Factorization Norms Sasho Nikolov U of Toronto Outline Hereditary Discrepancy Applications Factorization Norms 1 Reminder of definitions Set system where collection of subsets of ID: 1029042
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1. Hereditary Discrepancy & Factorization NormsSasho Nikolov (U of Toronto)
2. OutlineHereditary DiscrepancyApplicationsFactorization Norms1
3. Reminder of definitionsSet system: , where collection of subsets of Discrepancy: Matrix discrepancy: for matrix ,Definitions agree for incidence matrix of 2
4. Is discrepancy robust?If we “build” a set system from low/high discrepancy set systems, does the discrepancy stay low/high?E.g.: and ; 3
5. Hereditary discrepancy [LSV 86]Restriction: for , Hereditary discrepancy:Matrices: , where ranges over submatrices of 4
6. Example and Note that If , then For , 5
7. Robustness of hereditary discrepancyLet be set systems with sets on a common universe Union: Dual: , where Product: 6
8. OutlineHereditary DiscrepancyApplicationsFactorization Norms7
9. Applications: rounding[LSV 86] For any matrix and any , such thatExercise: when , Apply exercise to bits of written in binaryUse in approximation algorithms, e.g. bin packing [R13] [HR 17]:Solve an LP relaxation to get , and apply theorem to get an integral is approximately feasible; fix constraint violations 8
10. Applications: well-distributed sets family of geometric sets in E.g. all axis-aligned boxes in Given a probability measure on , find a set of points, that minimizes the continuous discrepancyUnder mild technical conditions, there is a , , s.t. 9Analogous to herdisc
11. “Proof” in picturesStart with a set with points and tiny Colour and remove blue-coloured points times to get In any : fraction of points in fraction of red points in 10
12. Applications: differential privacyData set: a (multi-)set from the population Each corresponds to one personCounting queries: for each , approximate E.g., set of female smokers, set of male non-smokers with PhD, …Captures many official statistics, released by, e.g., censusCan we approximate without revealing any person’s membership in ?Differential privacy [DMNS ‘06]: a randomized algorithm is differentially private if the distributions of and are similar when . 11Vancouver Courier
13. Differential privacy cont’d[DN 03] An algorithm that answers sufficiently many random queries with sufficient accuracy allows reconstructing almost all of .“Blatantly non-private”, violates differential privacyWhat is minimum error necessary to answer for given under differential privacy?I.e. what is the minimum of over differentially private ?[NTZ 13] The optimal error satisfies 12
14. Reconstruction[MN 12] For any there is a s.t., for any data set , answers to with error allow reconstructing at least points of .Idea: If and , but , then there is a partial colouring of with discrepancy Define by At least elements of have colour If every has a partial colouring with discrepancy , then 13Reconstruction contradicts differential privacyThis gives the lower bound on error. Upper bound later.
15. OutlineHereditary DiscrepancyApplicationsFactorization Norms14
16. Basic bounds on herdiscSome matrix norms: max norm of any column of max norm of any row of : a random colouring worksFor any submatrix , : best bound for the Komlós problem [B 98]For any submatrix , 15While useful, neither of these bounds is tight for all
17. Combining the boundsAnother matrix norm:[NT 15, MN 15] For any matrix ,Neither bound can be improved.[DNTT 18] A different factorization norm gives a approximation. Works for discrepancy under other norms. 16
18. Properties of can be computed in polynomial time: apx alg for There is a differentially private algorithm for counting queries with error where incidence matrix of 17All these properties hold approximately for herdiscImprovements in this conference.[ENU 20]
19. Application: Tusnády’s problemRecall axis-aligned boxes in In the 1980s, Tusnády asked to determine WLOG, , so Incidence matrix of is , where incidence matrix of Then So, . 18[N 17] Upper bound can be improved to
20. Proof of upper boundHow is the upper bound proved?[B 98] For any semi-norm on and any matrix ,, where Take s.t. Use [B 98] with by Chernoff + Union bound 19
21. Tighter bound?Argument gives , for [DNTT 18] Define . Then is efficiently computableExtends to measuring discrepancy in any norm, with the approximation poly-logarithmic in the dimension. 20
22. Some open problemsResolve Tusnády’s problemThe best known upper bound is also not poly-time constructiveDetermine the best possible approximation factor for herdisc[AHG 14] 2-apx is NP-hardDetermine how far herdisc can be from optimal error in differential privacyDevelop a better understanding of discrepancy under other norms, e.g. the matrix operator norm (non-commutative ) 21