Rectangles Are Nonnegative Juntas
(An approach to communication lower bounds)
Mika Göös, Shachar Lovett, Raghu Meka,
Thomas Watson, and David Zuckerman
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
1/8
Alice
Bob
x ∈ {0, 1}n
y ∈ {0, 1}n
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
2/8
Alice
Bob
x ∈ {0, 1}n
y ∈ {0, 1}n
Compute: F ( x, y)
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
2/8
Composed functions f ◦ gn
f
z1
z2
z3
f
Compose with gn
z4
z5
g
g
g
g
g
x 1 y1 x 2 y2 x 3 y3 x 4 y4 x 5 y5
Examples:
• Set-disjointness: OR ◦ ANDn
• Inner-product: XOR ◦ ANDn
• Equality: AND ◦ ¬XORn
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
3/8
Composed functions f ◦ gn
f
z1
z2
z3
f
Compose with gn
z4
z5
g
g
g
g
g
x 1 y1 x 2 y2 x 3 y3 x 4 y4 x 5 y5
In general:
g : {0, 1}b × {0, 1}b → {0, 1} is a small gadget
Alice holds x ∈ ({0, 1}b )n
Bob holds y ∈ ({0, 1}b )n
Inputs x and y encode z := gn ( x, y)
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
3/8
Composed functions f ◦ gn
f
z1
z2
z3
f
Compose with gn
z4
g
z5
g
g
g
g
x 1 y1 x 2 y2 x 3 y3 x 4 y4 x 5 y5
Holy grailg(Conjecture):
: {0, 1}b × {0, 1}b → {0, 1} is a small gadget
In general:
n
Simulate cost-d
randomised
protocol
Alice
holds x ∈ ({
0, 1}b )nfor f ◦ g
using height-d randomised decision
tree for f
Bob holds y ∈ ({0, 1}b )n
cc
n
dt
i.e., Inputs
BPPx and
( f y◦encode
g ) ≈z BPP
:= gn ( x,(yf))
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
3/8
Composed functions f ◦ gn
ple:
Bad examm
ust be
OR
z1
z2
z3
OR
Gadget
fully!
chosen care
z4
z5
OR
OR
OR
OR
OR
x 1 y1 x 2 y2 x 3 y3 x 4 y4 x 5 y5
Holy grailg(Conjecture):
: {0, 1}b × {0, 1}b → {0, 1} is a small gadget
In general:
n
Simulate cost-d
randomised
protocol
Alice
holds x ∈ ({
0, 1}b )nfor f ◦ g
using height-d randomised decision
tree for f
Bob holds y ∈ ({0, 1}b )n
cc
n
dt
i.e., Inputs
BPPx and
( f y◦encode
g ) ≈z BPP
:= gn ( x,(yf))
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
3/8
Composed functions f ◦ gn
ple:
Bad examm
ust be
OR
z1
z2
z3
OR
Gadget
fully!
chosen care
z4
z5
OR
OR
OR
OR
OR
x 1 y1 x 2 y2 x 3 y3 x 4 y4 x 5 y5
Our result:g : {0, 1}b × {0, 1}b → {0, 1} is a small gadget
In general:
n
Simulate cost-d
randomised
protocol
Alice
holds x ∈ ({
0, 1}b )nfor f ◦ g
using height-d randomised decision
tree for f
Bob holds y ∈ ({0, 1}b )n
. . . degree-d conical junta . . .
