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Your question, as I understand it, is this: Given a linearly separable data set, if a perceptron is trained on any subset of that data set, is the perceptron guaranteed to have a) 100% training accuracy and b) 100% test accuracy on the remaining data not in the training set?

Answer:

a) Since the data set is linearly separable, any subset of the data is also linearly separable. Thus, the perceptron is guaranteed to converge to a perfect solution on the training set. (https://en.wikipedia.org/wiki/Perceptron#Convergence)

b) No. Consider the following classification problem as a counterexample. The graph below shows a very simple linearly separable data set consisting of two classes of data points — red points and blue points. The black line highlights the linear separableness of the data.

Linearly separable data set

Now, since any subset of the points can be used for our training set, let’s suppose that the following two points ended up being our training set, taken from the data shown above.

Training set

Furthermore, suppose that the perceptron separates the classes in the following manner.

Training set, separated

Now the remaining points in the data set for validation or testing are the following, with the partition from training shown again as the same black line.

Entire data set again, with trained separation

Note that the test points are now falling on opposite sides of the line, so that the perceptron will fail on classifying both of those data points, because it did not select the best partition during training.

Your question, as I understand it, is this: Given a linearly separable data set, if a perceptron is trained on any subset of that data set, is the perceptron guaranteed to have a) 100% training accuracy and b) 100% test accuracy on the remaining data not in the training set?

a) Since the data set is linearly separable, any subset of the data is also linearly separable. Thus, the perceptron is guaranteed to converge to a perfect solution on the training set. (https://en.wikipedia.org/wiki/Perceptron#Convergence)

b) No. Consider the following classification problem as a counterexample. The graph below shows a very simple linearly separable data set consisting of two classes of data points — red points and blue points. The black line highlights the linear separableness of the data.

Linearly separable data set

Now, since any subset of the points can be used for our training set, let’s suppose that the following two points ended up being our training set, taken from the data shown above.

Training set

Furthermore, suppose that the perceptron separates the classes in the following manner.

Training set, separated

Now the remaining points in the data set for validation or testing are the following, with the partition from training shown again as the same black line.

Entire data set again, with trained separation

Note that the test points are now falling on opposite sides of the line, so that the perceptron will fail on classifying both of those data points, because it did not select the best partition during training.

Your question, as I understand it, is this: Given a linearly separable data set, if a perceptron is trained on any subset of that data set, is the perceptron guaranteed to have a) 100% training accuracy and b) 100% test accuracy on the remaining data not in the training set?

Answer:

a) Since the data set is linearly separable, any subset of the data is also linearly separable. Thus, the perceptron is guaranteed to converge to a perfect solution on the training set. (https://en.wikipedia.org/wiki/Perceptron#Convergence)

b) No. Consider the following classification problem as a counterexample. The graph below shows a very simple linearly separable data set consisting of two classes of data points — red points and blue points. The black line highlights the linear separableness of the data.

Linearly separable data set

Now, since any subset of the points can be used for our training set, let’s suppose that the following two points ended up being our training set, taken from the data shown above.

Training set

Furthermore, suppose that the perceptron separates the classes in the following manner.

Training set, separated

Now the remaining points in the data set for validation or testing are the following, with the partition from training shown again as the same black line.

Entire data set again, with trained separation

Note that the test points are now falling on opposite sides of the line, so that the perceptron will fail on classifying both of those data points, because it did not select the best partition during training.

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Your question, as I understand it, is this: Given a linearly separable data set, if a perceptron is trained on any subset of that data set, is the perceptron guaranteed to have a) 100% training accuracy and b) 100% test accuracy on the remaining data not in the training set?

a) Since the data set is linearly separable, any subset of the data is also linearly separable. Thus, the perceptron is guaranteed to converge to a perfect solution on the training set. (https://en.wikipedia.org/wiki/Perceptron#Convergence)

b) No. Consider the following classification problem as a counterexample. The graph below shows a very simple linearly separable data set consisting of two classes of data points — red points and blue points. The black line highlights the linear separableness of the data.

Linearly separable data set

Now, since any subset of the points can be used for our training set, let’s suppose that the following two points ended up being our training set, taken from the data shown above.

Training set

Furthermore, suppose that the perceptron separates the classes in the following manner.

Training set, separated

Now the remaining points in the data set for validation or testing are the following, with the partition from training shown again as the same black line.

Entire data set again, with trained separation

Note that the test points are now falling on opposite sides of the line, so that the perceptron will fail on classifying both of those data points, because it did not select the best partition during training.