research-article

Online discrepancy minimization for stochastic arrivals

Authors:

Makrand SinhaAuthors Info & Claims

SODA '21: Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

Pages 2842 - 2861

Published: 21 March 2021 Publication History

Abstract

In the stochastic online vector balancing problem, vectors v₁, v₂, …, v_T chosen independently from an arbitrary distribution in Rⁿ arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm.

We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC 20). In particular, for the Komlós problem where ||v_t||₂ ≤ 1 for each t, our algorithm achieves Õ(1) discrepancy with high probability, improving upon the previous Õ(n^3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(log^d+4T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log^2d+1 T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves Õ(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails.

Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multicolor discrepancy.

References

[1]

[ALS20] Ryan Alweiss, Yang P. Liu, and Mehtaab Sawhney. Discrepancy minimization via a self-balancing walk. CoRR, abs/2006.14009, 2020.

[2]

[Ban98] Wojciech Banaszczyk. Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Struct. Algorithms, 12(4):351--360, 1998.

Digital Library

[3]

[Ban10] Nikhil Bansal. Constructive Algorithms for Discrepancy Minimization. In Proceedings of FOCS 2010, pages 3--10, 2010.

Digital Library

[4]

[Ban12] Wojciech Banaszczyk. On series of signed vectors and their rearrangements. Random Struct. Algorithms, 40(3):301--316, 2012.

Digital Library

[5]

[Bár79] Imre Bárány. On a Class of Balancing Games. J. Comb. Theory, Ser. A, 26(2):115--126, 1979.

[6]

[BDG16] Nikhil Bansal, Daniel Dadush, and Shashwat Garg. An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound. In Proceedings of FOCS 2016, pages 788--799, 2016.

[7]

[BDGL18] Nikhil Bansal, Daniel Dadush, Shashwat Garg, and Shachar Lovett. The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues. In Proceedings of STOC 2018, pages 587--597, 2018.

Digital Library

[8]

[Bec81] József Beck. Balanced two-colorings of finite sets in the square I. Combinatorica, 1(4):327--335, 1981.

[9]

[BG17] Nikhil Bansal and Shashwat Garg. Algorithmic discrepancy beyond partial coloring. In Proceedings of STOC 2017, pages 914--926, 2017.

Digital Library

[10]

[BJM⁺20] Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Online Discrepancy Minimization for Stochastic Arrivals. CoRR, abs/2007.10622, 2020.

[11]

[BJSS20] Nikhil Bansal, Haotian Jiang, Sahil Singla, and Makrand Sinha. Online vector balancing and geometric discrepancy. In Proceedings of STOC, 2020.

[12]

[BKPP18] Gerdus Benade, Aleksandr M. Kazachkov, Ariel D. Procaccia, and Christos-Alexandros Psomas. How to Make Envy Vanish Over Time. In Proceedings of EC 2018, pages 593--610, 2018.

Digital Library

[13]

[BS19] Nikhil Bansal and Joel H. Spencer. On-Line Balancing of Random Inputs. CoRR, abs/1903.06898, 2019.

[14]

[Cha01] Bernard Chazelle. The discrepancy method: randomness and complexity. Cambridge University Press, 2001.

[15]

[CST⁺14] William Chen, Anand Srivastav, Giancarlo Travaglini, et al. A panorama of discrepancy theory, volume 2107. Springer, 2014.

[16]

[DFGGR19] Raaz Dwivedi, Ohad N. Feldheim, Ori Gurel-Gurevich, and Aaditya Ramdas. The power of online thinning in reducing discrepancy. Probability Theory and Related Fields, 2019.

[17]

[DGLN16] Daniel Dadush, Shashwat Garg, Shachar Lovett, and Aleksandar Nikolov. Towards a constructive version of banaszczyk's vector balancing theorem. In Proceedings of APPROX/RANDOM, 2016.

[18]

[DNTT18] Daniel Dadush, Aleksandar Nikolov, Kunal Talwar, and Nicole Tomczak-Jaegermann. Balancing Vectors in Any Norm. In Proceedings of FOCS, 2018.

[19]

[ES18] Ronen Eldan and Mohit Singh. Efficient algorithms for discrepancy minimization in convex sets. Random Struct. Algorithms, 53(2):289--307, 2018.

[20]

[Gia97] Apostolos A Giannopoulos. On some vector balancing problems. Stud. Math., 122(3):225--234, 1997.

