Direct Product :Decoding & Testing

Russell Impagliazzo ( IAS & UCSD )Ragesh Jaiswal ( Columbia U. )Valentine Kabanets ( IAS & SFU )

Avi Wigderson ( IAS )

( based on [IJKW’08, IKW’09] )

Average-Hardness Amplification

f g

hard on fractionof inputs

hard on 1- fractionof inputs

(Nonuniform) Hardness on Average

f s

f is δ-hard (for size s), if every circuit (of size s) fails to compute f on δ inputs.

2n

{0,1}n

{0,1}n {0,1}n

Amplification via Repetition

Intuition

If on a random x one can compute f(x) on < ( 1- ) fraction of inputs,

then on k independent random ( x1,…, xk ), one can compute all ( f(x1),…, f(xk) ) on < ( 1- )k exp(- k) fraction of inputs.

Direct-Product (DP) Function

For f: U R, its k-wise DP function is fk : Uk Rk where:

fk ( x1, …, xk ) = ( f(x1), …, f(xk) )

Direct Product Theorem [Yao’82, Levin’87, GNW’95, Imp’95, IW’97,…]

If: f is - hard ( for size s )

Then: fk is exp(-k)- hard ( for size s * poly(,) )

2n

{0,1}n

2n

{0,1}nk

Direct Product as Error-Correcting Code

DP Encoding [Impagliazzo’02, Trevisan’03]

U Uk

ff k

011010

011010110

010

DP Code Parameters U Uk

f fk

• Local encoding• Local approximate decoding

• Poor distance… Distance amplification: “far-away” messages are mapped to “farther-away” codewords

• Superpoly rate… “Derandomized” DP Code with poly rate.

Direct-Product Code:Decoding

DP Theorem: Constructive Proof [Yao’82, Levin’87, GNW’95, Imp’95, IW’97,…]

Given: circuit C ( of size s*poly(,) )

exp(-k)-computing

fk

Construct: circuit C’

( of size s )

( 1- )-computing f.

2n

{0,1}nk

2n

{0,1}n

List-Decoding Lower Bound

Proof: Pick L = 1/ functions f1, …, fL .

Partition inputs into L blocks of size each.

Define C to agree with fik on block i.

Theorem: To decode from agreement , need the list size (1/).

DP Decoding: Previous Work

r1 X rk

C’

X

b1 b bk

if “enough” bi = f(ri),

then output b

b

• [GNW, IW97,…]: List-size > exp(1/).

• [IJK06]: poly(1/) list size for “large” , but still sub-optimal & complicated.

New Decoding Algorithm [IJKW 08]

Features:

• list-size O(1/) ( tight ! )

• simple algorithm (and analysis)

• generalizes to Derandomized DP Code

Given C that -computes fk , for > exp(- k)

B1 B2

A

Pick random k-set

x

Randomly partition: |A|=|B1|=k/2 Freeze these random choices

On input x, Pick random k-set (A,B2) containing x.

If C is consistent ( C(B1,A)|A = C(A,B2)|

A ), output C(A,B2)|x.

Else re-sample B2 ( O((1/) log 1/) times ).

Given C that -computes fk , for > exp(- k)

Pick random k-set

Randomly partition: |A|=|B1|=k/2

On input x, Pick random k-set (A,B2) containing x.

If C is consistent ( C(B1,A)|A = C(A,B2)|

A ), output C(A,B2)|x.

Else re-sample B2 ( O((1/) log 1/) times ).

Preprocessing

AlgoA,B1 (x):

Main Theorem: DP Decoding

Pick random k-set

Randomly partition: |A|=|B1|=k/2

Preprocessing

On input x, Pick random k-set (A,B2) containing x.If C is consistent ( C(B1,A)|A = C(A,B2)|A ),

output C(A,B2)|x.

Else re-sample B2 ( O((1/) log 1/) times ).

AlgoA,B1 (x):

Theorem: With probability (²) over (B1,A), the resulting circuit Algo (1- )-computes f.

Proof Ideas

Flowers, cores, petals

Flower: determined by S=(A,B)

Core: A

Core values: α=C(A,B)A

Consistent petals: { (A,B’) | C(A,B’)A = α }

[IJKW08]: Flower analysis

B

B4

AA B2

B3

B1

B5

Structure (Decoding)

Then:There are many (²/2) flowers determined by S=(A,B) that are nice.

