2. (1)
1 point possible (graded, results hidden)
If we again use the linear perceptron algorithm to train the classifier, what will happen?
Note: In the choices below ,“converge" means given a certain input, the algorithm will terminate with a fixed output within finite steps (assume is very large: the output of the algorithm will not change as we increase ). Otherwise we say the algorithm diverges (even for an extremely large , the output of the algorithm will change as we increase further).
The algorithm always converges and we get a classifier that perfectly classifies the training dataset.
The algorithm always converges and we get a classifier that does not perfectly classifies the training dataset.
The algorithm will never converge.
The algorithm might converge for some initial input of and certain sequence of the data, but will diverge otherwise. When it converges, we always get a classifier that does not perfectly classifies the training dataset.
The algorithm might converge for some initial input of and certain sequence of the data, but will diverge otherwise. When it converges, we always get a classifier that perfectly classifies the training dataset.
2. (2)
2 points possible (graded, results hidden)
We decide to run the kernel perceptron algorithm over this dataset using the quadratic kernel. The number of mistakes made on each point is displayed in the table below. (These points correspond to those in the plot above.)
Label -1 -1 -1 -1 -1 +1 +1 +1 +1 +1
Coordinates (0,0) (2,0) (1,1) (0,2) (3,3) (4,1) (5,2) (1,4) (4,4) (5,5)
Perceptron mistakes 1 65 11 31 72 30 0 21 4 15
Define the feature map of our quadratic kernel to be:
Assume all parameters are set to zero before running the algorithm.
Based on the table, what is the output of and ?
(Enter accurate to at least 2 decimal places.)
theta_0=
theta=
(Enter as a vector, enclosed in square brackets, and components separated by commas, e.g. type [0,1] for . Note that this sample vector input may not be of the same dimension of the answer. Enter each component accurate to at least 2 decimal places.)
2. (3)
1 point possible (graded, results hidden)
Based on the calculation of and , does the decision boundary correctly classify all the points in the training dataset?
2. (4)
1 point possible (graded, results hidden)
Recall for x=[x_1,x_2]^T
Define the kernel function
phi(x)=[x_1^2,sqrt(2)*x_1*x_2,x_2^2]
Write as a function of the dot product . To answer, let , and enter in terms of .
K(x,x^')
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2. (1)
The algorithm will never converge.
2. (2)
theta_0= 5.00
theta= [-24.00, -12.00, -16.00]
2. (3)
Yes, the decision boundary correctly classifies all the points in the training dataset.
2. (4)
K(x,x^') = x^T x^' = x_1^2 * x_1'^2 + 2 * x_1 * x_2 * x_1' * x_2' + x_2^2 * x_2'^2
2. (1) The correct answer is: The algorithm might converge for some initial input of θ and certain sequence of the data, but will diverge otherwise. When it converges, we always get a classifier that does not perfectly classifies the training dataset.
2. (2) Based on the table, the output of θ₀ is 0.36 and the output of θ is [0.08, 0.28, -0.28].
2. (3) No, the decision boundary does not correctly classify all the points in the training dataset.
2. (4) The value of K(x, x') is:
K(x, x') = x₁² + √2x₁x₂ + x₂²
1) Will never converge
To answer the first question, we need to understand the linear perceptron algorithm. The linear perceptron algorithm is an iterative algorithm used to train a classifier to separate linearly separable data. The algorithm tries to find a hyperplane that separates the data points of different classes. It starts with a random weight vector and updates it based on the misclassified data points until all points are correctly classified or a maximum number of iterations is reached.
In the case of the linear perceptron algorithm, it is not guaranteed that the algorithm will converge and find a weight vector that perfectly classifies the training dataset. The algorithm might diverge, meaning that even for an extremely large number of iterations, the weight vector will keep changing. Therefore, the correct answer is:
- The algorithm might converge for some initial input of eta and certain sequence of the data, but will diverge otherwise. When it converges, we always get a classifier that does not perfectly classify the training dataset.
Now, let's move on to the second question about the quadratic kernel perceptron algorithm.
The quadratic kernel perceptron algorithm is an extension of the linear perceptron algorithm that uses a quadratic kernel function to map the input data into a higher dimensional feature space before applying the perceptron algorithm. The kernel function is defined as K(x, x') = (1 + <x, x'>)^2.
In the given table, the number of mistakes made by the quadratic kernel perceptron algorithm on each point is displayed. We need to use this information to calculate the values of theta_0 and theta for the quadratic kernel perceptron algorithm.
Theta_0 is the bias term and can be calculated as the negative sum of the mistakes made on all the points. In this case, theta_0 = -(1 + 65 + 11 + 31 + 72 + 30 + 0 + 21 + 4 + 15) = -250.
Theta, the weight vector, is calculated based on the mistakes made on each point. We can initialize theta as a zero vector and update it by adding the product of the mistake and the corresponding feature map value for each point. Using the feature map definition, we have:
Feature map of (0,0) = [0^2, sqrt(2)*0*0, 0^2] = [0, 0, 0]
Theta = theta + mistake * feature map = [0, 0, 0] + 1 * [0, 0, 0] = [0, 0, 0]
Following the same calculation for all the points, we get the following values for theta:
Theta = [0, 0, 0]
Therefore, the output of theta_0 is -250 and the output of theta is [0, 0, 0].
Now, let's move on to the third question about the decision boundary and the classification of the points in the training dataset.
To determine if the decision boundary correctly classifies all the points in the training dataset, we need to evaluate the sign of the decision function for each point. The decision function is defined as:
f(x) = sign(theta_0 + sum(mistake_i * K(x_i, x)))
For a correct classification, the sign of f(x) should be equal to the actual label of the point. We can check this for each point in the training dataset using the calculated values of theta_0, theta, and the quadratic kernel function.
Finally, let's move on to the fourth question about expressing K(x, x') in terms of the dot product.
According to the definition of the feature map phi(x) = [x_1^2, sqrt(2)*x_1*x_2, x_2^2], we can express the dot product of two feature maps as:
Phi(x) · Phi(x') = [x_1^2, sqrt(2)*x_1*x_2, x_2^2] · [x_1'^2, sqrt(2)*x_1'*x_2', x_2'^2]
= x_1^2 * x_1'^2 + sqrt(2)*x_1*x_2 * sqrt(2)*x_1'*x_2' + x_2^2 * x_2'^2
= x_1^2 * x_1'^2 + 2*x_1*x_2*x_1'*x_2' + x_2^2 * x_2'^2
Therefore, K(x, x') can be expressed in terms of the dot product as:
K(x, x') = (1 + x · x')^2 = (1 + x_1*x_1' + x_2*x_2')^2
I hope this helps clarify the concepts and solve the given questions. Let me know if there's anything else I can assist you with!