A Deterministic and Polynomial Modified Perceptron Algorithm

We construct a modi ed perceptron algorithm that is deterministic, polynomial and also as fast as previous known algorithms. The algorithm runs in time O(mn log n log(1/ρ)), where m is the number of examples, n the number of dimensions and ρ is approximately the size of the margin. We also construct a non-deterministic modi ed perceptron algorithm running in time O(mn log n log(1/ρ)). 1 A Deterministic and Polynomial Modi ed Perceptron Algorithm 1.1 Historical and Technological Exposition The Perceptron Algorithm was introduced by Rosenblatt in [12] and has been well-studied by mathematicians and computer scientists since then. For convenience, we will in this paper discuss the version of the algorithm that, given a set of points (constraints) A = ∪iai from Rn nds, if any, a normal z to a hyperplane through origo such that z·ai > 0 for every i. (Note that we do not have any zero rows in the matrix A). This is so to say, a hyperplane through origo such that all points are on the very same side of the hyperplane. The original algorithm can easily be described as follows: Algorithm 1.1 The Perceptron Algorithm Input: A set of points (constraints) A = ∪iai from Rn. Output: A normal z to a hyperplane such that z · ai > 0 for every i, if there is such a solution.