Optimal parallel repetition for projection games on low threshold - - PowerPoint PPT Presentation

▶
optimal parallel repetition for projection games on low
SMART_READER_LITE
LIVE PREVIEW

Optimal parallel repetition for projection games on low threshold - - PowerPoint PPT Presentation

Jan 28, 2024 162 likes •849 views

Optimal parallel repetition for projection games on low threshold rank graphs Madhur Tulsiani, John Wright, Yuan Zhou TTIC CMU CMU Two-prover one-round games Two-prover one-round games Bipartite graph Two-prover one-round games Bipartite

slide-1
SLIDE 1

Optimal parallel repetition for projection games on low threshold rank graphs

Madhur Tulsiani, John Wright, Yuan Zhou

TTIC CMU CMU

slide-2
SLIDE 2

Two-prover one-round games

slide-3
SLIDE 3

Two-prover one-round games

Bipartite graph

slide-4
SLIDE 4

Two-prover one-round games

Bipartite graph

  • 1. Sample
slide-5
SLIDE 5

Two-prover one-round games

Bipartite graph

  • 1. Sample
  • 2. Give u to P1 and v to P2
slide-6
SLIDE 6

Two-prover one-round games

Bipartite graph

  • 1. Sample
  • 2. Give u to P1 and v to P2
  • 3. Receive answers a and b
slide-7
SLIDE 7

Two-prover one-round games

Bipartite graph

  • 1. Sample
  • 2. Give u to P1 and v to P2
  • 3. Receive answers a and b
  • 4. Accept iff
slide-8
SLIDE 8

Two-prover one-round games

Bipartite graph

  • 1. Sample
  • 2. Give u to P1 and v to P2
  • 3. Receive answers a and b
  • 4. Accept iff
slide-9
SLIDE 9

Two-prover one-round games

Bipartite graph

  • 1. Sample
  • 2. Give u to P1 and v to P2
  • 3. Receive answers a and b
  • 4. Accept iff

(biregular)

slide-10
SLIDE 10

Projection games

Bipartite graph

(biregular)

slide-11
SLIDE 11

Projection games

Bipartite graph Tests of the form

(biregular)

slide-12
SLIDE 12

Projection games

Bipartite graph Tests of the form

(biregular)

is a projection game if for every b, only exists one a to satisfy

slide-13
SLIDE 13

Parallel repetition

Games

slide-14
SLIDE 14

Parallel repetition

Games Play as follows:

slide-15
SLIDE 15

Parallel repetition

Games Play as follows:

  • 1. Sample
slide-16
SLIDE 16

Parallel repetition

Games Play as follows:

  • 1. Sample
  • 2. Give to P1 and to P2
slide-17
SLIDE 17

Parallel repetition

Games Play as follows:

  • 1. Sample
  • 2. Give to P1 and to P2
  • 3. Receive answers
slide-18
SLIDE 18

Parallel repetition

Games Play as follows:

  • 1. Sample
  • 2. Give to P1 and to P2
  • 3. Receive answers
  • 4. Accept iff both

and .

slide-19
SLIDE 19

Parallel repetition

Games Play as follows:

  • 1. Sample
  • 2. Give to P1 and to P2
  • 3. Receive answers
  • 4. Accept iff both

and .

slide-20
SLIDE 20

Strong parallel repetition

is the parallel rep. of times with itself

slide-21
SLIDE 21

Strong parallel repetition

Main Q: How does relate to ?

is the parallel rep. of times with itself

slide-22
SLIDE 22

Strong parallel repetition

Main Q: How does relate to ? If , does ?

is the parallel rep. of times with itself

slide-23
SLIDE 23

Strong parallel repetition

Main Q: How does relate to ? If , does ?

is the parallel rep. of times with itself

NO!!

slide-24
SLIDE 24

Strong parallel repetition

Main Q: How does relate to ? If , does ?

is the parallel rep. of times with itself

NO!!

slide-25
SLIDE 25

Strong parallel repetition

Main Q: How does relate to ? If , does ? does ?

is the parallel rep. of times with itself

NO!!

slide-26
SLIDE 26

Strong parallel repetition

Main Q: How does relate to ? If , does ? does ? (known as strong parallel repetition)

is the parallel rep. of times with itself

NO!!

slide-27
SLIDE 27

Strong parallel repetition

Main Q: How does relate to ? If , does ? does ? (known as strong parallel repetition)

is the parallel rep. of times with itself

NO!! NO!!

slide-28
SLIDE 28

Strong parallel repetition

Main Q: How does relate to ? If , does ? does ? (known as strong parallel repetition) no strong parallel repetition in general

is the parallel rep. of times with itself

NO!! NO!!

slide-29
SLIDE 29

Parallel repetition theorems

If ,

slide-30
SLIDE 30

Parallel repetition theorems

If , then [Raz]

slide-31
SLIDE 31

Parallel repetition theorems

If , then [Raz] if G is a projection game [Rao]

slide-32
SLIDE 32

Parallel repetition theorems

If , then [Raz] if G is a projection game [Rao] (tight due to the Odd Cycle Game [Raz])

slide-33
SLIDE 33

Parallel repetition theorems

If , then [Raz] if G is a projection game [Rao] (tight due to the Odd Cycle Game [Raz]) no strong parallel repetition for projection games

slide-34
SLIDE 34

Expanding games

Bipartite graph

slide-35
SLIDE 35

Expanding games

Bipartite graph G’s spectrum via the SVD:

slide-36
SLIDE 36

Expanding games

Bipartite graph G’s spectrum via the SVD: G is an expanding game if is small.