A flower is nice if it has• correct core ( C(S) = fk (S) ), • many (/2) consistent petals.

Also: In a random nice flower, • almost all consistent petals are mostly correct

(C ¼ fk )

B

B4

AA B2

B3

B1

B5

Assume: C ²-agrees with fk

Consistency Correctness

Correctness of Decoding

There are many (²/4) flowers determined by S=(A,B) that have:

• correct core ( C(S) = fk (S) ), • many (/4) consistent petals, • almost all consistent petals are mostly correct

(C ¼ fk )

Preprocessing likely picks a nice

flower

AlgoA,B likely does not time-out

AlgoA,B (x) likely equals f(x)

Proof of DP Structure: Averaging & Symmetry arguments

Then:There are many (²/2) flowers determined by S=(A,B) that are nice.

A flower is nice if it has• correct core ( C(S) = fk (S) ), • many (/2) consistent petals.

Also: In a random nice flower, • almost all consistent petals are mostly correct

(C ¼ fk )

B

B4

AA B2

B3

B1

B5

Assume: C ²-agrees with fk

Averaging

Symmetry

Proof of DP Structure: Many nice flowers (Averaging)

PrA,B [ ( (A,B) correct ) & ( A has /2 correct extensions (A,B’) ) ] =

PrA,B [(A,B) correct ] - PrA,B [( (A,B) correct ) & ( A has < /2 correct extensions B’ ) ]

Proof of DP Structure: Many nice flowers (Averaging)

PrA,B [ ( (A,B) correct ) & ( A has /2 correct extensions (A,B’) ) ]

PrA,B [(A,B) correct ] - PrA,B [( (A,B) correct ) | ( A has < /2 correct extensions B’ ) ]

- /2 = /2.

Proof of DP Structure: Consistency implies correctness (Symmetry)

Idea: A highly incorrect set S’ can’t be a consistent petal in a random flower with correct core

BS’

A

Proof of DP Structure: Consistency implies correctness

Idea: A highly incorrect set S’ can’t be a consistent petal in a random flower with correct core

BS’

A

Proof of DP Structure: Consistency implies correctness

Idea: A highly incorrect set S’ can’t be a consistent petal in a random flower with correct core.

BS’

A

• f(A) = C(B,A)A &• C(B,A)A = C(S’)A &• C(S’)A f(A).

Contradiction !

Direct-Product Testing

Testing

C : Uk Rk

Is C = fk, for some f : U R ?

Fact: C = fk iff for all pairs of intersecting k-sets (S,S’), with A=SS’, C(S)|A = C(S’)|A

Local Testing: V-Test [Goldreich Safra, Dinur Reingold]

C : Uk Rk

Test for one random pair of intersecting k-sets (S,S’), with A=SS’, if C(S)|A = C(S’)|A

AS S’

DP Testing: More formally …

On C : Uk Rk Test makes few queries, and

(1) Accepts if C = fk. (2) Rejects if C is “far away” from any

fk(2’) Pr [ Test accepts C ] > C fk on > () of inputs.

- Minimize # queries ( 2 ? 3 ? )- Analyze small ( < 1/k ? <

exp(-k) ? )

DP Testing History

Given C : Uk Rk, is C = gk ? #queries

acc.prob.Goldreich-Safra 00 20 .99Dinur-Reingold 06 2 .99Dinur-Goldenberg 08 2 1/kα

Dinur-Goldenberg 08 need > 2 1/kIKW 09 3 exp(-kα)IKW 09* 2 1/kα

* Derandomization

*

Analysis of V-Test

V-Test [GS00,FK00,DR06,DG08]

Randomly pick two k-sets S1 =(B1,A) and S2

=(A,B2)

(with |A| = k1/2 ).

B1 B2

AS1 S2

Accept if C( S1 )A = C( S2 )A

Flowers, cores, petals

Flower: determined by S=(A,B)

Core: A

Core values: α=C(A,B)A

Consistent petals: { (A,B’) | C(A,B’)A = α }

B

B4

AA B2

B3

B1

B5

Structure (Testing)

There are many (²/2) flowers determined by S=(A,B) such that:

There is g : U R so that on almost all consistent petals Bi , C (Bi) gk (Bi).

