Approximating Concavely Parameterized Optimization Problems

We consider an abstract class of optimization problems that are parameterized concavely in a single parameter, and show that the solution path along the parameter can always be approximated with accuracy \varepsilon 0 by a set of size O(1/\sqrt{\varepsilon}) . A lower bound of size \Omega (1/\sqrt{\varepsilon}) 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/\sqrt{\varepsilon}) . Finally, we provide an implementation of the oracle for soft-margin support vector machines, and a parameterized semi-definite program for matrix completion.