Inputs x and y encode z := gn ( x, y)
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
3/8
Main structure theorem
Conical d-junta:
Nonnegative combination of d-conjunctions
E XAMPLE : 0.4 · z1 z̄2 + 0.66 · z2 z̄3 + 0.35 · z3 z̄1
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
4/8
Main structure theorem
Conical d-junta:
Nonnegative combination of d-conjunctions
E XAMPLE : 0.4 · z1 z̄2 + 0.66 · z2 z̄3 + 0.35 · z3 z̄1
z1
0
1
z2
z3
0
1
0
1
0
0
Mika Göös (Univ. of Toronto)
1
; z̄1 z̄2 + z1 z3
1
Rectangles Are Nonnegative Juntas
15th June 2015
4/8
Main structure theorem
Conical d-junta:
Nonnegative combination of d-conjunctions
E XAMPLE : 0.4 · z1 z̄2 + 0.66 · z2 z̄3 + 0.35 · z3 z̄1
Junta Theorem:
( f is any partial function)
g is inner-product on Θ(log n) bits
Π is cost-d randomised protocol for f ◦ gn
⇒
There exists a conical d-junta h s.t. ∀z ∈ {0, 1}n :
Pr
(x,y )∼( gn )−1 (z)
Mika Göös (Univ. of Toronto)
[ Π(x, y ) accepts ] ≈ h(z)
Rectangles Are Nonnegative Juntas
15th June 2015
4/8
Main structure theorem
Conical d-junta:
Nonnegative combination of d-conjunctions
E XAMPLE : 0.4 · z1 z̄2 + 0.66 · z2 z̄3 + 0.35 · z3 z̄1
Junta Theorem:
( f is any partial function)
g is inner-product on Θ(log n) bits
Π is cost-d randomised protocol for f ◦ gn
⇒
There exists a conical d-junta h s.t. ∀z ∈ {0, 1}n :
Pr
(x,y )∼( gn )−1 (z)
[ Π(x, y ) accepts ] ≈ h(z)
Cf: • Polynomial approximation [Razborov, Sherstov, Shi–Zhu,. . . ]
• Sherali–Adams vs. LPs [Chan–Lee–Raghavendra–Steurer]
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
4/8
Junta Theorem in pictures
01111101101010001101100110110
01001001010101110101000100111
11011110101001101010110011000
01111010111101010101010001001
00011101100110111010110000010
01010111100001000000111001011
00011001001101101011010001110
11010111110001100010010110010
01010000101100100110111010011
00010011100111001011001100001
01010100110100111111110001100
11111011010101011010001000110
00101111001000111111101110110
10011001100110001101011000011
11010110001101001101011110111
00111010010011111000010000110
01001010110111101001010110011
11000001110110000110101011111
00111010010010001000001110100
10010001000001101000001011000
10110110100100101100101000010
Communication matrix of f ◦ gn
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
5/8
Junta Theorem in pictures
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Communication matrix of gn
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
5/8
Junta Theorem in pictures
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Encode z ∈ {0, 1}n randomly:
(x, y ) ∼ ( gn )−1 (z)
Communication matrix of gn
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
5/8
Junta Theorem in pictures
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Encode z ∈ {0, 1}n randomly:
(x, y ) ∼ ( gn )−1 (z)
Want to understand
Pr[ Π(x, y ) accepts ]
Communication matrix of gn
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
5/8
Junta Theorem in pictures
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Encode z ∈ {0, 1}n randomly:
(x, y ) ∼ ( gn )−1 (z)
Want to understand
Pr[ (x, y ) ∈ R ]
Communication matrix of gn
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
5/8
Junta Theorem in pictures
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Encode z ∈ {0, 1}n randomly:
(x, y ) ∼ ( gn )−1 (z)
Main Theorem:
∃ conical d-junta h,
Pr[ (x, y ) ∈ R ] ≈ h(z)
Communication matrix of gn
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
5/8
Junta Theorem in pictures
z
z
z
z
z
z
z
z
z
z
z
z
z
z
(x, y ) ∼ ( gn )−1 (z)
Main Theorem:
∃ conical d-junta h,
Pr[ (x, y ) ∈ R ] ≈ h(z)
Communication matrix of gn
Mika Göös (Univ. of Toronto)
Encode z ∈ {0, 1}n randomly:
Proof: Partition R into
“conjunctions” R0 :
gn ( R0 ) = 1 1 0 ∗ ∗ ∗ ∗ ∗ ∗
Rectangles Are Nonnegative Juntas
15th June 2015
5/8
Corollaries—Simulation Theorems
Communication-to-query simulation for NP:
NPcc ( f ◦ gn ) = NPdt ( f ) · Θ(b)
. . . recall b = Θ(log n)
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
6/8
Corollaries—Simulation Theorems
Communication-to-query simulation for NP:
NPcc ( f ◦ gn ) = NPdt ( f ) · Θ(b)
. . . recall b = Θ(log n)
Conical d-junta:
0.4 · z1 z̄2 + 0.66 · z2 z̄3 + 0.35 · z3 z̄1
;
d-DNF:
Mika Göös (Univ. of Toronto)
z1 z̄2 ∨
z2 z̄3 ∨
Rectangles Are Nonnegative Juntas
z3 z̄1
15th June 2015
6/8
Corollaries—Simulation Theorems
Communication-to-query simulation for NP:
NPcc ( f ◦ gn ) = NPdt ( f ) · Θ(b)
. . . recall b = Θ(log n)
Also covered!
