Direct-Product testing Parallel Repetitions And Foams Avi Wigderson IAS

1. Direct-Product testing Parallel Repetitions And Foams Avi Wigderson IAS

2. Parallel Repetition of Games and Periodic Foams

3. Isoperimetric problem: Minimize surface area given volume. One bubble. Best solution: Sphere

4. Many bubbles Isoperimetric problem: Minimize surface area given volume. Why? Physics, Chemistry, Engineering, Math… Best solution?: Consider R3 Lord Kelvin 1873 Optimal… Wearie-Phelan1994 Even better

5. Our Problem Minimum surface area of body tiling Rd with period Zd ? Volume=1 d=2 area: Choe’89: Optimal! >4 4

6. [Kindler,O’Donnell, Rao,Wigderson] Bounds in d dimensions ≤OPT≤ ≤ OPT ≤ “Spherical Cubes” exist! Probabilistic construction! (simpler analysis [Alon-Klartag]) OPEN: Explicit?

7. Randomized Rounding Round points in Rd to points in Zd such that for every x,y 1. 2. x y 1 Bound does not depend on d

8. Spine Surface blocking all cycles that wrap around Torus

9. Probabilistic construction of spine Step 1 Randomly construct B in [0,1)d , which in expectation satisfies B Step 2 Sample independent translations of B until [0,1)d is covered, adding new boundaries to spine.

10. PCPs & Linear equations over GF(2) m linear equations: Az = b in n variables: z1,z2,…,zn Given (A,b) 1) Does there exist z satisfying all m equations? Easy – Gaussian elimination 2) Does there exist z satisfying ≥ .9m equations? NP-hard 3) Does there exist z satisfying ≥ .5m equations? Easy – YES! [Hastad] δ>0, it is NP-hard to distinguish (A,b) which are not (½+δ)-satisfiable, from those (1-δ)-satisfiable!

11. Linear equations as Games Game G Draw j  [m] at random Xij Yij Alice Bob αj βj Check if αj +βj = bj Pr [YES] ≤ 1-δ 2n variables: X1,X2,…,Xn, Y1,Y2,…,Yn m linear equations: Xi1 +Yi1 = b1 Xi2 +Yi2 = b2 ….. Xim +Yim = bm Promise: no setting of the Xi,Yi satisfy more than (1-δ)m of all equations

12. Hardness amplification byparallel repetition Game Gk Draw j1,j2,…jk [m] at random Xij1Xij2 XijkYij1Yij2 Yijk Alice Bob αj1αj2 αjkβj1βj2 βjk Check if αjt +βjt = bjt t [k] Pr[YES] ≤ (1-δ2)k [Raz,Holenstein,Rao] Pr[YES] ≥ (1-δ2)k 2n variables: X1,X2,…,Xn, Y1,Y2,…,Yn m linear equations: Xi1 +Yi1 = b1 Xi2 +Yi2 = b2 ….. Xim +Yim = bm Promise: no setting of the Xi,Yi satisfy more than (1-δ)m of all equations [Feige-Kindler-O’Donnell] Spherical Cubes  X [Raz] [KORW]Spherical Cubes 

13. Hardness amplification byother means? Xi1 +Yi1 = b1 Xi2 +Yi2 = b2 ….. Xim +Yim = bm Promise: no setting of the Xi,Yi satisfy more than (1-δ)m of all equations Amplification Xij1…XijkYij1…Yijk Alice Bob αj1…αjkβj1…βjk Test: αjt +βjt = bjt t ? Pr[YES] ≤ (1-δ2)k [Raz,Holenstein,Rao] Pr[YES] ≥ (1-δ2)k [Raz] Major open question: Is there Test’ s.t. Pr[YES] ≤ (1-δ)k ? [Khot] Unique games conjecture Idea: force each player to answer consistently -e.g.make Alice commit to one assignment of Xi’s [Impagliazzo-Kabanets-W] NewTest’ with Pr[YES] ≤ (1-δ)√k

14. Direct-product testingPart of - local testing of codes- property testing- discrete rigidity / stabilityRelated to- local decoding of codes- Yao’s XOR lemma

15. Direct Product: Definition • For f : U  R, the k -wise direct productfk : Uk Rk is fk (x1,…, xk) = ( f(x1), …, f(xk) ). [Impagliazzo’02, Trevisan’03]: DP Code TT (fk)is DP Encoding ofTT ( f ) Rate and distance of DP Code are “bad”, but the code is still very useful in Complexity …

16. Direct-Product Testing • Given an oracle C : Uk Rk Test makes few queries to C, and (1) Accept if C = fk. (2) Reject if C is “far away” from anyfk • (2’) [Inverse Thm] Pr [Test accepts C ] >  •  Cfk on > t() of inputs. • - Minimize #queries e.g. 2, 3,.. ? • Analyze small  e.g. < 1/k ,  < exp(-k) ? • Reduce rate/Derandomize e.g.|C| = poly (|U|) ?

