The problem involves finding the parameters $\theta_0$ and $\theta$ corresponding to the maximum margin separator and then calculating the value of the margin attained.

Given: $\theta_0 = -5$ $\theta = [1, 1]$

Maximum Margin Separator

1. Understanding the Margin: The margin for a linear classifier is defined as the distance from the decision boundary (the hyperplane) to the nearest data point. For a linear separator $\theta^T x + \theta_0 = 0$, the margin $\gamma$ can be expressed as: $\gamma = \frac{1}{\|\theta\|}$ where $\|\theta\|$ is the Euclidean norm (or magnitude) of the vector $\theta$.

2. Calculating the Norm: For the given $\theta = [1, 1]$: $\|\theta\| = \sqrt{1^2 + 1^2} = \sqrt{2}$

3. Margin Calculation: The margin $\gamma$ is: $\gamma = \frac{1}{\|\theta\|} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

4. Detailed Explanation:

   • The linear separator is defined by the equation $\theta^T x + \theta_0 = 0$, which can be written as $1 \cdot x_1 + 1 \cdot x_2 - 5 = 0$ or $x_1 + x_2 = 5$.
   • The maximum margin is the perpendicular distance from this hyperplane to the nearest point.
   • The margin is inversely proportional to the norm of $\theta$. Therefore, $\gamma = \frac{1}{\|\theta\|} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

Thus, the value of the margin attained is: $\boxed{\frac{\sqrt{2}}{2} \approx 0.707}$

Summary

• $\theta_0 = -5$
• $\theta = [1, 1]$
• The margin attained is $\frac{\sqrt{2}}{2}$, which is approximately 0.707 when rounded to two decimal places.

Would you like any more details or have any questions?

Here are some questions you might want to ask:

1. How do you find the Euclidean norm of a vector?
2. What is the significance of maximizing the margin in SVM?
3. How does the perceptron algorithm differ from SVM?
4. Can the margin be negative?
5. What are the applications of support vector machines?

Tip: When working with linear classifiers, remember that maximizing the margin helps improve the classifier's generalization performance.