A Composition Theorem for Conical Juntas Mika G o os University - PowerPoint PPT Presentation

A Composition Theorem for Conical Juntas Mika G¨ o¨ os University of Toronto IBM Almaden T.S. Jayram G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 1 / 12

Motivation – Randomised communication A ND -O R trees [KS’87] Set-disjointness 1 Ω ( n ) [Raz’91] O R n ◦ A ND 2 [B J KS’04] (info) Tribes [ J KS’03] (info) 2 Ω ( n ) A ND √ n ◦ O R √ n ◦ A ND 2 [HJ’13] [ J KR’09] (info) k n /2 O ( k ) A ND n 1/ k ◦ · · · ◦ O R n 1/ k ◦ A ND 2 [LS’10] (info) G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 2 / 12

Motivation – Randomised communication A ND -O R trees [KS’87] Set-disjointness 1 Ω ( n ) [Raz’91] O R n ◦ A ND 2 [B J KS’04] (info) Tribes [ J KS’03] (info) 2 Ω ( n ) A ND √ n ◦ O R √ n ◦ A ND 2 [HJ’13] [ J KR’09] (info) k n /2 O ( k ) A ND n 1/ k ◦ · · · ◦ O R n 1/ k ◦ A ND 2 [LS’10] (info) O ( n 0.753... ) [Snir’85] Ω ( √ n ) log n A ND 2 ◦ · · · ◦ O R 2 ◦ A ND 2 [JKZ’10] � Gap! G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 2 / 12

New tool – Conical juntas Communication-to-query theorem [GLMWZ’15]: For every boolean function f : { 0, 1 } n → { 0, 1 } , ǫ ( f ◦ IP log n ) ≥ Ω ( deg + BPP cc ǫ ( f )) f f Compose with IP z 1 z 2 z 3 z 4 z 5 IP IP IP IP IP x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 3 / 12

New tool – Conical juntas Communication-to-query theorem [GLMWZ’15]: For every boolean function f : { 0, 1 } n → { 0, 1 } , ǫ ( f ◦ IP log n ) ≥ Ω ( deg + BPP cc ǫ ( f )) Conical juntas: Nonnegative combination of conjunctions 1 2 x 1 + 1 2 x 2 + 1 x 1 x 2 + 1 O R 2 : 2 ¯ 2 x 1 ¯ x 2 Approximate conical junta degree deg + ǫ ( f ) is the least degree of a conical junta h such that ∀ x ∈ { 0, 1 } n : | f ( x ) − h ( x ) | ≤ ǫ . G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 3 / 12

New tool – Conical juntas Communication-to-query theorem [GLMWZ’15]: For every boolean function f : { 0, 1 } n → { 0, 1 } , ǫ ( f ◦ IP log n ) ≥ Ω ( deg + BPP cc ǫ ( f )) Conical juntas: Nonnegative combination of conjunctions 1 2 x 1 + 1 2 x 2 + 1 x 1 x 2 + 1 O R 2 : 2 ¯ 2 x 1 ¯ x 2 Approximate conical junta degree deg + ǫ ( f ) is the least degree of a conical junta h such that ∀ x ∈ { 0, 1 } n : | f ( x ) − h ( x ) | ≤ ǫ . Previous talk: 0-1 coefficients = Unambiguous DNFs G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 3 / 12

New tool – Big picture ← Open problem BPP cc BPP dt Classical world WAPP cc WAPP dt ← [GLMWZ’15] approx. rank + approx. junta deg ← Open problem BQP cc BQP dt Quantum world AWPP cc AWPP dt ← [She’09, SZ’09] approx. poly deg approx. rank G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 4 / 12

This work: A Composition Theorem for Conical Juntas That is: Want to understand approximate conical junta degree of f ◦ g in terms of f and g G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 5 / 12

Our results – Applications ¯ ∧ ∨ M ¯ ¯ ∧ ∧ ∧ ∧ M M M = ¯ ¯ ¯ ¯ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ M M M M M M M M M N AND ◦ 4 ( O R ◦ A ND ) ◦ 2 M AJ ◦ 3 3 • deg + 1/ n ( N AND ◦ k ) ≥ Ω ( n 0.753... ) Query: • deg + 1/ n ( M AJ ◦ k ≥ Ω ( 2.59... k ) 3 ) G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 6 / 12

