Sören Laue

We consider an abstract class of optimization problems that are parameterized concavely in a single parameter, and show that the solution path along the param-eter can always be approximated with accuracy ε> 0 by a set of sizeO(1/ ε). A lower bound of size Ω(1/ ε) shows that the upper bound is tight up to a constant factor. We also devise an algorithm that calls a step-size oracle and computes an approximate path of size O(1/ ε). Finally, we provide an implementation of the oracle for soft-margin support vector machines, and a parameterized semi-definite program for matrix completion.