Thorem of SVM part 2

Introduction

日本語 ver

This post is written about theorem of SVM.
This post is part 2. Part 1 is written about deriving the objective function. I will write about dual problem. The objective function which deriving in Part 1 is called the main problem. Typically, We optimize not the main problem, but the dual problem. I will write about deriving the dual problem from the main problem.

If you until look up part 1, please look up
Theorem of SVM part 1.

I implement SVM. It post is
Implement linear SVM
Implement kernel SVM

Overview

• Main problem
• Dual problem 
• Lagurange function
• maximize L about dual variable, minimize L about primal variable
• minimize L about primal variable, maximize L about dual variable.

Main problem
I will review the main problem in Part 1.

$$\min_{w,b} \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i$$

$$~~s.t~~ \forall i \in N, y_i (w^T \phi(x_i) + b) \geq 1 - \epsilon_i ,~~~, \forall i \in N~\epsilon \geq 0$$

This problem is called soft margin. I will handle soft margin of the main problem.

Dual problem is naturally derived from the main problem.

Dual problem

Optimization problem of SVM is a kind of convex quadratic optimization problem. Convex quadratic optimization problem is a kind of optimizatio plroblem. It is easy to solve convex quadratic optimization problem. It is found the global optimal solution, because the objectice function is convex.


I will drive dual problem. Firstly, I change the main problem into

$$\min_{w,b} \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i$$

$$~~s.t~~ \forall i \in N, -\{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \}\leq 0,~~~, \forall i \in N~ -\epsilon \leq 0$$

Lagurange function
Secondly, I will introduce new vector $\alpha,\mu$ which has element of non-negative paremeter to define Laguranju function. Lagrange function is

$$L(w,b,\epsilon,\alpha,\mu) := \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i -\sum_{i \in {1,2..n}} \alpha_i \{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \} - \sum_{i \in {1,2,..,n}} \mu_i \epsilon_i$$

Here, $\alpha = (\alpha_1,\alpha_2,...,\alpha_n)$ and $\mu = (\mu_1,\mu_2,..,\mu_n)$.
The paremeter $w,b,\epsilon$ is called primal variable and paremeter $\alpha,\mu$ is called dual variable.

maximize L about dual variable, minimize L about primal variable


Now, I define $P(w,b,\epsilon)$ as maximizing $\alpha,\mu$ of L.

$$P(w,b,\epsilon):= \max_{\alpha \geq 0, \mu \geq 0} L(w,b,\epsilon,\alpha,\mu)$$

I minimize P about primal variable.

$$\min_{w,b,\epsilon} P(w,b,\epsilon) = \min_{w,b,\epsilon}\max_{\alpha \geq 0, \mu \geq 0} L(w,b,\epsilon,\alpha,\mu)$$

Optimizing this problem is same as optimizing the main problem.
I will explain it.

I change $\min_{w,b,\epsilon} P(w,b,\epsilon)$ into

$$\min_{w,b,\epsilon} P(w,b,\epsilon) = \min_{w,b,\epsilon}\max_{\alpha \geq 0, \mu \geq 0} L(w,b,\epsilon,\alpha,\mu)$$
$$= \min_{w,b} \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i + \max_{\alpha \geq 0,\mu \geq 0} \{-\sum_{i \in {1,2..n}} \alpha_i \{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \} - \sum_{i \in {1,2,..,n}} \mu_i \epsilon_i \}$$

because, first item and second item is not relate $\alpha,\mu$.

The main problem is

$$\min_{w,b} \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i$$

$$~~s.t~~ \forall i \in N, -\{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \}\leq 0,~~~, \forall i \in N~ -\epsilon \leq 0$$


Now,
If $\exists ~i \in N ~~s.t.~~ -\{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \} >  0$,  or $-\epsilon > 0$, We can not optimize $\min_{w,b,\epsilon} P(w,b,\epsilon)$, because We can increase $P(w,b,\epsilon)$ to $\infty$ by increase $\alpha,\mu$.

If $\forall ~i \in N, -\{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \}\leq 0$ and $~ -\epsilon \leq 0$ ,

$$\max_{\alpha \geq 0,\mu \geq 0} \{-\sum_{i \in {1,2..n}} \alpha_i \{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \} - \sum_{i \in {1,2,..,n}} \mu_i \epsilon_i \} = 0$$

then, $$\min_{w,b} \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i + \max_{\alpha \geq 0,\mu \geq 0} \{-\sum_{i \in {1,2..n}} \alpha_i \{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \} - \sum_{i \in {1,2,..,n}} \mu_i \epsilon_i \}$$
$$= \min_{w,b} \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i$$

this is the main problem.

minimize L about primal variable, maximize L about dual variable.

