Parallel Repetition in Projection Games and a Concentration Bound

Transcription

1 Parallel Repetition in Projection Games and a Concentration Bound Anup Rao (Institute for Advanced Study) Presented by: Ran Raz (Weizmann Institute of Science)

2 Projection Games Player 1 Player 2

3 Projection Games Game G defined by: Player 1 Player 2

4 Projection Games Game G defined by: Distro: X,Y Player 1 Player 2

5 Projection Games Game G defined by: Distro: X,Y Projections: Pxy Player 1 Player 2

6 Projection Games Game G X,Y defined by: Distro: X,Y Projections: Pxy Player 1 Player 2

7 Projection Games Game G X,Y defined by: Distro: X,Y Projections: Pxy x Player 1 y Player 2

8 Projection Games Game G X,Y defined by: Distro: X,Y Projections: Pxy a(x) b(y) x Player 1 y Player 2

9 Projection Games Pxy(a) = b? X,Y Game G defined by: Distro: X,Y Projections: Pxy a(x) b(y) x Player 1 y Player 2

10 Projection Games Pxy(a) = b? X,Y Game G defined by: Distro: X,Y Projections: Pxy a(x) x Player 1 b(y) y Player 2 Val(G) = success probability of best strategy A(G) = #bits needed to represent answers

11 Example: MAX Cut Game

12 Example: MAX Cut Game

13 Example: MAX Cut Game

14 Example: MAX Cut Game

15 Example: MAX Cut Game

16 Example: MAX Cut Game Player 1

17 Example: MAX Cut Game Player 1 Player 2

18 Example: MAX Cut Game Player 1 Player 2

19 Example: MAX Cut Game Player 1 Player 2

20 Example: MAX Cut Game Player 1 Player 2

21 Example: MAX Cut Game Player 1 Player 2

22 Example: MAX Cut Game Player 1 Best strategy: play according to MAX Cut A(G) = 1 Player 2

23 Parallel Repetition of Game Player 1 Player 2

24 Parallel Repetition of Game X1,Y1 X2,Y2 Xn,Yn Player 1 Player 2

25 Parallel Repetition of Game X1,Y1 X2,Y2 Xn,Yn x 1,x 2,...,x n y 1,y2,...,y n Player 1 Player 2

26 Parallel Repetition of Game X1,Y1 X2,Y2 a1,a 2,...,a n x 1,x 2,...,x n Xn,Yn y 1,y2,...,y n b 1,b2,...,bn Player 1 Player 2

27 Parallel Repetition of Game X1,Y1 X2,Y2 Check all conditions Px1y1(a1) = b1? a1,a 2,...,a n x 1,x 2,...,x n Xn,Yn y 1,y2,...,y n b 1,b2,...,bn Px2y2(a2) = b2? Pxnyn(an) = bn? Player 1 Player 2

28 Parallel Repetition of Game X1,Y1 X2,Y2 Check all conditions Px1y1(a1) = b1? G n is another projection game Xn,Yn a1,a 2,...,a n x 1,x 2,...,x n y 1,y2,...,y n b 1,b2,...,bn Px2y2(a2) = b2? Pxnyn(an) = bn? Player 1 Player 2

29 Parallel Repetition of Game X1,Y1 X2,Y2 Check all conditions Px1y1(a1) = b1? G n is another projection game Xn,Yn a1,a 2,...,a n x 1,x 2,...,x n Player 1 y 1,y2,...,y n b 1,b2,...,bn Player 2 Does Px2y2(a2) parallel = b2? repetition reduce the value of Pxnyn(an) projection = bn? games?

30 Bounds If val(g) < 1 ɛ Reference [Raz1995] [Holenstein2007] [This Work] [Raz2008] Result (1 ɛ 32 ) Ω(n/A(G)) (1 ɛ 3 ) Ω(n/A(G)) (1 ɛ 2 ) Ω(n) (1 ɛ 2 ) O(n)

31 Relation To Hardness of Approximation PCP Thm[ALMSS]: NP-Hard to distinguish Value(G)=1 from Value(G)=0.9, even for constant A(G) PCP Thm + Parallel Rep. Thm: NP-Hard to distinguish Value(G)=1 from Value(G)=0.1 Most hardness results obtained by reducing to distinguishing Value(G)=1 from value(g)=0.1.

