From 35be1d6ddcd6c55ef0f295e80a4b83b1d71e3b22 Mon Sep 17 00:00:00 2001
From: endy <endymecy@sina.cn>
Date: Tue, 24 Jan 2017 14:32:44 +0800
Subject: [PATCH] lbfgs

---
 .../L-BFGS/imgs/2.24.png"                     | Bin 0 -> 26684 bytes
 .../L-BFGS/imgs/2.25.svg"                     |  38 ++++++++++
 .../L-BFGS/imgs/2.26.svg"                     |  65 +++++++++++++++++
 .../L-BFGS/imgs/2.27.svg"                     |  66 ++++++++++++++++++
 .../L-BFGS/lbfgs.md"                          |  16 +++++
 5 files changed, 185 insertions(+)
 create mode 100644 "\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/imgs/2.24.png"
 create mode 100644 "\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/imgs/2.25.svg"
 create mode 100644 "\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/imgs/2.26.svg"
 create mode 100644 "\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/imgs/2.27.svg"

diff --git "a/\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/imgs/2.24.png" "b/\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/imgs/2.24.png"
new file mode 100644
index 0000000000000000000000000000000000000000..b6eaef91c088ae916326b243369066eb707380b6
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diff --git "a/\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/lbfgs.md" "b/\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/lbfgs.md"
index e880d2a..0930379 100644
--- "a/\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/lbfgs.md"
+++ "b/\346\234\200\344\274\230\345\214\226\347\256\227\346\263\225/L-BFGS/lbfgs.md"
@@ -141,6 +141,8 @@
 
 ## 2.6 OWL-QN算法
 
+### 2.6.1 L1 正则化
+
 &emsp;&emsp;在机器学习算法中，使用损失函数作为最小化误差，而最小化误差是为了让我们的模型拟合我们的训练数据，此时，
 若参数过分拟合我们的训练数据就会有过拟合的问题。正则化参数的目的就是为了防止我们的模型过分拟合训练数据。此时，我们会在损失项之后加上正则化项以约束模型中的参数：
 
@@ -169,6 +171,20 @@ $$J(x) = l(x) + C ||x||_{2}$$
 &emsp;&emsp;这里有一个问题是，`L1`正则化项不可微，所以无法像求`L-BFGS`那样去求。微软提出了`OWL-QN`(`Orthant-Wise Limited-Memory Quasi-Newton`)算法，该算法是基于`L-BFGS`算法的可用于求解`L1`正则的算法。
 简单来讲，`OWL-QN`算法是指假定变量的象限确定的条件下使用`L-BFGS`算法来更新，同时，使得更新前后变量在同一个象限中(使用映射来满足条件)。
 
+### 2.6.2 OWL-QN算法的具体过程
+
+- <b>1 次微分</b>
+
+&emsp;&emsp;设$f:I\rightarrow R$是一个实变量凸函数，定义在实数轴上的开区间内。这种函数不一定是处处可导的，例如绝对值函数$f(x)=|x|$。但是，从下面的图中可以看出（也可以严格地证明），对于定义域中的任何$x_0$，我们总可以作出一条直线，它通过点($x_0$, $f(x_0)$)，并且要么接触f的图像，要么在它的下方。
+这条直线的斜率称为函数的次导数。
+
+<div  align="center"><img src="imgs/2.24.png" width = "500" height = "400" alt="2.24" align="center" /></div><br>
+
+&emsp;&emsp;凸函数$f:I\rightarrow R$在点$x_0$的次导数，是实数`c`使得：
+
+<div  align="center"><img src="imgs/2.25.svg" width = "500" height = "400" alt="2.25" align="center" /></div><br>
+
+
 # 3 源码解析
 
 ## 3.1 BreezeLBFGS