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Two Questions About The Perceptron Algorithm

Last Modified: 2012-06-27
If anyone has any insight into any of the 2 questions below, I'd greatly appreciate it.

1) We know that the online perceptron algorithm can be used to learn a linear threshold function: w1*x1 + w2*x2 + w3*x3 >= 0". What if, there's a linear threshold function, and we already KNOW that:
a) the weights of this linear threshold function are always positive
b) the sum of the weights of our linear threshold function do not exceed a constant W.

How can we use the Perceptron algorithm to learn this particular linear threshold function? (As in, how would we modify the original Perceptron algorithm to learn this particular function)? And what is the most number of mistakes we can make with this Perceptron algorithm?

2) When learning using the perceptron algorithm, imagine there's one example where the wrong label is shown (eg: output should be 1, but it falsely told the algorithm that the output is 0). How does this change the maximum number of mistakes we make with the Perceptron?
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