Tight Bounds for Approximate Carathéodory and Beyond

Vahab Mirrokni, Renato Paes Leme, Adrian Vladu, Sam Chiu-wai Wong
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:2440-2448, 2017.

Abstract

We present a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope’s vertices. Our result provides a constructive proof for the Approximate Carathéodory Problem, which states that any point inside a polytope contained in the $\ell_p$ ball of radius $D$ can be approximated to within $\epsilon$ in $\ell_p$ norm by a convex combination of $O\left(D^2 p/\epsilon^2\right)$ vertices of the polytope for $p \geq 2$. While for the particular case of $p=2$, this can be achieved by the well-known Perceptron algorithm, we follow a more principled approach which generalizes to arbitrary $p\geq 2$; furthermore, this naturally extends to domains with more complicated geometry, as it is the case for providing an approximate Birkhoff-von Neumann decomposition. Secondly, we show that the sparsity bound is tight for $\ell_p$ norms, using an argument based on anti-concentration for the binomial distribution, thus resolving an open question posed by Barman. Experimentally, we verify that our deterministic optimization-based algorithms achieve in practice much better sparsity than previously known sampling-based algorithms. We also show how to apply our techniques to SVM training and rounding fractional points in matroid and flow polytopes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v70-mirrokni17a, title = {Tight Bounds for Approximate {C}arath{\'e}odory and Beyond}, author = {Vahab Mirrokni and Renato Paes Leme and Adrian Vladu and Sam Chiu-wai Wong}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {2440--2448}, year = {2017}, editor = {Precup, Doina and Teh, Yee Whye}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v70/mirrokni17a/mirrokni17a.pdf}, url = {https://proceedings.mlr.press/v70/mirrokni17a.html}, abstract = {We present a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope’s vertices. Our result provides a constructive proof for the Approximate Carathéodory Problem, which states that any point inside a polytope contained in the $\ell_p$ ball of radius $D$ can be approximated to within $\epsilon$ in $\ell_p$ norm by a convex combination of $O\left(D^2 p/\epsilon^2\right)$ vertices of the polytope for $p \geq 2$. While for the particular case of $p=2$, this can be achieved by the well-known Perceptron algorithm, we follow a more principled approach which generalizes to arbitrary $p\geq 2$; furthermore, this naturally extends to domains with more complicated geometry, as it is the case for providing an approximate Birkhoff-von Neumann decomposition. Secondly, we show that the sparsity bound is tight for $\ell_p$ norms, using an argument based on anti-concentration for the binomial distribution, thus resolving an open question posed by Barman. Experimentally, we verify that our deterministic optimization-based algorithms achieve in practice much better sparsity than previously known sampling-based algorithms. We also show how to apply our techniques to SVM training and rounding fractional points in matroid and flow polytopes.} }
Endnote
%0 Conference Paper %T Tight Bounds for Approximate Carathéodory and Beyond %A Vahab Mirrokni %A Renato Paes Leme %A Adrian Vladu %A Sam Chiu-wai Wong %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-mirrokni17a %I PMLR %P 2440--2448 %U https://proceedings.mlr.press/v70/mirrokni17a.html %V 70 %X We present a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope’s vertices. Our result provides a constructive proof for the Approximate Carathéodory Problem, which states that any point inside a polytope contained in the $\ell_p$ ball of radius $D$ can be approximated to within $\epsilon$ in $\ell_p$ norm by a convex combination of $O\left(D^2 p/\epsilon^2\right)$ vertices of the polytope for $p \geq 2$. While for the particular case of $p=2$, this can be achieved by the well-known Perceptron algorithm, we follow a more principled approach which generalizes to arbitrary $p\geq 2$; furthermore, this naturally extends to domains with more complicated geometry, as it is the case for providing an approximate Birkhoff-von Neumann decomposition. Secondly, we show that the sparsity bound is tight for $\ell_p$ norms, using an argument based on anti-concentration for the binomial distribution, thus resolving an open question posed by Barman. Experimentally, we verify that our deterministic optimization-based algorithms achieve in practice much better sparsity than previously known sampling-based algorithms. We also show how to apply our techniques to SVM training and rounding fractional points in matroid and flow polytopes.
APA
Mirrokni, V., Leme, R.P., Vladu, A. & Wong, S.C.. (2017). Tight Bounds for Approximate Carathéodory and Beyond. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:2440-2448 Available from https://proceedings.mlr.press/v70/mirrokni17a.html.

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