Approximate counting, the Lovasz local lemma, and inference in graphical models
In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula φ when the width is logarithmic in the maximum degree. This closes an exponent...
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| Format: | Article |
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Association for Computing Machinery (ACM)
2018
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| Online Access: | http://hdl.handle.net/1721.1/116220 https://orcid.org/0000-0001-7047-0495 |
| _version_ | 1826217380100964352 |
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| author | Moitra, Ankur |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Moitra, Ankur |
| author_sort | Moitra, Ankur |
| collection | MIT |
| description | In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula φ when the width is logarithmic in the maximum degree. This closes an exponential gap between the known upper and lower bounds. Moreover our algorithm extends straightforwardly to approximate sampling, which shows that under Lovász Local Lemma-like conditions it is not only possible to find a satisfying assignment, it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments. Our approach is a significant departure from earlier techniques in approximate counting, and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables. Finally, we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models. |
| first_indexed | 2024-09-23T17:02:45Z |
| format | Article |
| id | mit-1721.1/116220 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T17:02:45Z |
| publishDate | 2018 |
| publisher | Association for Computing Machinery (ACM) |
| record_format | dspace |
| spelling | mit-1721.1/1162202022-10-03T10:01:26Z Approximate counting, the Lovasz local lemma, and inference in graphical models Moitra, Ankur Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Department of Mathematics Moitra, Ankur In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula φ when the width is logarithmic in the maximum degree. This closes an exponential gap between the known upper and lower bounds. Moreover our algorithm extends straightforwardly to approximate sampling, which shows that under Lovász Local Lemma-like conditions it is not only possible to find a satisfying assignment, it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments. Our approach is a significant departure from earlier techniques in approximate counting, and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables. Finally, we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models. National Science Foundation (U.S.). Faculty Early Career Development Program (Award CCF-1453261) National Science Foundation (U.S.). Computing and Communication Foundation (CCF-1565235) Alfred P. Sloan Foundation. Fellowship David & Lucile Packard Foundation Fellowship Alfred P. Sloan Foundation. Fellowship Edmond F. Kelley Research Award Google Research Award Nihon Denki Kabushiki Kaisha (MIT NEC grant) 2018-06-11T18:40:07Z 2018-06-11T18:40:07Z 2017-06 2018-05-29T13:54:24Z Article http://purl.org/eprint/type/ConferencePaper 9781450345286 http://hdl.handle.net/1721.1/116220 Moitra, Ankur. “Approximate Counting, the Lovasz Local Lemma, and Inference in Graphical Models.” Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017 (2017). https://orcid.org/0000-0001-7047-0495 http://dx.doi.org/10.1145/3055399.3055428 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Association for Computing Machinery (ACM) arXiv |
| spellingShingle | Moitra, Ankur Approximate counting, the Lovasz local lemma, and inference in graphical models |
| title | Approximate counting, the Lovasz local lemma, and inference in graphical models |
| title_full | Approximate counting, the Lovasz local lemma, and inference in graphical models |
| title_fullStr | Approximate counting, the Lovasz local lemma, and inference in graphical models |
| title_full_unstemmed | Approximate counting, the Lovasz local lemma, and inference in graphical models |
| title_short | Approximate counting, the Lovasz local lemma, and inference in graphical models |
| title_sort | approximate counting the lovasz local lemma and inference in graphical models |
| url | http://hdl.handle.net/1721.1/116220 https://orcid.org/0000-0001-7047-0495 |
| work_keys_str_mv | AT moitraankur approximatecountingthelovaszlocallemmaandinferenceingraphicalmodels |