B

B4

AA B2

B3

B1

B5

Assume: V-Test accepts with prob ²

“Locally” C is DP

Harmonious Flowers

There are many (²/2) harmonious flowers determined by S=(A,B).

Harmonious flower:• many (/2) consistent

petals,• on consistent petals,

V-test accepts almost certainly ( 1-poly(²) ).

B

B4

AA B2

B3

B1

B5

E

C(A, B1 )E ¼ C(A, B2 )E, with |E| = |

A| Assume: V-Test accepts with prob ²

Proof by symmetry arguments

(as in Decoding)

Harmony DP structure

Harmonious flower:• many (/2) consistent

petals,• on consistent petals, V-test

accepts almost certainly ( 1-poly(²) ).

B

B4

AA B2

B3

B1

B5

Main Lemma: Define g(x) = PLURALITY { C( S’ )x | consistent petals S’ , x S’ }.Then C(S’) ¼ gk (S’) for almost all (1-poly(²)) consistent petals S’.

Assume otherwise.A random B1 in Cons has many “minority” elements x where C(B1)x g(x).

A random E ½ B1 has many “minority” elements [Chernoff]

A random B2=(E,D2) is likely s.t. C(B2)E ¼ g(E) [def of g]

Then C(B1)E C(B2)E, Hence no harmony !

B

A D2

D1

E

B1

Proof Sketch of Main Lemma

B2

Decoding vs. Testing

Decoding Testing

There are many harmonious flowers with:• many consistent petals,• restricted to consistent

petals, V-Test accepts almost surely.

V-Test ²-accepts C

Conclude: C(S’) ¼ gk (S’) for almost all consistent petals S’ of the flower.

There are many nice flowers with:• correct core,• many consistent

petals.

C ²-computes fk

Define: g(x) = PLURALITY { C( S )x }consistent petals S :

x2 S

Conclude: g(x) = f(x) for almost all inputs x.

Consistency Correctness

Harmony DP

DP Testing: The Z-Test

Local DP structure

Field of flowers (Ai,Bi)

Each with its ownLocal DP function gi

Global g ? BAA

B2 AA B

i AAB3 AA

B1 AA

Is there GLOBAL DP function g ?

Yes, if ² > 1/ka [DG08] [we re-prove it] ( can “glue together” many flowers )

No, if ² < 1/k [DG08] But, with one extra query, get ² = exp( - ka) !

Z-TestRandomly pick k-sets S1 =(B1,A1), S2=(A1,B2),

S3=(B2,A2)

( |A1| = |A2| = m = k1/2). B1

B2

A1

A2

S1

S2

S3

Accept if C( S1 )A1= C( S2 )A1

and C( S2 )B2 =

C( S3 )B2

Derandomization

Subspace DP Code [IJKW 08]

U = ( Fq )m T Uk T = { 8-dim affine

subspaces of U } ( k = q8 )

• Same list-decoding algo

(from = 1/poly(k) agreement)

• Same DP Test (V-Test) ( for = 1/poly(k) acc prob )Corollary: Polynomial-rate locally

(approximately) list-decodable and locally testable code.

Independent vs. Subspace DP Code All k-sets All d-dim subspaces

• -approx list-decodable from agreement:

exp ( - k ) 1/poly( k )

• 2-query testable, acc prob > 1/poly( k ).

3-query testable, acc prob > exp (-k1/2 )

• Analysis: sampling properties of DP graphs

Chernoff Chebyshev (full independence) (2-wise independence)

Summary ( Derandomized ) DP Code Decoding

and Testing

Analysis of V-Test :Either reject C, or verify that “locally” C = gk ( for some g ),

and get g(x1), …, g(xk) for independent random xi‘s.

Application to 2-Query PCP: Parallel k-repetition for restricted games.

Other ResultsYao’s XOR binary ECC:

fk (x1, …, xk) = f(x1) … f(xk)

- approximately locally list-decodable from agreement ½ + , > exp(- k), with list size O(1/2) (tight)

Open Questions Other examples of derandomized DP

Codes ? Better parameters ???

Derandomized 2-PCP : Re-proving / improving [Moshkovitz-Raz’08, Dinur-Harsha’09] using similar techniques.

( recent progress [Dinur-Meir’09] )