yuan zhou carnegie mellon university joint works with boaz barak, fernando g.s.l. brandão, aram w....
TRANSCRIPT
![Page 1: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/1.jpg)
Yuan ZhouCarnegie Mellon University
Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan
O'Donnell and David Steurer
![Page 2: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/2.jpg)
Constraint Satisfaction Problems
• Given:– a set of variables: V– a set of values: Ω– a set of "local constraints": E
• Goal: find an assignment σ : V -> Ω to maximize #satisfied constraints in E
• α-approximation algorithm: always outputs a solution of value at least α*OPT
![Page 3: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/3.jpg)
Example 1: Max-Cut
• Vertex set: V = {1, 2, 3, ..., n}• Value set: Ω = {0, 1}• Typical local constraint: (i, j) э E wants σ(i) ≠
σ(j)
• Alternative description:– Given G = (V, E), divide V into two parts,– to maximize #edges across the cut
• Best approx. alg.: 0.878-approx. [GW'95]• Best NP-hardness: 0.941 [Has'01, TSSW'00]
![Page 4: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/4.jpg)
Example 2: Balanced Seperator
• Vertex set: V = {1, 2, 3, ..., n}• Value set: Ω = {0, 1}• Minimize #satisfied local constraints: (i, j) э E : σ(i) ≠ σ(j)• Global constraint: n/3 ≤ |{i : σ(i) = 0}| ≤
2n/3
• Alternative description:– given G = (V, E)– divide V into two "balanced" parts,– to minimize #edges across the cut
![Page 5: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/5.jpg)
Example 2: Balanced Seperator (cont'd)
• Vertex set: V = {1, 2, 3, ..., n}• Value set: Ω = {0, 1}• Minimize #satisfied local constraints: (i, j) э E : σ(i) ≠ σ(j)• Global constraint: n/3 ≤ |{i : σ(i) = 0}| ≤
2n/3
• Best approx. alg.: sqrt{log n}-approx. [ARV'04]
• Only (1+ε)-approx. alg. is ruled out even assuming 3-SAT does not have subexp time alg. [AMS'07]
![Page 6: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/6.jpg)
Example 3: Unique Games• Vertex set: V = {1, 2, 3, ..., n}• Value set: Ω = {0, 1, 2, ..., q - 1}• Maximize #satisfied local constraints: (i, j) э E : σ(i) - σ(j) = c (mod q)
• Unique Games Conjecture (UGC) [Kho'02, KKMO'07]
No poly-time algorithm, given an instance where optimal solution satisfies (1-ε) constraints, finds a solution satisfying ε constraints
• Stronger than (implies) "no constant approx. alg."
![Page 7: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/7.jpg)
Example 3: Unique Games (cont'd)
• Vertex set: V = {1, 2, 3, ..., n}• Value set: Ω = {0, 1, 2, ..., q - 1}• Maximize #satisfied local constraints: (i, j) э E : σ(i) - σ(j) = c (mod q)
• UG(ε): to tell whether an instance has a solution satisfying (1-ε) constraints, or no solution satisfying ε constraints
• Unique Games Conjecture (UGC). UG(ε) is hard for sufficiently large q
![Page 8: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/8.jpg)
Example 3: Unique Games (cont'd)
• Implications of UGC– For large class of problems, BASIC-SDP
(semidefinite programming relaxation) achieves optimal approximation ratio
Max-Cut: 0.878-approx. Vertex-Cover: 2-approx. Max-CSP [KKMO '07, MOO '10, KV '03, Rag '08]
![Page 9: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/9.jpg)
Open questions
• Is UGC true?
• Are the implications of UGC true?– Is Max-Cut hard to approximate better than
0.878?
– Is Balanced Seperator hard to approximate with in constant factor?
![Page 10: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/10.jpg)
SDP Relaxation hierarchies
• A systematic way to write tighter and tighter SDP relaxations
• Examples– Sherali-Adams+SDP [SA'90]– Lasserre hierarchy [Par'00, Las'01]
…
?
