dc.contributor.author | Daskalakis, Konstantinos | |
dc.contributor.author | Kamath, Gautam Chetan | |
dc.contributor.author | Tzamos, Christos | |
dc.date.accessioned | 2017-07-25T17:58:54Z | |
dc.date.available | 2017-07-25T17:58:54Z | |
dc.date.issued | 2015-10 | |
dc.identifier.isbn | 978-1-4673-8191-8 | |
dc.identifier.uri | http://hdl.handle.net/1721.1/110840 | |
dc.description.abstract | An (n, k)-Poisson Multinomial Distribution (PMD) is the distribution of the sum of n independent random vectors supported on the set Bk={e1,...,ek} of standard basis vectors in Rk. We prove a structural characterization of these distributions, showing that, for all ε > 0, any (n, k)-Poisson multinomial random vector is ε-close, in total variation distance, to the sum of a discretized multidimensional Gaussian and an independent (poly(k/ε), k)-Poisson multinomial random vector. Our structural characterization extends the multi-dimensional CLT of Valiant and Valiant, by simultaneously applying to all approximation requirements ε. In particular, it overcomes factors depending on log n and, importantly, the minimum Eigen value of the PMD's covariance matrix. We use our structural characterization to obtain an ε-cover, in total variation distance, of the set of all (n, k)-PMDs, significantly improving the cover size of Daskalakis and Papadimitriou, and obtaining the same qualitative dependence of the cover size on n and ε as the k=2 cover of Daskalakis and Papadimitriou. We further exploit this structure to show that (n, k)-PMDs can be learned to within ε in total variation distance from Õk(1/ε) samples, which is near-optimal in terms of dependence on ε and independent of n. In particular, our result generalizes the single-dimensional result of Daskalakis, Diakonikolas and Servedio for Poisson binomials to arbitrary dimension. Finally, as a corollary of our results on PMDs, we give a Õk(1/ε2) sample algorithm for learning (n, k)-sums of independent integer random variables (SIIRVs), which is near-optimal for constant k. | en_US |
dc.description.sponsorship | Alfred P. Sloan Foundation (Fellowship) | en_US |
dc.description.sponsorship | Microsoft Research Faculty Fellowship | en_US |
dc.description.sponsorship | National Science Foundation (U.S.) (Award CCF-0953960 (CAREER)) | en_US |
dc.description.sponsorship | National Science Foundation (U.S.). Division of Computing and Communication Foundations (CCF-1101491) | en_US |
dc.description.sponsorship | Simons Award for Graduate Students in Theoretical Computer Science | en_US |
dc.language.iso | en_US | |
dc.publisher | Institute of Electrical and Electronics Engineers (IEEE) | en_US |
dc.relation.isversionof | http://dx.doi.org/10.1109/FOCS.2015.77 | en_US |
dc.rights | Creative Commons Attribution-Noncommercial-Share Alike | en_US |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-sa/4.0/ | en_US |
dc.source | arXiv | en_US |
dc.title | On the Structure, Covering, and Learning of Poisson Multinomial Distributions | en_US |
dc.type | Article | en_US |
dc.identifier.citation | Daskalakis, Constantinos, Gautam Kamath, and Christos Tzamos. “On the Structure, Covering, and Learning of Poisson Multinomial Distributions.” 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (October 2015). | en_US |
dc.contributor.department | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory | en_US |
dc.contributor.department | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science | en_US |
dc.contributor.mitauthor | Daskalakis, Konstantinos | |
dc.contributor.mitauthor | Kamath, Gautam Chetan | |
dc.contributor.mitauthor | Tzamos, Christos | |
dc.relation.journal | 2015 IEEE 56th Annual Symposium on Foundations of Computer Science | en_US |
dc.eprint.version | Author's final manuscript | en_US |
dc.type.uri | http://purl.org/eprint/type/ConferencePaper | en_US |
eprint.status | http://purl.org/eprint/status/NonPeerReviewed | en_US |
dspace.orderedauthors | Daskalakis, Constantinos; Kamath, Gautam; Tzamos, Christos | en_US |
dspace.embargo.terms | N | en_US |
dc.identifier.orcid | https://orcid.org/0000-0002-5451-0490 | |
dc.identifier.orcid | https://orcid.org/0000-0003-0048-2559 | |
dc.identifier.orcid | https://orcid.org/0000-0002-7560-5069 | |
mit.license | OPEN_ACCESS_POLICY | en_US |