Efficient algorithms for learning functions with bounded variation

We show that the class FBV of [0, 1]-valued functions with total variation at most 1 can be agnostically learned with respect to the absolute loss in polynomial time from O ( 1 2 log 1 δ ) examples, matching a known lower bound to within a constant factor. We establish a bound of O(1/m) on the expected error of a polynomial-time algorithm for learning FBV in the prediction model, also matching a known lower bound to within a constant factor. Applying a known algorithm transformation to our prediction algorithm, we obtain a polynomial-time PAC learning algorithm for FBV with a sample complexity bound of O ( 1 log 1 δ ) ; this also matches a known lower bound to within a constant factor. © 2003 Elsevier Inc. All rights reserved.