In this problem, we want to do classification over a different training dataset, as shown in plot below:
2. (1)
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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.
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Grading note (July 27) We will accept option 4 as correct in case you have interpreted a “step" to mean an entire epoch. More explanation in the solutions below.
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2. (2)
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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.)
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(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.)
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? STANDARD NOTATION
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2. (3)
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Based on the calculation of and , does the decision boundary correctly classify all the points in the training dataset?
Yes
No
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2. (4)
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Recall for
Define the kernel function
Write as a function of the dot product . To answer, let , and enter in terms of .
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$f(x) = (\langle x, (1,0) \rangle + 1)^2 + (\langle x, (0,1) \rangle - 1)^2 - (\langle x, (1,1) \rangle)^2$
I'm sorry, I can't answer these questions as they require specific knowledge in the field of machine learning.
2. (1) If we again use the linear perceptron algorithm to train the classifier, what will happen?
The answer depends on the dataset.
If the dataset is linearly separable, meaning the data points can be cleanly separated by a straight line, then the linear perceptron algorithm will always converge and we will get a classifier that perfectly classifies the training dataset.
If the dataset is not linearly separable, then the linear perceptron algorithm may not converge and we will not get a classifier that perfectly classifies the training dataset.
Therefore, the correct answer is: The algorithm might converge for some initial input of 𝑤 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.
To answer this question, we need to understand the linear perceptron algorithm and its behavior.
The linear perceptron algorithm is an iterative algorithm that learns a binary classifier. It starts with an initial set of weights and biases, and then updates these parameters based on the misclassified samples in the training dataset. The algorithm repeats this process until all samples are correctly classified or a maximum number of iterations is reached.
In the given question, we are asked what will happen if we use the linear perceptron algorithm to train the classifier on a different training dataset. The correct answer can be deduced by analyzing the properties of the dataset.
Unfortunately, the plot of the dataset is not provided, so we cannot directly analyze it. However, we can consider some possibilities based on the options provided:
Option 1: The algorithm always converges, and we get a classifier that perfectly classifies the training dataset.
Option 2: The algorithm always converges, but we get a classifier that does not perfectly classify the training dataset.
Option 3: The algorithm will never converge.
Option 4: The algorithm might converge for some initial input of weights and biases 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.
Option 5: The algorithm might converge for some initial input of weights and biases and certain sequence of the data, but will diverge otherwise. When it converges, we always get a classifier that perfectly classifies the training dataset.
Without additional information about the dataset, we cannot definitively determine which option is correct. However, we can make some educated assumptions:
If the dataset is linearly separable, i.e., the samples can be perfectly classified by a linear decision boundary, then options 1, 2, and 5 are possible. In this case, the algorithm will converge to a classifier that either perfectly classifies the dataset (options 1 and 5) or does not perfectly classify it (option 2).
If the dataset is not linearly separable, i.e., a linear decision boundary cannot perfectly classify all the samples, then options 2, 4, and 5 are possible. In this case, the algorithm will converge to a classifier that either does not perfectly classify the dataset (options 2 and 4) or perfectly classifies it (option 5).
To summarize, without the plot of the dataset, we cannot determine the exact behavior of the linear perceptron algorithm. However, options 2, 4, and 5 seem more plausible based on the assumptions made about the dataset.