From 610e937db9457f73bfafcf98dbb38239333c24cd Mon Sep 17 00:00:00 2001
From: SJ2050cn <sj2050@vip.163.com>
Date: Sun, 21 Nov 2021 15:17:37 +0800
Subject: [PATCH] Finish derivativing the formulas of softmax method.

---
 homework_04_logistic_regression/homework/README.md | 12 ++++++++----
 1 file changed, 8 insertions(+), 4 deletions(-)

diff --git a/homework_04_logistic_regression/homework/README.md b/homework_04_logistic_regression/homework/README.md
index 79298f9..16d9756 100644
--- a/homework_04_logistic_regression/homework/README.md
+++ b/homework_04_logistic_regression/homework/README.md
@@ -22,11 +22,13 @@ g(z_i)=g(\theta_i^T \mathbf{x})=\frac{e^{\theta_i^T\mathbf{x}}}{\sum\limits_{j=1
 $$
 构造似然函数，若有$m$个训练样本：
 $$
-\begin{align}
+\begin{equation}
+\begin{split}
 L(\Theta)&=p(\mathbf{y}|\mathbf{X};\Theta) \\
 & = \prod\limits_{i=1}^{m} p(y^{i}|\mathbf{x}^{i};\Theta) \\
 & = \prod_{i=1}^m h_{\theta_i}(\mathbf{x})
-\end{align}
+\end{split}
+\end{equation}
 $$
 对似然函数取对数，转换为：
 $$
@@ -41,10 +43,12 @@ $$
 $$
 转换后的似然函数对$\theta$求偏导，在这里我们以只有一个训练样本的情况为例：
 $$
-\begin{align}
+\begin{equation}
+\begin{split}
 \frac{\partial}{\partial\theta_k}l(\Theta)&=\frac{\partial l(\Theta)}{\partial{z_k}}\cdot \frac{\partial z_k}{\partial \theta_k} \\
 &=(y_k-h_{\theta_k}(\mathbf{x}))\mathbf{x}
-\end{align}
+\end{split}
+\end{equation}
 $$
 上式中$y_k$的表达式如下：
 $$