Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications

Rishabh Iyer, Jeff A. Bilmes
Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence, PMLR R10:405-415, 2012.

Abstract

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a dierence between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the multiplicative inapproximability of minimizing the dierence between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the dierence between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR10-iyer12a, title = {Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications}, author = {Iyer, Rishabh and Bilmes, Jeff A.}, booktitle = {Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence}, pages = {405--415}, year = {2012}, editor = {de Freitas, Nando and Murphy, Kevin}, volume = {R10}, series = {Proceedings of Machine Learning Research}, month = {14--18 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r10/main/assets/iyer12a/iyer12a.pdf}, url = {https://proceedings.mlr.press/r10/iyer12a.html}, abstract = {We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a dierence between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the multiplicative inapproximability of minimizing the dierence between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the dierence between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications %A Rishabh Iyer %A Jeff A. Bilmes %B Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2012 %E Nando de Freitas %E Kevin Murphy %F pmlr-vR10-iyer12a %I PMLR %P 405--415 %U https://proceedings.mlr.press/r10/iyer12a.html %V R10 %X We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a dierence between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the multiplicative inapproximability of minimizing the dierence between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the dierence between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features. %Z Reissued by PMLR on 04 October 2026.
APA
Iyer, R. & Bilmes, J.A.. (2012). Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications. Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R10:405-415 Available from https://proceedings.mlr.press/r10/iyer12a.html. Reissued by PMLR on 04 October 2026.

Related Material