17. DP Testing History • Given an oracle C : Uk Rk, is C¼ gk? #queries acc prob  Goldreich-Safra’00*20 .99 Dinur-Reingold‘062 .99 Dinur-Goldenberg‘082 1/kα Dinur-Goldenberg’082 1/k Impagliazzo-Kabanets-W‘083 exp(-kα) Impagliazzo-Kabanets-W‘08* 2 1/kα / *Derandomization

18. Consistency tests

19. V-Test[GS00,FK00,DR06,DG08] Pick two random k-setsS1 = (B1,A), S2 = (A,B2) withm = k1/2common elements A. Check if C(S1)A = C(S2)A [DG08]: IfV-Testaccepts with probability ² > 1/kα, then there is g : U R s.t. C ¼gkon at least ²/4fraction of k-sets. [IKW09]: Derandomize [DG08]: V-Testfails for ²<1/k B1 B2 S1 S2 A

20. Z-Test Pick three random k-setsS1 =(B1, A1), S2=(A1,B2), S3=(B2, A2) with|A1| = |A2| = m = k1/2. Check if C(S1)A1= C(S2)A1andC(S2)B2 = C(S3)B2 Theorem [IKW09]: IfZ-Testaccepts with probability² > exp(-kα), then there is g : U R s.t. C ¼gkon at least ²/4fraction of k-sets. B1 A1 S1 S2 B2 A2 S3

21. Proof Ideas

22. Proof steps • 1. Pr [ Test accepts C ] >   structure • 2. Structure  local agreement • 3. local agreement  global agreement • Agreement: there is g : U R • s.t. C ¼gkon at least • ²/4fraction of k-sets.

23. Flowers, cores, petals Flower: determined by S=(A,B) Core: A Core values: α=C(A,B)A Petals: ConsA,B = { (A,B’) | C(A,B’)A=α} In a flower, all petals agree on core values! [IJKW08]: Flower analysis for DP-decoding. Symmetry arguments! B1 B B2 A A B3 B5 B4

24. V-Test ) Structure (similar to [FK, DG]) Suppose V-Test accepts with probability ². • ConsA,B={ (A,B’) | C(A,B’)A= C(A,B)A } • Largeness:Many • (²/2) flowers (A,B)have • many (²/2) petals ConsA,B • Harmony:In every large • flower, almost all pairs of • overlapping sets in Consare • almost perfectly consistent. B1 B B2 A A B3 B5 B4

25. V-Test: Harmony Almost all B1 = (E,D1) andB2 = (E,D2) in Cons (with |E|=|A|) satisfy C(A, B1)E C(A, B2)E Proof: Symmetry between A and E (few errors in AuE) Chernoff: ²¼ exp(-kα) D1 B E Implication: Restricted to Cons, an approx V-Test on E accepts almost surely: Unique Decode! A D2 A E

26. Harmony ) Local DP Main Lemma: Assume(A,B) is harmonious. Define g(x) = Plurality { C(A,B’)x | B’2 Cons & x 2 B’ } Then C(A,B’)B’¼ gk (B’), for almost all B’ 2 Cons D1 Intuition: g = g(A,B) is the unique (approximate) decoding of C on Cons(A,B) B E D2 A A Idea: Symmetry arguments. Largness guarantees that random selections are near-uniform. x B’ Challenge: Our analysis gets stuck in²¼ exp(-√k) Can one get ²¼ exp(-k)??

27. Local DP structure across Uk Field of flowers (Ai,Bi) For each, gi s.t C(S)¼gik (S) if S2 Cons(Ai,Bi) Global g? B2 B1 B3 Bi B A3 A2 A1 Ai A A A A A A

28. From local DP to global DPQ: How to “glue” local solutions ?A: If a typical S has two disjoint large, harmonious A’s² > 1/kα high probability (2 queries) [DG]² > exp(-kα)Z-test (3 queries) [IKW]

29. DerandomizationDP code whose length ispoly (|U|), instead of |U|k

30. m-subsets  A Inclusion graphs are Samplers Most lemmas analyze sampling properties of U k-subsets elements S Cons x Subsets: Chernoff bounds – exponential error Subspaces: Chebychev bounds – polynomial error

31. Derandomized DP Test Derandomized DP: U=(Fq)d Encodefk (S), S subspaceof const dimension (as [IJKW08] ) Theorem (Derandomized V-Test): If derandomized V-Test accepts C with probability ² > poly(1/k), then there is a function g : U  R such that C (S) ¼ gk (S) on poly(²) of subspaces S. Corollary: Polynomial rate testable DP-code with [DG] parameters!

32. Summary Spherical cubes exist Power of consistency

33. Counterexample [DG] For every x 2 U pick a random gx: U  R For every k-subset S pick a random x(S) 2 S Define C(S) = gx(S)(S) C(S1)A=C(S2)A “iff” x(S1)=x(S2) V-test passes with high prob: ² = Pr[C(S1)A=C(S2)A] ~ m/k2 No global g if ² < 1/k2 B1 B2 S1 S2 A