The following text is from Hal Daumé III's "A Course in Machine Learning" online text book (Page-41).
I understand that, D = size of the input vector.
(1) What kind of Perceptron algorithm is this? Binary/Multi-class? Online/Offline?
(2) What is y here? Bias/weight/Sample/class_label?
(3) What is the rationale of testing ya<=0 for updating weights?
EDIT.
y is class_label.
Answers to your questions:
1 - This is a binary perceptron algorithm, working on an offline batch.
2 - as you wrote - Y is the labels vector. each label can be either be 1 or -1.
3 - The rational of testing if y*a<=0 is to check if the perceptron classified a certain sample correctly. If not - the weights of the perceptron are modified.
A bit more about the 3rd question
The idea behind the perceptron algorithm is as follows:
a. we iterate over the samples MaxIter times.
b. The perceptron classifies each sample by multiplying it with the weights vector W and adding a bias b. The result is assigned into the variable a.
c. The prediction for each sample can be either 1 or -1. It is calculated by sign(a). At this stage we check the correctness of the classification.
if y*a>0 that means that y=sign(a). In other words the predicted classification is correct, and we move on to the next sample.
If however y*a<=0, that means that the perceptron failed to predict the correct label. In this case, the algorithm changes the perceptron's weights in a way that they'll be more compatible to the sample which we failed to classify.