Analytical Approach to Parallel Repetition

1. Analytical Approach to Parallel Repetition IritDinur David Steurer Weizmann Institute Cornell University Banff, July 2013

2. constraint graph bipartite -regular (for simplicity) random: game no communication between A & B Bob Alice strategy strategy win: label cover given: game find: projection constraint

3. parallel repeated game random: game Bob Alice strategy strategy win: bound in terms of and goal:

4. previous bounds (long history, notoriety) parallel repetition theorem improved: Holenstein’07, Rao’08] [Raz’95, (for projection games) (tight even for games with XOR constraints) [Raz’08] main application: hardness amplification for label cover vs approximation is NP-hard (basis of inapproximability results) What happens if … or ?

5. parallel repetition theorem improved: Holenstein’07, Rao’08] [Raz’95, (for projection games) our results analytical framework to analyze parallel repetition (contrast to previous information-theoretic approach) new bounds low value: (for projection constraints) few repetitions: (for projection constraints, )

6. new bounds low value: (for projection constraints) few repetitions: (for projection constraints, ) implications optimal np-hardness for set cover (and better np-hardness for label cover) -approximation, via [Moshkovitz–Raz, Feige, Moshkovitz] Raz’s parallel-repetition counterexample tight even for small some have but (answers question of O’Donnell)

7. proof overview show game parameter with 1.for all (relaxation) 2. for all (multiplicativity) 3. for all (approximation) proof of parallel-repetition bound 1. 2. 3.

8. proof overview show game parameter with 1.for all (relaxation) 2. for all (multiplicativity) 3. for all (approximation) basic sdp satisfies 1 & 2 but not 3 [Feige–Lovász’92](no efficient param. can satisfy 3) our parameter is the analog over cone of completely positive matrices (instead of cone of p.s.d. matrices) “Hellinger value” is very similar [Barak-Hardt-Haviv-Rao-Regev-S.’08]

9. analytical setup constraint graph

10. analytical setup constraint graph

11. analytical setup label-extended graph constraint graph

12. analytical setup label-extended graph is assignment if and for all For assignment , = prob. that random -neighbor “demands” linear operator = adjacency matrix of label-extended graph success probability for assignments bilinear form over assignments

13. analytical setup label-extended graph tensor product = parallel repetition

14. analytical setup good proxy for claim: proof: assignment for V (because is projecting) squared game sample sample two neighbors win if collision

15. warm-up theorem Suppose constraint graph is expanding (often wlog) Then, implies two steps assignment nonnegative assignment relaxation & multiplicativity approximation (rounding) trivial game

16. assignment nonnegative assignment trivial game for every assignment for all games does not help to win in parallel repetition

17. assignment nonnegative assignment Wlog: Otherwise, can take smaller (using )

18. assignment nonnegative assignment Wlog: Have: What can we do? norms within

19. assignment nonnegative assignment Wlog: Have: norms within

20. assignment nonnegative assignment Wlog: Have: apply to columns of exists column of h with norms within apply to columns of

21. assignment nonnegative assignment Explicitly: Wlog: is “deterministic” ( for at most one per ) Otherwise, write as distribution over such functions with fixed . Use convexity of . (u,a) 2 4 0 0 = ½ + ¼ + ¼ (u,b) 1 0 4 0 (u,c) 1 0 0 4

22. assignment nonnegative assignment Wlog: is “deterministic” ( for at most one per )

23. assignment nonnegative assignment Wlog: is “deterministic” ( for at most one per ) can extract an assignment How good is this assignment?

24. assignment nonnegative assignment can extract an assignment expander! (squared constraint graph) -correlated with expander  -close to constant function!

25. assignment nonnegative assignment can extract an assignment assignment has value expander! (squared constraint graph) value of assignment! -correlated with expander  -close to constant function!

26. extensions  non-expanding constraint graphs corresponds to operator norm for compare against family of trivial games use Cheeger-style rounding to extract “partial assignments” use correlated sampling to combine them low-value regime () use low-correlation version of Cheeger(like for d-to-1 games [S.’10])  few repetitions () show: intermediate non-negative function is close to 0/1 careful rounding to exploit near-integrality

27. open questions operator-theoretic viewpoint applications for other PCP constructions? combination with information-theoretic approach? Thank you! Question?