32 Unique Games Pxy(a) = b? X,Y Game G defined by: Distro: X,Y Permutations: Pxy a(x) x Player 1 b(y) y Player 2 Val(G) = success probability of best strategy A(G) = #bits needed to represent answers

33 Unique Games Conjecture [Khot] Conjecture: For every ɛ, exists m(ɛ) s.t. NP-hard to distinguish unique game with A(G) <mand val(g) > 1 ɛ from game with val(g) ɛ. Many Optimal/Almost Optimal hardness results proved assuming UGC MAX 2-Lin [K], MAX-Cut[K,KKMO,MOO], Vertex Cover [KR], Approximate Coloring [DMR], Sparsest Cut [CKK+,KV], MAX 2- SAT [A]

34 Is the UGC True? Conjecture: For every ɛ, exists m(ɛ) s.t. NP-hard to distinguish unique game with A(G) <mand val(g) > 1 ɛ from game with val(g) ɛ. Reference Result [Raz1995] (1 ɛ 32 ) Ω(n/A(G)) [Holenstein2007] (1 ɛ 3 ) Ω(n/A(G))

35 Is the UGC True? Conjecture: For every ɛ, exists m(ɛ) s.t. NP-hard to distinguish unique game with A(G) <mand val(g) > 1 ɛ from game with val(g) ɛ. Bounds are problematic to amplify gap Reference Result [Raz1995] [Holenstein2007] (1 ɛ 32 ) Ω(n/A(G)) (1 ɛ 3 ) Ω(n/A(G))

36 Is the UGC True? Conjecture: For every ɛ, exists m(ɛ) s.t. NP-hard to distinguish unique game with A(G) <mand val(g) > 1 ɛ from game with val(g) ɛ. Bounds are problematic to amplify gap Reference [Raz1995] [Holenstein2007] Result (1 ɛ 32 ) Ω(n/A(G)) If val(g) 1 ɛ, val(g n ) (1 ɛ ) n (1 ɛ 3 ) Ω(n/A(G))

37 Is the UGC True? Conjecture: For every ɛ, exists m(ɛ) s.t. NP-hard to distinguish unique game with A(G) <mand val(g) > 1 ɛ from game with val(g) ɛ. Bounds are problematic to amplify gap Reference [Raz1995] Result (1 ɛ 32 ) Ω(n/A(G)) If val(g) 1 ɛ, val(g n ) (1 ɛ ) n [Holenstein2007] (1 ɛ 3 ) Ω(n/A(G)) If val(g) < 1 ɛ, val(g n ) (1 ɛ 3 ) Ω(n/A(G))

38 Is the UGC True? Conjecture: For every ɛ, exists m(ɛ) s.t. NP-hard to distinguish unique game with A(G) <mand val(g) > 1 ɛ from game with val(g) ɛ. Reference [This Work] Result (1 ɛ 2 ) Ω(n) Suffices to prove: Conjecture: For every ɛ, exists m(ɛ) s.t. NP-hard to distinguish unique game with A(G) <mand val(g) > 1 ɛ 2.1 from game with val(g) 1 ɛ.

39 Concentration Bound Theorem: If val(g) < γ, probability players win > γ + δ fraction of games = exp( Ω(δ 4 n)). Applications to testing Bell Inequalities in Quantum Mechanics

40 Techniques Player 1 Player 2

41 Techniques X1,Y1 X2,Y2 Xn,Yn Player 1 Player 2

42 Techniques X1,Y1 X2,Y2 Xn,Yn x 1,x 2,...,x n y 1,y2,...,y n Player 1 Player 2

43 Techniques X1,Y1 X2,Y2 a1,a 2,...,a n x 1,x 2,...,x n Xn,Yn y 1,y2,...,y n b 1,b2,...,bn Player 1 Player 2

44 Techniques X1,Y1 X2,Y2 Check all conditions Px1y1(a1) = b1? a1,a 2,...,a n x 1,x 2,...,x n Xn,Yn y 1,y2,...,y n b 1,b2,...,bn Px2y2(a2) = b2? Pxnyn(an) = bn? Player 1 Player 2

45 Techniques New Bound Requires only one conceptual idea over Raz/Holenstein

46 Techniques New Bound Requires only one conceptual idea over Raz/Holenstein Reduction Best Strategy for G n Impossibly good strategy for G

47 Techniques X1,Y1 X2,Y2 X3,Y3 X4,Y4 An-k An-1 An Xn-k,Yn-k Xn-1,Yn-1 Xn,Yn Bn-k Bn-1 Bn

48 Techniques X1,Y1 X2,Y2 X3,Y3 X4,Y4 An-k An-1 An Xn-k,Yn-k FIX Xn-1,Yn-1 Xn,Yn Bn-k Bn-1 Bn

49 Techniques X1,Y1 X2,Y2 W:= event that players win last k games X3,Y3 X4,Y4 An-k Xn-k,Yn-k Bn-k An-1 An FIX Xn-1,Yn-1 Xn,Yn Bn-1 Bn

50 Techniques X1,Y1 X2,Y2 W:= event that players win last k games X3,Y3 X4,Y4 Need: exists b with An-k Xn-k,Yn-k Bn-k Pr[Bn-k,...,Bn =b W] large An-1 FIX Xn-1,Yn-1 Bn-1 An Xn,Yn Bn

51 Techniques X1,Y1 X2,Y2 W:= event that players win last k games X3,Y3 X4,Y4 Need: exists b with An-k Xn-k,Yn-k Bn-k Pr[Bn-k,...,Bn =b W] large An-1 FIX Xn-1,Yn-1 Bn-1 Earlier proofs (averaging): Pr[Bn-k,...,Bn =b W] > 2 -ka(g) An Xn,Yn Bn

52 Techniques X1,Y1 W:= event that Pxy(A)= B X2,Y2 X3,Y3 X4,Y4 Xn-k,Yn-k A FIX Xn-1,Yn-1 B Xn,Yn

53 Techniques X1,Y1 W:= event that Pxy(A)= B X2,Y2 X3,Y3 X4,Y4 After fixing, A independent of B Xn-k,Yn-k A FIX Xn-1,Yn-1 B Xn,Yn

54 Techniques X1,Y1 W:= event that Pxy(A)= B X2,Y2 X3,Y3 After fixing, A independent of B X4,Y4 Xn-k,Yn-k Claim: Exists b, A FIX Xn-1,Yn-1 B Pr[B=b W] > Pr[W]/100 Xn,Yn

55 Techniques Claim: If A,B independent, exists b, Pr[B=b A=B] > Pr[A=B]/100

56 Techniques Claim: If A,B independent, exists b, Pr[B=b A=B] > Pr[A=B]/100 Call b light if Pr[A=b]< Pr[A=B]/100

57 Techniques Claim: If A,B independent, exists b, Pr[B=b A=B] > Pr[A=B]/100 Call b light if Pr[A=b]< Pr[A=B]/100 Pr[A=B B is light] < Pr[A=B]/100

58 Techniques Claim: If A,B independent, exists b, Pr[B=b A=B] > Pr[A=B]/100 Call b light if Pr[A=b]< Pr[A=B]/100 Pr[A=B B is light] < Pr[A=B]/100 Light B Heavy B b1 b2 b3 b4 b5 b6

59 Techniques Claim: If A,B independent, exists b, Pr[B=b A=B] > Pr[A=B]/100 Call b light if Pr[A=b]< Pr[A=B]/100 Pr[A=B B is light] < Pr[A=B]/100 Light B Heavy B b1 b2 b3 b4 b5 At most 100/Pr[A=B] b s b6

60 Techniques Claim: If A,B independent, exists b, Pr[B=b A=B] > Pr[A=B]/100 Call b light if Pr[A=b]< Pr[A=B]/100 Pr[A=B B is light] < Pr[A=B]/100 Light B Heavy B b1 b2 b3 b4 A=B b5 At most 100/Pr[A=B] b s b6

61 Questions?