Approximating Concavely Parameterized Optimization Problems
Abstract
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.
Cite
Text
Giesen et al. "Approximating Concavely Parameterized Optimization Problems." Neural Information Processing Systems, 2012.Markdown
[Giesen et al. "Approximating Concavely Parameterized Optimization Problems." Neural Information Processing Systems, 2012.](https://mlanthology.org/neurips/2012/giesen2012neurips-approximating/)BibTeX
@inproceedings{giesen2012neurips-approximating,
title = {{Approximating Concavely Parameterized Optimization Problems}},
author = {Giesen, Joachim and Mueller, Jens and Laue, Soeren and Swiercy, Sascha},
booktitle = {Neural Information Processing Systems},
year = {2012},
pages = {2105-2113},
url = {https://mlanthology.org/neurips/2012/giesen2012neurips-approximating/}
}