rectangle size
NP
MA
P
BPP
WAPP
smooth rectangle
approx rank+
Mika Göös (Univ. of Toronto)
SBP
corruption
PostBPP
ext. discrepancy
Rectangles Are Nonnegative Juntas
PP
discrepancy
15th June 2015
6/8
Corollaries—Simulation Theorems
Communication-to-query simulation for NP:
NPcc ( f ◦ gn ) = NPdt ( f ) · Θ(b)
. . . recall b = Θ(log n)
rectangle size
NP
P
MA
[Raz–McKenzie]
[Polynomial apx.]
BPP
WAPP
smooth rectangle
approx rank+
Mika Göös (Univ. of Toronto)
SBP
corruption
PostBPP
ext. discrepancy
Rectangles Are Nonnegative Juntas
PP
discrepancy
15th June 2015
6/8
Resolving open problems
Query lower bound ; Communication lower bound
1
From [Böhler–Glaßer–Meister’06]:
SBPcc is not closed under intersection
SBP: Small bounded-error computations
yes-inputs accepted with prob. ≥ α
no-inputs accepted with prob. ≤ α/2
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
7/8
Resolving open problems
Query lower bound ; Communication lower bound
1
From [Böhler–Glaßer–Meister’06]:
SBPcc is not closed under intersection
2
From [Klauck’03]:
Corruption does not characterise MAcc
i.e., MAcc ( SBPcc
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
7/8
Resolving open problems
Query lower bound ; Communication lower bound
1
From [Böhler–Glaßer–Meister’06]:
SBPcc is not closed under intersection
2
From [Klauck’03]:
Corruption does not characterise MAcc
i.e., MAcc ( SBPcc
3
From [Kol–Moran–Shpilka–Yehudayoff’14]:
No efficient error amplification for e-rank+
4
From [Yannakakis’88]: (Subsequent work)
Clique vs. Independent Set problem
i.e., coNPcc ( F ) UPcc ( F )
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
7/8
Summary
Main result: Junta Theorem
Let g = inner-product on b = Θ(log n) bits
Let (x, y ) ∼ ( gn )−1 (z)
=⇒ Pr[ Π(x, y ) accepts ] ≈ Conical junta of z
Open problems
More applications of Junta Theorem?
Simulation theorems for BPP?
Improve gadget size down to b = O(1)
(Would give new proof of Ω(n) bound for set-disjointness)
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
8/8
Summary
Main result: Junta Theorem
Let g = inner-product on b = Θ(log n) bits
Let (x, y ) ∼ ( gn )−1 (z)
=⇒ Pr[ Π(x, y ) accepts ] ≈ Conical junta of z
Open problems
More applications of Junta Theorem?
Simulation theorems for BPP?
Improve gadget size down to b = O(1)
(Would give new proof of Ω(n) bound for set-disjointness)
Cheers!
Mika Göös (Univ. of Toronto)
Rectangles Are Nonnegative Juntas
15th June 2015
8/8