[21]

[Glu89] Efim Davydovich Gluskin. Extremal properties of orthogonal parallelepipeds and their applications to the geometry of banach spaces. Mathematics of the USSR-Sbornik, 64(1):85, 1989.

[22]

[JKS19] Haotian Jiang, Janardhan Kulkarni, and Sahil Singla. Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization. arXiv:1910.01073, 2019.

[23]

[LM15] Shachar Lovett and Raghu Meka. Constructive Discrepancy Minimization by Walking on the Edges. SIAM J. Comput., 44(5):1573--1582, 2015.

[24]

[LRR17] Avi Levy, Harishchandra Ramadas, and Thomas Rothvoss. Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method. In Proceedings of IPCO 2017, pages 380--391, 2017.

[25]

[Mat09] Jiri Matousek. Geometric discrepancy: An illustrated guide, volume 18. SSBM, 2009.

[26]

[MN15] Jiří Matoušek and Aleksandar Nikolov. Combinatorial discrepancy for boxes via the γ₂ norm. In Proceedings of SoCG 2015, pages 1--15, 2015.

[27]

[MNT14] Jiří Matoušek, Aleksandar Nikolov, and Kunal Talwar. Factorization norms and hereditary discrepancy. CoRR, abs/1408.1376, 2014.

[28]

[Nik17] Aleksandar Nikolov. Tighter bounds for the discrepancy of boxes and polytopes. CoRR, abs/1701.05532, 2017.

[29]

[Rot14] Thomas Rothvoß. Constructive Discrepancy Minimization for Convex Sets. In Proceedings of FOCS 2014, pages 140--145, 2014.

Digital Library

[30]

[Spe77] Joel Spencer. Balancing games. J. Comb. Theory, Ser. B, 23(1):68--74, 1977.

[31]

[Spe85] Joel Spencer. Six standard deviations suffice. Trans. Am. Math. Soc., 289(2):679--706, 1985.

[32]

[Spe87] Joel H. Spencer. Ten lectures on the probabilistic method, volume 52. Society for Industrial and Applied Mathematics Philadelphia, 1987.

[33]

[Ver18] Roman Vershynin. High-Dimensional Probability, volume 1. Cambridge University Press, 2018.

[34]

[Wai19] Martin J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2019.

Cited By

Alweiss RLiu YSawhney MKhuller SVassilevska Williams V(2021)Discrepancy minimization via a self-balancing walkProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3450994(14-20)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3450994

Online discrepancy minimization for stochastic arrivals
1. Theory of computation

Recommendations

Online vector balancing and geometric discrepancy
STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

We consider an online vector balancing question where T vectors, chosen from an arbitrary distribution over [−1,1]ⁿ, arrive one-by-one and must be immediately given a ± sign. The goal is to keep the discrepancy—the ℓ_∞-norm of any signed prefix-sum—as ...
Constructive Algorithms for Discrepancy Minimization
FOCS '10: Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science

Given a set system $(V,\mathcal{S})$, $V=\{1,\ldots,n\}$ and $\mathcal{S}=\{S_1,\ldots,S_m\}$, the minimum discrepancy problem is to find a 2-coloring $\mathcal{X}:V \right arrow \{-1,+1\}$, such that each set is colored as evenly as possible, i.e. find ...
Deterministic discrepancy minimization
ESA'11: Proceedings of the 19th European conference on Algorithms

We derandomize a recent algorithmic approach due to Bansal [2] to efficiently compute low discrepancy colorings for several problems. In particular, we give an efficient deterministic algorithm for Spencer's six standard deviations result [13], and to a ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SODA '21: Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

January 2021

3063 pages

ISBN:9781611976465

Program Chair:
Dániel Marx
CISPA Helmholtz Center for Information Security, Germany

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 21 March 2021

Check for updates

Qualifiers

Research-article

Conference

SODA '21

Sponsor:

SIGACT

SODA '21: ACM-SIAM Symposium on Discrete Algorithms

January 10 - 13, 2021

Virginia, Virtual Event

Acceptance Rates

Overall Acceptance Rate 411 of 1,322 submissions, 31%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

1
Total Citations
View Citations
27
Total Downloads

Downloads (Last 12 months)5
Downloads (Last 6 weeks)1

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Alweiss RLiu YSawhney MKhuller SVassilevska Williams V(2021)Discrepancy minimization via a self-balancing walkProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3450994(14-20)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3450994

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents