Submodular Maximization via Gradient Ascent: The Case of Deep Submodular Functions
We study the problem of maximizing deep submodular functions (DSFs) [13, 3] subject to a matroid constraint. DSFs are an expressive class of submodular functions that include, as strict subfamilies, the facility location, weighted coverage, and sums of concave composed with modular functions. We use...
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| Published in: | Advances in neural information processing systems Vol. 2018; p. 7989 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
United States
01.12.2018
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| ISSN: | 1049-5258 |
| Online Access: | Get more information |
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| Summary: | We study the problem of maximizing deep submodular functions (DSFs) [13, 3] subject to a matroid constraint. DSFs are an expressive class of submodular functions that include, as strict subfamilies, the facility location, weighted coverage, and sums of concave composed with modular functions. We use a strategy similar to the continuous greedy approach [6], but we show that the multilinear extension of any DSF has a natural and computationally attainable concave relaxation that we can optimize using gradient ascent. Our results show a guarantee of
with a running time of
(
) plus time for pipage rounding [6] to recover a discrete solution, where
is the rank of the matroid constraint. This bound is often better than the standard 1 - 1
guarantee of the continuous greedy algorithm, but runs much faster. Our bound also holds even for fully curved (
= 1) functions where the guarantee of 1 -
degenerates to 1 - 1
where
is the curvature of
[37]. We perform computational experiments that support our theoretical results. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 1049-5258 |