sub>"-->z13 z12--g-->a12 z22--g-->a22 z13--g-->a13 z23--g-->a23 a13-.->y1-.->j a23-.->y2-.->j subgraph x x1((x<sub>1</sub>)) x2((x<sub>2</sub>)) end subgraph 第一层 direction LR z12(("z<sub>1</sub><sup>(2)</sup>")) a12(("a<sub>1</sub><sup>(2)</sup>")) z22(("z<sub>2</sub><sup>(2)</sup>")) a22(("a<sub>2</sub><sup>(2)</sup>")) end subgraph 第二层 z13(("z<sub>1</sub><sup>(3)</sup>")) a13(("a<sub>1</sub><sup>(3)</sup>")) z23(("z<sub>2</sub><sup>(3)</sup>")) a23(("a<sub>2</sub><sup>(3)</sup>")) end subgraph y y1((ŷ<sub>1</sub>)) y2((ŷ<sub>2</sub>)) end j(("J(θ)"))如图(省略了偏置),输入数据为 \(\boldsymbol x=\begin{bmatrix}x_1\\x_2\end{bmatrix}\) ,实际输出为 \(\boldsymbol y=\begin{bmatrix}y_1\\y_2\end{bmatrix}\)
这张图上表示了所有的运算,例如:
\[a_1^{\left(2\right)}=g\left(z_1^{\left(2\right)}\right) \]\[z_2^{\left(2\right)}=\left(\Theta_1^{\left(1\right)}\right)_2x_1+\left(\Theta_2^{\left(1\right)}\right)_2x_2 \]同时,此图认为预测输出为 \(\hat y_1=a_1^{\left(3\right)}\) ,即有误差(注意此处不是定义而是结论):
\[\delta_1^{\left(3\right)}=\hat y_1-y_1=a_1^{\left(3\right)}-y_1 \]下面我们将上列函数改写成对应元素的写法,先作定义:
-
\(L\) :被 \(\Theta\) 作用的层
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\(m\) : \(L\) 层单元数量,用 \(j\) 进行遍历(即 \(j\in\left\{1,2,\cdots,m\right\}\) )
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\(n\) : \(L+1\) 层单元数量,用 \(i\) 进行遍历
推导
综上可得,若 \(L\) 是倒数第二层,则给出定义:
\[\begin{align*}\delta_i^{\left(L+1\right)} &=\frac{\partial J}{\partial z_i^{\left(L+1\right)}}\\ &=\frac{\partial J}{\partial a_i^{\left(L+1\right)}}&&\cdot \frac{\partial a_i^{\left(L+1\right)}}{\partial z_i^{\left(L+1\right)}}\\ &=\left(\frac{-y_i}{a_i^{\left(L+1\right)}}+\frac{1-y_i}{1-a_i^{\left(L+1\right)}}\right)&&\cdot g^\prime z_i^{\left(L+1\right)}\\ &=\left(\frac{-y_i}{a_i^{\left(L+1\right)}}+\frac{1-y_i}{1-a_i^{\left(L+1\right)}}\right)&&\cdot a_i^{\left(L+1\right)}\left(1-a_i^{\left(L+1\right)}\right)\\ &=a_i^{\left(L+1\right)}-y_i \end{align*}\]将同一层 \(\delta_i^{\left(L+1\right)}\) 合并为矩阵得( \(\boldsymbol\delta,\boldsymbol a,\boldsymbol y\) 都是列向量):
\[\boldsymbol\delta^{\left(L+1\right)}=\boldsymbol a^{\left(L+1\right)}-\boldsymbol y \]下面推隐含层,以第一个单元为例:
\[\begin{align*} \delta_1^{\left(2\right)}&=\frac{\partial J}{\partial z_1^{\left(2\right)}}\\ &=\frac{\partial J}{\partial z_1^{\left(3\right)}}&& \cdot\frac{\partial z_1^{\left(3\right)}}{\partial a_1^{\left(2\right)}}&& \cdot\frac{\partial a_1^{\left(2\right)}}{\partial z_1^{\left(2\right)}}&&+ \frac{\partial J}{\partial z_2^{\left(3\right)}}&& \cdot\frac{\partial z_2^{\left(3\right)}}{\partial a_1^{\left(2\right)}}&& \cdot\frac{\partial a_1^{\left(2\right)}}{\partial z_1^{\left(2\right)}}\\ &=\delta_1^{\left(3\right)}&& \cdot\left(\Theta_1^{\left(2\right)}\right)_1&& \cdot g^\prime z_1^{\left(2\right)}&&+ \delta_2^{\left(3\right)}&& \cdot\left(\Theta_1^{\left(2\right)}\right)_2&& \cdot g^\prime z_1^{\left(2\right)} \end{align*}\]令:
\[\left\{\begin{align*} \boldsymbol\delta^{\left(L\right)}&=\begin{bmatrix}\delta_1^{\left(L\right)}\\\delta_2^{\left(L\right)}\\\vdots\\\delta_n^{\left(L\right)}\end{bmatrix}\\ \boldsymbol\Theta_i^{\left(L\right)}&=\begin{bmatrix} \left(\Theta_i^{\left(L\right)}\right)_1& \left(\Theta_i^{\left(L\right)}\right)_2& \cdots& \left(\Theta_i^{\left(L\right)}\right)_n \end{bmatrix}\end{align*}\right.\]可将上式化为矩阵:
\[\delta_1^{\left(2\right)} =\boldsymbol\Theta_1^{\left(2\right)}\boldsymbol\delta^{\left(3\right)} \cdot g^\prime z_1^{\left(2\right)}\]结论
由上,可写出递推普式:
\[\delta_j^{\left(L\right)} =\boldsymbol\Theta_j^{\left(L\right)}\boldsymbol\delta^{\left(L+1\right)}\cdot g^\prime z_j^{\left(L\right)}\]其中最后一层:
\[\boldsymbol\delta^{\left(Last\right)}=\boldsymbol a^{\left(Last\right)}-\boldsymbol y \]