UG(ε)
r rounds SDP relaxation in roughly time
)(rOn
BASIC-SDP
GW SDP for Maxcut (0.878-approx.)ARV SDP for Balanced Seperator
![Page 11: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/11.jpg)
How many rounds of tighening suffice?• Upperbounds
– rounds of SA+SDP suffice for UG(ε) [ABS'10,
BRS'11]
• Lowerbounds [KV'05, DKSV'06, RS'09, BGHMRS '12]
(also known as constructing integrality gap instances)
– rounds of SA+SDP needed for UG(ε)
– rounds of SA+SDP needed for better-than-0.878 approx for Max-Cut
– rounds for SA+SDP needed for constant approx. for Balanced Seperator
)1(n
))logexp((log )1(n
)1()log(log n
))logexp((log )1(n
![Page 12: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/12.jpg)
Our Results
• We study the performance of Lasserre SDP hierarchy against known lowerbound instances for SA+SDP hierarchy, and show that
• 8-round Lasserre solves the Unique Games lowerbound instances [BBHKSZ'12]
• 4-round Lasserre solves the Balanced Seperator lowerbound instances [OZ'12]
• Constant-round Lasserre gives better-than-0.878 approximation for Max-Cut lowerbound instances [OZ'12]
![Page 13: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/13.jpg)
Proof overview
• Integrality gap instance– SDP completeness: a good vector solution– Integral soundness: no good integral
solution
• A common method to construct gaps (e.g. [RS'09])
– Use the instance derived from a hardness reduction
– Lift the completeness proof to vector world– Use the soundness proof directly
![Page 14: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/14.jpg)
Proof overview (cont'd)
• Our goal: to prove there is no good vector solution– Rounding algorithms?
• Instead, – we bound the value of the dual of the SDP– interpret the dual of the SDP as a proof
system ("Sum-of-squares proof system")– lift the soundness proof to the proof
system
![Page 15: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/15.jpg)
Remarks• Using a connection between SDP hierarchies
and algebraic proof systems, we refute all known UG lowerbound instances and many instances for its related problems
• We provide new insight in designing integrality gap instances -- should avoid soundness proofs that can be lifted to the powerful Sum-of-Squares proof system
• We show that Lasserre is strictly stronger than other hierarchies on UG and its related problems (as it was believed to be)
![Page 16: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/16.jpg)
Outline of the rest of the talk
• Sum-of-Squares proof system and Lasserre hierarchy
• Lift the soundness proofs to the SoS proof system
![Page 17: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/17.jpg)
Sum-of-Squares proof system and Lasserre hierarchy
![Page 18: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/18.jpg)
Polynomial optimization
• Maximize/Minimize• Subject to
all functions are low-degree n-variate polynomial functions
• Max-Cut example: Maximize
s.t.
)(xp
0)(,0)(,0)( 21 xqxqxq m0)(,0)(,0)( '21 xrxrxr m
2)(E jiE(i,j)
xx
ixx ii ,0)1(
![Page 19: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/19.jpg)
Polynomial optimization (cont'd)
• Maximize/Minimize• Subject to
all functions are low-degree n-variate polynomial functions
• Balanced Seperator example: Minimize
s.t.
)(xp
0)(,0)(,0)( 21 xqxqxq m0)(,0)(,0)( '21 xrxrxr m
32
31 ][E,][E
,0)1(
ii
ii
ii
xx
ixx
2)(E jiE(i,j)
xx
![Page 20: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/20.jpg)
Certifying no good solution
• Maximize• Subject to
• To certify that there is no solution better than , simply say that the following equations & inequalities are infeasible
)(xp
0)(,0)(,0)( 21 xqxqxq m0)(,0)(,0)( '21 xrxrxr m
)(xp
0)(,0)(,0)( 21 xqxqxq m0)(,0)(,0)( '21 xrxrxr m
![Page 21: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/21.jpg)
The Sum-of-Squares proof system
• To show the following equations & inequalities are infeasible,
• Show that
• where is a sum of squared polynomials, including 's
• A degree-d "Sum-of-Squares" refutation, where
0)(,0)(,0)( 21 xqxqxq m0)(,0)(,0)( '21 xrxrxr m
)}deg(),deg(){deg(max hqfd iii
)()()(1...1
xhxqxfmi
ii
)(xh)(xri
![Page 22: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/22.jpg)
Example 1
• To refute
• We simply write
• A degree-2 SoS refutation
2)1()2()1(1 xxxx
0)1(
2
xx
x
![Page 23: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/23.jpg)
Example 2: Max-Cut on triangle graph
• To refute
• We "simply" write ... ...
