Scalar and Matrix Chernoff Bounds from ℓ∞-Independence

Tali Kaufman; Rasmus Kyng; Federico Soldá

doi:10.1137/1.9781611977073.147

Scalar and Matrix Chernoff Bounds from ℓ∞-Independence

Tali Kaufman
, Rasmus Kyng
, Federico Soldá

Department of Computer Science

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

4 Scopus citations

Abstract

We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube {0, 1}n. Motivated by recent tools developed for the study of mixing times of Markov chains on discrete distributions, we say that a distribution is ℓ∞-independent when the infinity norm of its influence matrix is bounded by a constant. We show that any distribution which is ℓ∞-infinity independent satisfies a matrix Chernoff bound that matches the matrix Chernoff bound for independent random variables due to Tropp. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18).

Original language	English
Title of host publication	ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
Publisher	Association for Computing Machinery
Pages	3732-3753
Number of pages	22
ISBN (Electronic)	9781611977073
DOIs	https://doi.org/10.1137/1.9781611977073.147
State	Published - 2022
Event	33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 - Alexander, United States Duration: 9 Jan 2022 → 12 Jan 2022

Publication series

Name	Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume	2022-January
ISSN (Print)	1071-9040
ISSN (Electronic)	1557-9468

Conference

Conference	33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
Country/Territory	United States
City	Alexander
Period	9/01/22 → 12/01/22

Bibliographical note

Publisher Copyright:
Copyright © 2022 by SIAM.

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10.1137/1.9781611977073.147

Cite this

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title = "Scalar and Matrix Chernoff Bounds from ℓ∞-Independence",

abstract = "We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube \{0, 1\}n. Motivated by recent tools developed for the study of mixing times of Markov chains on discrete distributions, we say that a distribution is ℓ∞-independent when the infinity norm of its influence matrix is bounded by a constant. We show that any distribution which is ℓ∞-infinity independent satisfies a matrix Chernoff bound that matches the matrix Chernoff bound for independent random variables due to Tropp. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18).",

author = "Tali Kaufman and Rasmus Kyng and Federico Sold{\'a}",

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Kaufman, T, Kyng, R & Soldá, F 2022, Scalar and Matrix Chernoff Bounds from ℓ∞-Independence. in ACM-SIAM Symposium on Discrete Algorithms, SODA 2022. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 2022-January, Association for Computing Machinery, pp. 3732-3753, 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Alexander, United States, 9/01/22. https://doi.org/10.1137/1.9781611977073.147

Scalar and Matrix Chernoff Bounds from ℓ∞-Independence. / Kaufman, Tali; Kyng, Rasmus; Soldá, Federico.
ACM-SIAM Symposium on Discrete Algorithms, SODA 2022. Association for Computing Machinery, 2022. p. 3732-3753 (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 2022-January).

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

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N2 - We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube {0, 1}n. Motivated by recent tools developed for the study of mixing times of Markov chains on discrete distributions, we say that a distribution is ℓ∞-independent when the infinity norm of its influence matrix is bounded by a constant. We show that any distribution which is ℓ∞-infinity independent satisfies a matrix Chernoff bound that matches the matrix Chernoff bound for independent random variables due to Tropp. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18).

AB - We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube {0, 1}n. Motivated by recent tools developed for the study of mixing times of Markov chains on discrete distributions, we say that a distribution is ℓ∞-independent when the infinity norm of its influence matrix is bounded by a constant. We show that any distribution which is ℓ∞-infinity independent satisfies a matrix Chernoff bound that matches the matrix Chernoff bound for independent random variables due to Tropp. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18).

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Scalar and Matrix Chernoff Bounds from ℓ∞-Independence

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