Next I define $D(\alpha,\mu)$ as optimize primal variable.

$$D(\alpha,\mu) := \min_{w,b,\epsilon,\alpha,\mu} L(w,b,\epsilon,\alpha,\mu)$$

I miximize $D(\alpha,\mu)$ about $\alpha,\mu$.
$$\max_{\alpha,\mu} D(\alpha,\mu) = \max_{\alpha,\mu} \min_{w,b,\epsilon,\alpha,\mu} L(w,b,\epsilon,\alpha,\mu)$$

This problem is called the dual problem.

About $\min_{w,b,\epsilon,\alpha,\mu} L(w,b,\epsilon,\alpha,\mu)$,
Partial differential is

$$\frac{\partial L}{\partial w} = w - \sum_{i \in {1,2,..,n}} \alpha_i y_i x_i = 0$$
$$\frac{\partial L}{\partial b} = - \sum_{i \in {1,2,..,n}} \alpha_i y_i = 0$$
$$\frac{\partial L}{\partial \epsilon} = C - \alpha_i - \mu_i = 0$$

then, L is
\begin{eqnarray*}
L &=& \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i -\sum_{i \in {1,2..n}} \alpha_i \{y_i (w^T \phi(x_i) + b) -1 + \epsilon_i \} - \sum_{i \in {1,2,..,n}} \mu_i \epsilon_i\\
&=& \frac{1}{2}||W||^2 + C\sum_{i \in N} \epsilon_i -\sum_{i \in {1,2,..,n}} \alpha_i y_i w^T \phi(x) + \alpha_i y_i b - \alpha_i + \alpha_i \epsilon_i - \sum_{i \in {1,2,..,n}} \mu_i \epsilon_i\\
&=& \frac{1}{2}||W||^2 - \sum_{i \in {1,2,..,n}} \alpha_i y_i w^T x_i - b \sum_{i \in {1,2,..,n}} \alpha_i y_i + \sum_{i \in {1,2,..,n}} \alpha_i + \sum_{i \in {1,2,..,n}} (C - \alpha_i - \mu_i) \epsilon_i
\end{eqnarray*}

I substitute partial differentials.
$$-\frac{1}{2} \sum_{i,j \in {1,2,..,n}} \alpha_i \alpha_j y_i y_j \phi(x_i)^T \phi(x_j) + \sum_{i \in {1,2,..,n}} \alpha_i$$

Now,
\begin{eqnarray*}
C - \alpha_i - \mu_i &=& 0 \\
C - \alpha_i &=& \mu \geq 0\\
C - \alpha_i & \geq & 0
\end{eqnarray*}

I get this dual problem.

$$\max_{\alpha} -\frac{1}{2} \sum_{i,j \in {1,2,..,n}} \alpha_i \alpha_j y_i y_j \phi(x_i)^T \phi(x_j) + \sum_{i \in {1,2,..,n}} \alpha_i$$
$$s.t. \sum_{i \in {1,2,..,n}} \alpha_i y_i = 0 ~~~ , 0 \leq \alpha_i \leq C$$

$$\min_{\alpha} \frac{1}{2} \sum_{i,j \in {1,2,..,n}} \alpha_i \alpha_j y_i y_j \phi(x_i)^T \phi(x_j) - \sum_{i \in {1,2,..,n}} \alpha_i$$
$$s.t. \sum_{i \in {1,2,..,n}} \alpha_i y_i = 0 ~~~ , 0 \leq \alpha_i \leq C$$


Reference
https://www.amazon.co.jp/%E3%82%B5%E3%83%9D%E3%83%BC%E3%83%88%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E3%83%9E%E3%82%B7%E3%83%B3-%E6%A9%9F%E6%A2%B0%E5%AD%A6%E7%BF%92%E3%83%97%E3%83%AD%E3%83%95%E3%82%A7%E3%83%83%E3%82%B7%E3%83%A7%E3%83%8A%E3%83%AB%E3%82%B7%E3%83%AA%E3%83%BC%E3%82%BA-%E7%AB%B9%E5%86%85-%E4%B8%80%E9%83%8E/dp/4061529064