slide-37
SLIDE 37

Expanding games

Bipartite graph G’s spectrum via the SVD: G is an expanding game if is small. G has low threshold rank if is small for some small .

slide-38
SLIDE 38

Strong PR for Expanding Games

Let be a projection game with 2nd largest singular value . If , then [Raz and Rosen]

slide-39
SLIDE 39

Strong PR for Expanding Games

Let be a projection game with 2nd largest singular value . If , then Q1: Does this have a tight depencence on ? [Raz and Rosen]

slide-40
SLIDE 40

Strong PR for Expanding Games

Let be a projection game with 2nd largest singular value . If , then Q1: Does this have a tight depencence on ? Q2: What about games with low threshold rank? [Raz and Rosen]

slide-41
SLIDE 41

Main result

Let be a projection game with k-th largest singular value . If , then

slide-42
SLIDE 42

Main result

Let be a projection game with k-th largest singular value . If , then Improves on Raz and Rosen when k = 2.

slide-43
SLIDE 43

Main result

Let be a projection game with k-th largest singular value . If , then Improves on Raz and Rosen when k = 2. Optimal for all fixed k.

slide-44
SLIDE 44

Proof strategy

Parallel repetition framework of [Dinur Steurer]

+

Strong Cheeger’s inequality due to [KLLGT13]

slide-45
SLIDE 45

Proof strategy

Parallel repetition framework of [Dinur Steurer]

+

Strong Cheeger’s inequality due to [KLLGT13]

slide-46
SLIDE 46

[Dinur Steurer 2014]

New framework for proving parallel repetition theorems using linear algebra.

slide-47
SLIDE 47

[Dinur Steurer 2014]

New framework for proving parallel repetition theorems using linear algebra. New proof that if G is a projection game [originally from Rao]

slide-48
SLIDE 48

[Dinur Steurer 2014]

New framework for proving parallel repetition theorems using linear algebra. New proof that if G is a projection game [originally from Rao] Connects PR to Cheeger’s inequality.

slide-49
SLIDE 49

Cheeger’s inequality

Technique for finding non-expanding sets in graphs.

slide-50
SLIDE 50

Cheeger’s inequality

Technique for finding non-expanding sets in graphs. Given a graph , let be the second-smallest eigenvalue of its Laplacian. Then

slide-51
SLIDE 51

Cheeger’s inequality

Technique for finding non-expanding sets in graphs. Given a graph , let be the second-smallest eigenvalue of its Laplacian. Then (conductance of )

slide-52
SLIDE 52

Cheeger’s inequality

Technique for finding non-expanding sets in graphs. Given a graph , let be the second-smallest eigenvalue of its Laplacian. Then

slide-53
SLIDE 53

Cheeger’s inequality

Technique for finding non-expanding sets in graphs. Given a graph , let be the second-smallest eigenvalue of its Laplacian. Then ( loses a square )

slide-54
SLIDE 54

[Dinur Steurer 2014]

Proof that if G is a projection game uses Cheeger’s inequality.

slide-55
SLIDE 55

[Dinur Steurer 2014]

Proof that if G is a projection game uses Cheeger’s inequality. ( loses a square )

slide-56
SLIDE 56

[Dinur Steurer 2014]

Proof that if G is a projection game uses Cheeger’s inequality. ( loses a square ) ( because of Cheeger’s inequality)

slide-57
SLIDE 57

[Dinur Steurer 2014]

Proof that if G is a projection game uses Cheeger’s inequality. Better Cheeger’s inequality ⇒ Better PR ( loses a square ) ( because of Cheeger’s inequality)

slide-58
SLIDE 58

[Dinur Steurer 2014]

Proof that if G is a projection game uses Cheeger’s inequality. Strong Cheeger’s inequality ⇒ Strong PR ( loses a square ) ( because of Cheeger’s inequality)

slide-59
SLIDE 59

Proof strategy

Parallel repetition framework of [Dinur Steurer]

+

Strong Cheeger’s inequality due to [KLLGT13]

slide-60
SLIDE 60

Proof strategy

Parallel repetition framework of [Dinur Steurer]

+

Strong Cheeger’s inequality due to [KLLGT13]

slide-61
SLIDE 61

[KLLGT13] A strong Cheeger’s inequality for graphs with low threshold rank.

slide-62
SLIDE 62

[KLLGT13] A strong Cheeger’s inequality for graphs with low threshold rank. Given a graph , let be the eigenvalues of its Laplacian. Then for every ,

slide-63
SLIDE 63

[KLLGT13] A strong Cheeger’s inequality for graphs with low threshold rank. ( when is constant. No square lost!) Given a graph , let be the eigenvalues of its Laplacian. Then for every ,

slide-64
SLIDE 64

Main result

Let be a projection game with k-th largest singular value . If , then

slide-65
SLIDE 65

Second result

Let be a projection game with 2nd largest singular value . If , then

slide-66
SLIDE 66

Second result

Let be a projection game with 2nd largest singular value . If , then Improves on [Raz and Rosen]

slide-67
SLIDE 67

Second result

Let be a projection game with 2nd largest singular value . If , then Improves on [Raz and Rosen] Simple proof: no Cheeger’s needed

slide-68
SLIDE 68

Thanks!