Abstract
We present a framework for speeding up the time it takes to sample from discrete distributions μ defined over subsets of size k of a ground set of n elements, in the regime where k is much smaller than n. We show that if one has access to estimates of marginals P_{S∼ μ} {i ∈ S}, then the task of sampling from μ can be reduced to sampling from related distributions ν supported on size k subsets of a ground set of only n^{1α}⋅ poly(k) elements. Here, 1/α ∈ [1, k] is the parameter of entropic independence for μ. Further, our algorithm only requires sparsified distributions ν that are obtained by applying a sparse (mostly 0) external field to μ, an operation that for many distributions μ of interest, retains algorithmic tractability of sampling from ν. This phenomenon, which we dub domain sparsification, allows us to pay a onetime cost of estimating the marginals of μ, and in return reduce the amortized cost needed to produce many samples from the distribution μ, as is often needed in upstream tasks such as counting and inference.
For a wide range of distributions where α = Ω(1), our result reduces the domain size, and as a corollary, the costpersample, by a poly(n) factor. Examples include monomers in a monomerdimer system, nonsymmetric determinantal point processes, and partitionconstrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a logconcave generating polynomial (corresponding to α = 1). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of logconcave polynomials; roughly speaking, we show that constantfactor approximation is enough for domain sparsification, improving over O(1/k) relative error established in prior work.
BibTeX  Entry
@InProceedings{anari_et_al:LIPIcs.ITCS.2022.5,
author = {Anari, Nima and Derezi\'{n}ski, Micha{\l} and Vuong, ThuyDuong and Yang, Elizabeth},
title = {{Domain Sparsification of Discrete Distributions Using Entropic Independence}},
booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
pages = {5:15:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772174},
ISSN = {18688969},
year = {2022},
volume = {215},
editor = {Braverman, Mark},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15601},
URN = {urn:nbn:de:0030drops156013},
doi = {10.4230/LIPIcs.ITCS.2022.5},
annote = {Keywords: Domain Sparsification, Markov Chains, Sampling, Entropic Independence}
}
Keywords: 

Domain Sparsification, Markov Chains, Sampling, Entropic Independence 
Collection: 

13th Innovations in Theoretical Computer Science Conference (ITCS 2022) 
Issue Date: 

2022 
Date of publication: 

25.01.2022 