Estimation of KL divergence between large-alphabet distributions

Yuheng Bu; Shaofeng Zou; Yingbin Liang; Venugopal V. Veeravalli

doi:10.1109/ISIT.2016.7541473

Estimation of KL divergence between large-alphabet distributions

Yuheng Bu, Shaofeng Zou, Yingbin Liang, Venugopal V. Veeravalli

Department of Electrical Engineering & Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

10 Scopus citations

Abstract

The problem of estimating the KL divergence between two unknown distributions is studied. The alphabet size k of the distributions can scale to infinity. The estimation is based on m and n independent samples respectively drawn from the two distributions. It is first shown that there does not exist any consistent estimator to guarantee asymptotic small worst-case quadratic risk over the set of all pairs of distributions. A restricted set that contains pairs of distributions with bounded ratio F(k) is further considered. An augmented plug-in estimator is proposed, and is shown to be consistent if and only if m = ω(k V log²(F(k)) and n = ω(kF(k)). Furthermore, if F(k) ≥ log²k and log²(F(k)) = o(k), it is shown that any consistent estimator must satisfy the necessary conditions: m = ω( k/log k V log²(F(k)) and n = ω( kF(k)/log k).

Original language	English (US)
Title of host publication	Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	1118-1122
Number of pages	5
Volume	2016-August
ISBN (Electronic)	9781509018062
DOIs	https://doi.org/10.1109/ISIT.2016.7541473
State	Published - Aug 10 2016
Event	2016 IEEE International Symposium on Information Theory, ISIT 2016 - Barcelona, Spain Duration: Jul 10 2016 → Jul 15 2016

Other

Other	2016 IEEE International Symposium on Information Theory, ISIT 2016
Country	Spain
City	Barcelona
Period	7/10/16 → 7/15/16

ASJC Scopus subject areas

Theoretical Computer Science
Information Systems
Modeling and Simulation
Applied Mathematics

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10.1109/ISIT.2016.7541473

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Estimation of KL divergence between large-alphabet distributions. / Bu, Yuheng; Zou, Shaofeng; Liang, Yingbin; Veeravalli, Venugopal V.

Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory. Vol. 2016-August Institute of Electrical and Electronics Engineers Inc., 2016. p. 1118-1122 7541473.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Bu, Y, Zou, S, Liang, Y & Veeravalli, VV 2016, Estimation of KL divergence between large-alphabet distributions. in Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory. vol. 2016-August, 7541473, Institute of Electrical and Electronics Engineers Inc., pp. 1118-1122, 2016 IEEE International Symposium on Information Theory, ISIT 2016, Barcelona, Spain, 7/10/16. https://doi.org/10.1109/ISIT.2016.7541473

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abstract = "The problem of estimating the KL divergence between two unknown distributions is studied. The alphabet size k of the distributions can scale to infinity. The estimation is based on m and n independent samples respectively drawn from the two distributions. It is first shown that there does not exist any consistent estimator to guarantee asymptotic small worst-case quadratic risk over the set of all pairs of distributions. A restricted set that contains pairs of distributions with bounded ratio F(k) is further considered. An augmented plug-in estimator is proposed, and is shown to be consistent if and only if m = ω(k V log2(F(k)) and n = ω(kF(k)). Furthermore, if F(k) ≥ log2k and log2(F(k)) = o(k), it is shown that any consistent estimator must satisfy the necessary conditions: m = ω( k/log k V log2(F(k)) and n = ω( kF(k)/log k).",

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N2 - The problem of estimating the KL divergence between two unknown distributions is studied. The alphabet size k of the distributions can scale to infinity. The estimation is based on m and n independent samples respectively drawn from the two distributions. It is first shown that there does not exist any consistent estimator to guarantee asymptotic small worst-case quadratic risk over the set of all pairs of distributions. A restricted set that contains pairs of distributions with bounded ratio F(k) is further considered. An augmented plug-in estimator is proposed, and is shown to be consistent if and only if m = ω(k V log2(F(k)) and n = ω(kF(k)). Furthermore, if F(k) ≥ log2k and log2(F(k)) = o(k), it is shown that any consistent estimator must satisfy the necessary conditions: m = ω( k/log k V log2(F(k)) and n = ω( kF(k)/log k).

AB - The problem of estimating the KL divergence between two unknown distributions is studied. The alphabet size k of the distributions can scale to infinity. The estimation is based on m and n independent samples respectively drawn from the two distributions. It is first shown that there does not exist any consistent estimator to guarantee asymptotic small worst-case quadratic risk over the set of all pairs of distributions. A restricted set that contains pairs of distributions with bounded ratio F(k) is further considered. An augmented plug-in estimator is proposed, and is shown to be consistent if and only if m = ω(k V log2(F(k)) and n = ω(kF(k)). Furthermore, if F(k) ≥ log2k and log2(F(k)) = o(k), it is shown that any consistent estimator must satisfy the necessary conditions: m = ω( k/log k V log2(F(k)) and n = ω( kF(k)/log k).

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