On learning discretized geometric concepts

We present a polynomial time online learning algorithm that learns any discretized geometric concept generated from any number of halfspaces with any number of known (to the learner) slopes in a constant dimensional space. In particular, our algorithm learns (from equivalence queries only) unions of discretized axis-parallel rectangles in a constant dimensional space in polynomial time. The algorithm also runs in polynomial time in l if the teacher lies on l counterexamples. We then show a PAC-learning algorithm for the above discretized geometric concept when the example oracle lies on the labels of the examples with a fixed probability p⩽½-1/r that runs in polynomial time also with r. We use these methods, as well as a bounded version of the finite injury priority method, to construct algorithms for learning several classes of rectangles. In particular we design efficient algorithms for learning several classes of unions of discretized axis-parallel rectangles in either arbitrary dimensional spaces or constant dimensional spaces