Direct Product : Decoding & Testing

1. Direct Product :Decoding & Testing Russell Impagliazzo ( IAS & UCSD ) RageshJaiswal ( Columbia U. ) Valentine Kabanets ( IAS & SFU ) AviWigderson ( IAS ) ( based on [IJKW’08, IKW’09] )

2. Average-Hardness Amplification g f hard on  fraction of inputs hard on 1- fraction of inputs

3. (Nonuniform) Hardness on Average {0,1}n {0,1}n f s {0,1}n 2n f is δ-hard (for size s), if every circuit (of size s)fails to compute f on δ inputs.

4. Amplification via Repetition Intuition If on a random x one can compute f(x) on < ( 1- ) fractionof inputs, then on k independent random (x1,…, xk), one can compute all ( f(x1),…, f(xk) ) on < ( 1- )k exp(- k) fractionof inputs.

5. Direct-Product (DP) Function For f: U  R, its k-wise DP function is fk: UkRkwhere: fk ( x1, …, xk) = ( f(x1), …, f(xk) )

6. Direct Product Theorem [Yao’82, Levin’87, GNW’95, Imp’95, IW’97,…] Then: fkis exp(-k)- hard ( for size s * poly(,) ) {0,1}nk {0,1}n 2n 2n If: f is - hard ( for size s )

7. Direct Product as Error-Correcting Code

8. DP Encoding [Impagliazzo’02, Trevisan’03] U Uk fk f 011 010 0 110 1 1 0 1 0 010

9. DP Code Parameters U Uk • Local encoding • Local approximate decoding fk f • Poor distance… • Distance amplification: • “far-away” messages are mapped to • “farther-away” codewords • Superpoly rate… • “Derandomized” DP Code with poly rate.

10. Direct-Product Code:Decoding

11. DP Theorem: Constructive Proof[Yao’82, Levin’87, GNW’95, Imp’95, IW’97,…] Construct: circuitC’ ( of size s ) ( 1- )-computing f. {0,1}nk {0,1}n 2n   2n Given:circuitC ( of size s*poly(,) ) exp(-k)-computingfk

12. List-Decoding Lower Bound Theorem: To decode from agreement , need the list size (1/). Proof: Pick L = 1/functions f1, …, fL . Partition inputs into L blocks of size  each. Define C to agree with fik on block i.

13. DP Decoding: Previous Work X X r1 rk C’ • [GNW, IW97,…]: List-size > exp(1/). • [IJK06]: poly(1/) list size for “large” , but still sub-optimal & complicated. b1 b bk if “enough” bi = f(ri), then output b b

14. New Decoding Algorithm [IJKW 08] • Features: • list-sizeO(1/)( tight ! ) • simple algorithm (and analysis) • generalizes to Derandomized DP Code

15. Given C that -computes fk, for  > exp(- k) x On input x, Pick random k-set (A,B2) containing x. B1 B2 Pick random k-set A If C is consistent( C(B1,A)|A = C(A,B2)|A ), output C(A,B2)|x. Else re-sample B2( O((1/) log 1/) times ). Randomly partition: |A|=|B1|=k/2 Freeze these random choices

16. Given C that -computes fk, for  > exp(- k) AlgoA,B1 (x): On input x, Pick random k-set (A,B2) containing x. Preprocessing Pick random k-set If C is consistent( C(B1,A)|A = C(A,B2)|A ), output C(A,B2)|x. Else re-sample B2( O((1/) log 1/) times ). Randomly partition: |A|=|B1|=k/2

17. Main Theorem: DP Decoding AlgoA,B1 (x): Preprocessing On input x, Pick random k-set (A,B2) containing x. Pick random k-set Randomly partition: |A|=|B1|=k/2 If C is consistent( C(B1,A)|A = C(A,B2)|A ), output C(A,B2)|x. Else re-sample B2( O((1/) log 1/) times ). Theorem:With probability  (²) over (B1,A), the resulting circuit Algo(1- )-computes f.