0)1(,0)1(,0)1( 332211 xxxxxx
2)()()( 213
232
221 xxxxxx
![Page 24: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/24.jpg)
)12)(1(
)3222)(1(
)12)(1(
)1()1()(
2)()()(
212133
313123122
3223
2211
232
221
22313221
213
232
221
xxxxxx
xxxxxxxx
xxxxxx
xxxxxxxxxxx
xxxxxx
Example 2: Max-Cut on triangle graph (cont'd)
• A degree-4 SoS refutation
![Page 25: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/25.jpg)
Relation between SoS proof system and Lasserre SDP hierarchy
![Page 26: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/26.jpg)
Finding SoS refutation by SDP
• A degree-d SoS refutation corresponds to solution of an SDP with variables
• The SDP is the same as the dual of -round Lasserre relaxation
• An SoS refutation => upperbound on the dual of optimum of Lasserre => upperbound on the value of Lasserre– e.g. 4-round Lasserre says that Max-Cut of
the triangle graph is at most 2 (BASIC-SDP gives 9/4)
)( dnO
)(d
Bounding SDP value by SoS refutation
![Page 27: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/27.jpg)
Remarks• Positivestellensatz. [Krivine'64, Stengle'73] If
the given equalities & inequalities are infeasible, there is always an SoS refutation (degree not bounded).
• The degree-d SoS proof system was first proposed by Grigoriev and Vorobjov in 1999
• Grigoriev showed degree is needed to refute unsatisfiable sparse -linear equations– later rediscovered by Schoenbeck in
Lasserre world
)(n2F
![Page 28: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/28.jpg)
SoS proofs (in contrast to refutations)
• Given assumptions
to prove that
• A degree-d SoS proof writes
where are sums of squared
polynomials
• Remark. Degree-d SoS proof => degree-d SoS refutation for
)(xp
0)(,0)(,0)( 21 xqxqxq m0)(,0)(,0)( '21 xrxrxr m
)()()()(...1
xhxqxfxpmi
ii
)}deg(),deg(){deg(max hqfd iii
0,)( xp
)(),( xhxgi
![Page 29: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/29.jpg)
Technical Part:Lift the proofs to SoS proof
system
![Page 30: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/30.jpg)
Components of the soundness proof
• Cauchy-Schwarz/Hölder's inequality• Hypercontractivity inequality• Smallsets expand in the noisy hypercube• Invariance Principle• Influence decoding
(of known UG instances)
![Page 31: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/31.jpg)
Hypercontractivity Inequality
• 2->4 hypercontractivity inequality: for low degree polynomial
we have
• Goal of an SoS proof: write
Note that 's are indeterminates
dSnSi
SiS xxf
||],[
)(
22
}1,1{
4
}1,1{])([E9])([E
xfxf
nn x
d
x
ixx
d hxfxfnn
2}2{}1{
4
}1,1{
22
}1,1{),,,(])([E])([E9
S
![Page 32: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/32.jpg)
Traditional proof of hypercontractivity
• 2->4 hypercontractivity inequality: for low degree polynomial
we have
• (Traditional) proof. Apply induction on d and n.– Let – g and h are (n-1)-variate polynomials,
dSnSi
SiS xxf
||],[
)(
22
}1,1{
4
}1,1{])([E9])([E
xfxf
nn x
d
x
hgxf 1
nhng )deg(,1)deg(
![Page 33: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/33.jpg)
Traditional proof of hypercontractivity (cont'd)
]446[E
])[(E][E33
13
1222
1444
1
41
4
hgxghxhgxhgx
hgxf
][E6][E][E 2244 hghg
][E][E6][E][E 4444 hghg
][E9][E96][E9][E9 42/)1(22/2222 hghg dddd
222 ])[E][(E9 hgd 22 ])[(E9 fd
(Cauchy-Schwartz)
(induction)
All equalities are polynomial identities about indeterminatesS
![Page 34: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/34.jpg)
SoS proof of hypercontractivity?
