The dual form of the Perceptron algorithm is used to learn a binary classifier, based on n training
points. It converges after k updates, and returns a vector α and a number b. For each of the following
statements, indicate whether it is necessarily true or possibly false, and give a brief justification.
(a) Each αi is either 0 or 1.
(b) sum{αi} = k.
(c) α has at most k nonzero coordinates.
(d) The training data must be linearly separable
(a) Each αi is either 0 or 1:
This statement is possibly false. The alpha values, αi, in the dual form of the Perceptron algorithm are derived from the updates made during the learning process. While it is true that the Perceptron algorithm updates the alpha values by 1, there is no constraint that the updated alpha values must be either 0 or 1. The updates can be positive or negative depending on the data and the iterations, which means αi can take on any real value.
(b) sum{αi} = k:
This statement is necessarily false. The sum of αi values does not always equal k. During the learning process, the alpha values are updated based on the misclassified points until convergence is achieved. The sum of the alpha values may not necessarily equal k.
(c) α has at most k nonzero coordinates:
This statement is necessarily true. The dual form of the Perceptron algorithm converges after k updates, meaning that after k updates, the algorithm has found a binary classifier that separates the training data correctly. This implies that there are at most k nonzero coordinates in the alpha vector, as each nonzero coordinate corresponds to a misclassified data point.
(d) The training data must be linearly separable:
This statement is necessarily true. The Perceptron algorithm, whether in its primal or dual form, can only learn a binary classifier if the training data is linearly separable. Linear separability means that it is possible to draw a hyperplane that separates the data points of different classes. If the training data is not linearly separable, the Perceptron algorithm will not converge and will not be able to find a binary classifier.