18. Proof Ideas

19. Flowers, cores, petals Flower:determined by S=(A,B) Core:A Core values:α=C(A,B)A Consistent petals: { (A,B’) | C(A,B’)A = α} [IJKW08]: Flower analysis B1 B B2 A A B3 B5 B4

20. Structure (Decoding) Assume:C²-agrees with fk • Then: • There are many (²/2) • flowers determined by S=(A,B) • that are nice. • A flower is nice if it has • correct core ( C(S) = fk (S) ), • many (/2) consistent petals. • Also: In a random nice flower, • almost all consistent petals are mostly correct (C ¼fk ) B1 B B2 A A B3 Consistency  Correctness B5 B4

21. Correctness of Decoding Preprocessing likely picks a nice flower • There are many (²/4) • flowers determined by S=(A,B) • that have: • correct core ( C(S) = fk (S) ), • many (/4) consistent petals, • almost all consistent petals are mostly correct (C ¼fk ) AlgoA,B likely does not time-out AlgoA,B(x) likely equals f(x)

22. Proof of DP Structure: Averaging & Symmetry arguments Assume:C²-agrees with fk • Then: • There are many (²/2) • flowers determined by S=(A,B) • that are nice. • A flower is nice if it has • correct core ( C(S) = fk (S) ), • many (/2) consistent petals. • Also: In a random nice flower, • almost all consistent petals are mostly correct (C ¼fk ) B1 B B2 A A Averaging B3 B5 Symmetry B4

23. Proof of DP Structure: Many nice flowers (Averaging) PrA,B [ ( (A,B) correct ) & ( A has/2 correct extensions (A,B’) ) ] = PrA,B [(A,B) correct ] - PrA,B [( (A,B) correct ) & ( A has</2 correct extensions B’ ) ]

24. Proof of DP Structure: Many nice flowers (Averaging) PrA,B [ ( (A,B) correct ) & ( A has/2 correct extensions (A,B’) ) ]  PrA,B [(A,B) correct ] - PrA,B [( (A,B) correct ) | ( A has</2 correct extensions B’ ) ]  -/2 = /2.

25. Proof of DP Structure: Consistency implies correctness (Symmetry) Idea: A highly incorrect set S’ can’t be a consistentpetal in a random flower with correct core B S’ A

26. Proof of DP Structure: Consistency implies correctness Idea: A highly incorrect set S’ can’t be a consistentpetal in a random flower with correct core B S’ A

27. Proof of DP Structure: Consistency implies correctness Idea: A highly incorrect set S’ can’t be a consistentpetal in a random flower with correct core. • f(A) = C(B,A)A & • C(B,A)A = C(S’)A & • C(S’)A  f(A). • Contradiction ! B S’ A

28. Direct-Product Testing

29. Testing C : UkRk IsC = fk, for some f : U  R? Fact:C = fkiff for all pairs of intersectingk-sets (S,S’), withA=SS’, C(S)|A = C(S’)|A

30. LocalTesting: V-Test[GoldreichSafra, DinurReingold] S S’ C : UkRk A Test for one random pair of intersectingk-sets (S,S’), withA=SS’, if C(S)|A = C(S’)|A

31. DP Testing: More formally … • (2’) Pr [Testaccepts C ] >  • Cfk on > () of inputs. • - Minimize #queries ( 2 ? 3 ? ) • Analyze small  (  < 1/k ?  < exp(-k) ? ) OnC : UkRkTest makes few queries, and (1) Accepts if C = fk. (2) Rejects if C is “far away” from any fk

32. DP Testing History * Given C : UkRk, is C =gk? #queries acc.prob. Goldreich-Safra 0020 .99 Dinur-Reingold 06 2 .99 Dinur-Goldenberg 08 2 1/kα Dinur-Goldenberg 08 need > 2 1/k IKW 093 exp(-kα) IKW 09*2 1/kα * Derandomization

33. Analysis of V-Test

34. V-Test[GS00,FK00,DR06,DG08] Randomly pick two k-setsS1 =(B1,A) andS2 =(A,B2) (with|A| = k1/2 ). B1 B2 S1 S2 A Accept if C( S1)A = C( S2)A

35. Flowers, cores, petals Flower:determined by S=(A,B) Core:A Core values:α=C(A,B)A Consistent petals: { (A,B’) | C(A,B’)A = α} B1 B B2 A A B3 B5 B4