• The square-root in the Cauchy-Schwartz step looks difficult for polynomials
• Solution: Prove a stronger statement -- two-function hypercontractivity inequality
• Theorem. Suppose
• then
eSnSi
SiS
dSnSi
SiS xxgxxf
||],[||],[
)(,)(
][E][E9][E 2222 2 gfgfed
![Page 35: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/35.jpg)
SoS proof of two-fcn hypercontractivity• Write 101101 , ggxgffxf
]4[E
])()[(E][E
101020
21
21
20
21
21
20
20
2101
2101
22
ggffgfgfgfgf
ggxffxgf
][E2][E2][E 20
21
21
20
20
21
21
20
21
21
20
20 gfgfgfgfgfgf
]33[E 20
21
21
20
21
21
20
20 gfgfgfgf
][E93][E93
][E][E9][E][E920
21
21
20
21
21
20
20
21
21
22
gfgf
gfgfeded
eded
0)( 20110 gfgfusing
(induction)
unroll the induction to get the SoS proof][E][E9
])[E][E])([E][(E922
21
20
21
20
2
2
gf
ggffed
ed
![Page 36: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/36.jpg)
Components of the soundness proof
• Cauchy-Schwarz/Hölder's inequality• Hypercontractivity inequality• Smallsets expand in the noisy hypercube• Invariance Principle• Influence decoding
(of known UG instances)
![Page 37: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/37.jpg)
Smallset expansion of noisy hypercube
• For , let
• Theorem. If
• then
• Traditional proof. Let be the projection operator onto the eigenspace of with eigenvalue . I.e. the space spanned by
Rf n }1,1{: )]([E)(1~
1 yfxfTxy
][E
,0))(1)((
f
xxfxf
)(11 )]()([E xfTxf
x
P
1T
}log:)({ 1 Sxx i
SiS
![Page 38: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/38.jpg)
Traditional proof of SSE of noisy hypercube (cont'd)
])([E)]()([E
)]()([E)]()([E
)]()([E
2
11
1
xfxfPxf
xfPTxfxfPTxf
xfTxf
xx
xx
x
])([E]))([(E])([E 24/144/33/4 xfxfPxfxxx
)]([E]))([(E)]([E 4/144/3 xfxfPxfxxx
)]([E]))([(E3)]([E 2/12log4/3 1
xfxfPxfxxx
)]([E])([E3)]([E 2/12log4/3 1
xfxfxfxxx
)]([E)]([E3 4/5log1
xfxfxx
(SoS friendly)
(Holder's)
(SoS friendly)
(SoS friendly)
(hypercontractivity)
(SoS friendly)
(poly. identity)
![Page 39: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/39.jpg)
Traditional proof of SSE of noisy hypercube (cont'd)
)]()([E 1 xfTxfx
4/5log
4/5log
1
1
3
)]([E)]([E3 xfxfxx
)(1 100/
(SoS friendly)
(take )
Key problem: fractional power involved in the Holder's step
Solution: Cauchy-Schwartz/Holders with no fractional power
![Page 40: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/40.jpg)
SoS-izable Cauchy-Schwartz
• Theorem. For any constant a > 0
where SoS is a sum of squared polynomials of degree at most 2
• Remark. and the equality holds when
• Proof. Skipped.
• Corollary. (Holder's) For any constant a > 0
• Proof. Apply C-S twice
SoSfg-gf aa ]E[]E[]E[ 2
1222
SoSgf-fgf abaab ]E[]E[]E[]E[ 3
214
4424
4
XX aa 2
12
Xa
![Page 41: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/41.jpg)
SoS proof of SSE
axba
xx
ab
xx
xfxfPxf
xfPxfxfPxf
214
4424
4
3
])([E]))([(E])([E
)]()([E)]()([E
aba
x
ab xfP 21
442
4 ]))([(E
aba
x
ab xfP 21
422log2
4 ]))([(E31
abaab
21
4
log44
1
3
4/543
log4/541 1
3
(Holder's)
(SoS friendly)
(take )
4/64/5 , ba
(hypercontractivity)
![Page 42: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/42.jpg)
SoS proof of SSE (cont'd)
])([E)]()([E
)]()([E
2
1
xfxfPxf
xfTxf
xx
x
)(1
4/543
log4/541 1
3
100/ (take )
![Page 43: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/43.jpg)
Components of the soundness proof
• Cauchy-Schwarz/Hölder's inequality• Hypercontractivity inequality• Smallsets expand in the noisy hypercube• Invariance Principle• Influence decoding
(of known UG instances)
![Page 44: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/44.jpg)
A few words on Invariance Principle• trickier • "bump function" is used in the original proof
--- not a polynomial!
• but... a polynomial substitution is enough for UG
![Page 45: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/45.jpg)
Max-Cut and Balanced Seperator• An SoS proof for "Majority Is Stablest" theorem
is needed for Max-Cut instances– We don't know how to get around the bump
function issue in the invariance step– Instead, we proved a weaker theorem: "2/pi
theorem" -- suffices to give better-than-0.878 algorithms for known Max-Cut instances
• Balanced Seperator. Key is to SoS-ize the proof for KKL theorem– Hypercontractivity and SSE is also useful
there – Some more issues to be handled
![Page 46: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/46.jpg)
Summary
• SoS/Lasserre hierarchy refutes all known UG instances and Balanced Seperator instances, gives better-than-0.878 approximation for known Max-Cut instances,– certain types of soundness proof does not
work for showing a gap of SoS/Lasserre hierarchy
![Page 47: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/47.jpg)
Open problems
• Show that SoS/Lasserre hierarchy fully refutes Max-Cut instances?– SoS-ize Majority Is Stablest theorem...
• More lowerbound instances for SoS/Lasserre hierarchy?
![Page 48: Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f1d5503460f94c3360e/html5/thumbnails/48.jpg)
Thank you!