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 theta and theta_0 ?
To determine the output of theta and theta_0, we need to understand the steps involved in running the kernel perceptron algorithm using the quadratic kernel.
1. Initialize theta and theta_0 to zero.
2. Iterate through each point in the dataset.
3. For each point, compute the quadratic kernel function between the current point and the labeled points seen so far.
- The quadratic kernel function can be defined as k(x, x') = (x·x' + 1)^2, where x and x' represent the input feature vectors.
4. If the sign of the dot product between the current point and the sum of theta and theta_0 is equal to the label, then the prediction is correct, and no update is required.
5. If the prediction is incorrect, update theta and theta_0 using the following formulas:
- theta = theta + label * feature_map(current_point)
- theta_0 = theta_0 + label
6. Continue iterating through all the points in the dataset until there are no more mistakes.
In the given table, we have the number of mistakes made on each point using the kernel perceptron algorithm. We need to utilize this information to determine the final values of theta and theta_0. Here's how we can do it:
1. Initialize theta and theta_0 to zero.
2. Iterate through each point in the dataset, considering the number of mistakes made on each point.
3. Update theta and theta_0 according to the number of mistakes made on each point:
- For each mistake made on a point, subtract the feature map of that point multiplied by the label from theta, and subtract the label from theta_0.
- Repeat this process according to the number of mistakes indicated in the table.
4. After updating for all the mistakes, the final values of theta and theta_0 will represent the output.
Let's calculate the output of theta and theta_0 based on the given table:
- Initial values: theta = [0, 0], theta_0 = 0
1. For the first point, there was 1 mistake.
- Update theta: theta = theta - 1 * feature_map(0, 0)
- Update theta_0: theta_0 = theta_0 - 1
2. For the second point, there were 65 mistakes.
- Update theta: theta = theta - (-1) * 65 * feature_map(2, 0)
- Update theta_0: theta_0 = theta_0 - (-1) * 65
3. Continue this process for all the points, updating theta and theta_0 according to the number of mistakes mentioned in the table.
After performing these calculations, the resulting values of theta and theta_0 will represent the output of the kernel perceptron algorithm applied to the given dataset using the quadratic kernel.
To determine the output of theta and theta_0, we need to understand how the kernel perceptron algorithm updates these parameters.
In the kernel perceptron algorithm, theta represents the weights associated with the support vectors, and theta_0 represents the bias term. These parameters are updated iteratively as new mistakes are made on the training data.
Let's go through the table to update the parameters:
1. Start with theta and theta_0 initialized as zero.
2. For each point in the dataset, update the parameters based on the mistakes made:
a. For points with label -1:
- The first mistake is made on the first point, so theta is updated by adding the coordinates of that point, i.e., (0,0).
- Increment theta_0 by 1.
b. For points with label +1:
- The first mistake is made on the sixth point, so theta is updated by subtracting the coordinates of that point, i.e., (4,1).
- Increment theta_0 by -1.
3. Repeat step 2 for all points in the dataset.
After going through the entire dataset, we end up with the final values of theta and theta_0.
From the table, we can see that the number of mistakes made on each point is displayed. However, it is not clear which mistakes correspond to which points. Without this information, it is difficult to determine the exact values of theta and theta_0.
Therefore, based on the information provided, we cannot determine the exact output of theta and theta_0.