36. Structure (Testing) Assume: V-Test accepts with prob² There are many(²/2) flowers determined by S=(A,B) such that: There is g : U  R so that on almost all consistent petals Bi , C (Bi) gk (Bi). B1 B B2 A A B3 B5 B4 “Locally” C is DP

37. Harmonious Flowers C(A, B1 )E¼ C(A, B2 )E, with |E| = |A| Assume: V-Test accepts with prob² • There are many(²/2) harmonious flowers determined by S=(A,B). • Harmonious flower: • many (/2) consistent petals, • on consistent petals, V-test accepts almost certainly ( 1-poly(²) ). B1 B B2 E A A B3 B5 B4 Proof by symmetry arguments (as in Decoding)

38. Harmony  DP structure • Harmonious flower: • many (/2) consistent petals, • on consistent petals, V-test accepts almost certainly ( 1-poly(²) ). B1 B B2 A A Main Lemma: Define g(x) = PLURALITY { C( S’ )x| consistent petals S’ , x S’ }. Then C(S’) ¼gk (S’) for almost all (1-poly(²)) consistent petals S’. B3 B5 B4

39. Proof Sketch of Main Lemma Assume otherwise. A random B1inConshas many “minority” elements xwhere C(B1)x g(x). A random E ½ B1has many “minority” elements [Chernoff] A random B2=(E,D2) is likely s.t. C(B2)E¼ g(E) [def of g] Then C(B1)E C(B2)E, Hence no harmony ! B1 D1 B D2 E A B2

40. Decoding vs. Testing

41. Decoding Testing C ²-computes fk V-Test²-accepts C • There are manynice flowers with: • correct core, • many consistent petals. • There are many harmonious flowers with: • many consistent petals, • restricted to consistent petals, V-Test accepts almost surely. Consistency  Correctness Harmony  DP Define: g(x) = PLURALITY { C( S )x } consistent petals S : x2 S Conclude:C(S’) ¼gk (S’) for almost all consistent petals S’ of the flower. Conclude: g(x) = f(x) for almost all inputsx.

42. DP Testing: The Z-Test

43. Local DP structure Field of flowers (Ai,Bi) Each with its own Local DP functiongi Global g ? B1 B3 B2 Bi B A A A A A A A A A A

44. Is there GLOBAL DP function g ? • Yes, if ² > 1/ka[DG08] [we re-prove it] • ( can “glue together” many flowers ) • No, if² < 1/k [DG08] • But,with one extra query, get ² = exp( - ka) !

45. Z-Test S1 Randomly pick k-setsS1 =(B1,A1), S2=(A1,B2), S3=(B2,A2) ( |A1| = |A2| = m = k1/2). B1 A1 S2 B2 A2 S3 Accept if C( S1)A1= C( S2)A1andC( S2 )B2 = C( S3)B2

46. Derandomization

47. Subspace DP Code [IJKW 08] T Uk T = { 8-dim affine subspacesof U } ( k = q8 ) U = ( Fq )m • Same list-decoding algo • (from  = 1/poly(k) agreement) • Same DP Test (V-Test) • ( for  = 1/poly(k) acc prob ) Corollary: Polynomial-rate locally (approximately) list-decodable and locally testable code.

48. Independent vs. Subspace DP Code All k-sets All d-dim subspaces • -approx list-decodable from agreement: • exp ( - k ) 1/poly(  k ) • 2-query testable, acc prob > 1/poly(  k ). • 3-query testable, • acc prob > exp (-k1/2 ) • Analysis:sampling properties of DP graphs • ChernoffChebyshev • (full independence) (2-wise independence)

49. Summary ( Derandomized ) DP Code Decoding and Testing Analysis of V-Test : Either reject C, or verify that “locally” C = gk( for some g ), and get g(x1), …, g(xk) for independent random xi‘s. Application to 2-Query PCP: Parallel k-repetition for restricted games.

50. Other Results Yao’s XOR binary ECC: fk (x1, …, xk) = f(x1)  …  f(xk) - approximately locally list-decodable from agreement ½ + ,  > exp(- k), with list size O(1/2) (tight)