Our results – Applications ¯ ∧ ∨ M ¯ ¯ ∧ ∧ ∧ ∧ M M M = ¯ ¯ ¯ ¯ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ M M M M M M M M M N AND ◦ 4 ( O R ◦ A ND ) ◦ 2 M AJ ◦ 3 3 • deg + 1/ n ( N AND ◦ k ) ≥ Ω ( n 0.753... ) Query: • deg + 1/ n ( M AJ ◦ k ≥ Ω ( 2.59... k ) 3 ) O ( 2.65 k ) ≥ BPP dt ( M AJ ◦ k 3 ) ≥ Ω ( 2.57 k ) Previously: [JKS’03, LNPV’06, Leo’13, MNSSTX’15] Note: Unamplifiability of error G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 6 / 12

Our results – Applications ¯ ∧ ∨ M ¯ ¯ ∧ ∧ ∧ ∧ M M M = ¯ ¯ ¯ ¯ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ M M M M M M M M M N AND ◦ 4 ( O R ◦ A ND ) ◦ 2 M AJ ◦ 3 3 • deg + 1/ n ( N AND ◦ k ) ≥ Ω ( n 0.753... ) Query: • deg + 1/ n ( M AJ ◦ k ≥ Ω ( 2.59... k ) 3 ) • BPP cc ( N AND ◦ k ) ≥ ˜ Ω ( n 0.753... ) Communication: • BPP cc ( M AJ ◦ k ≥ Ω ( 2.59 k ) 3 ) Note: Log-factor loss G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 6 / 12

Formalising the composition theorem Average degree adeg x ( h ) : = ∑ w C | C | · C ( x ) For h = ∑ w C C , adeg ( h ) : = max x adeg x ( h ) G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 7 / 12

Formalising the composition theorem Average degree adeg x ( h ) : = ∑ w C | C | · C ( x ) For h = ∑ w C C , adeg ( h ) : = max x adeg x ( h ) Simplification: Consider zero-error conical juntas adeg ( O R 2 ) = 3/2, adeg ( M AJ 3 ) = 8/3 Example: G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 7 / 12

Formalising the composition theorem Average degree adeg x ( h ) : = ∑ w C | C | · C ( x ) For h = ∑ w C C , adeg ( h ) : = max x adeg x ( h ) Simplification: Consider zero-error conical juntas adeg ( O R 2 ) = 3/2, adeg ( M AJ 3 ) = 8/3 Example: ————— First formalisation attempt ————— � ? � adeg ( f ◦ g ) ≥ adeg ( f ) · min adeg ( g ) , adeg ( ¬ g ) G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 7 / 12

Formalising the composition theorem Average degree adeg x ( h ) : = ∑ w C | C | · C ( x ) For h = ∑ w C C , adeg ( h ) : = max x adeg x ( h ) Simplification: Consider zero-error conical juntas adeg ( O R 2 ) = 3/2, adeg ( M AJ 3 ) = 8/3 Example: ————— First formalisation attempt ————— � ? � adeg ( f ◦ g ) ≥ adeg ( f ) · min adeg ( g ) , adeg ( ¬ g ) adeg ( O R 2 ◦ M AJ 3 ) = 3.92... < 4 Counter-example! G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 7 / 12

Formalisation – LP duality cf. [She’13, BT’15] adeg ( h ; b 0 , b 1 � – charge b i for reading an input bit that is i Primal / Dual for average degree of f � � min adeg ∑ w C C ; b 0 , b 1 subject to ∑ w C C ( x ) = f ( x ) , ∀ x w C ≥ 0, ∀ C max � Ψ , f � � Ψ , C � ≤ adeg ( C ; b 0 , b 1 ) , ∀ C subject to Ψ ( x ) ∈ R , ∀ x G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 8 / 12

Formalisation – Statement of theorem Regular certificates: Circumventing the counter-example Require that Ψ is balanced (has a primal meaning!) G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 9 / 12

Formalisation – Statement of theorem Regular certificates: Circumventing the counter-example Require that Ψ is balanced (has a primal meaning!) Require that Ψ 1 and Ψ 0 for f and ¬ f “share structure” G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 9 / 12

Formalisation – Statement of theorem Regular certificates: Circumventing the counter-example Require that Ψ is balanced (has a primal meaning!) Require that Ψ 1 and Ψ 0 for f and ¬ f “share structure” Composition Theorem Suppose we have regular LP certificates witnessing adeg ( g ) ≥ b 1 adeg ( f ; b 0 , b 1 ) ≥ a 1 adeg ( ¬ g ) ≥ b 0 adeg ( ¬ f ; b 0 , b 1 ) ≥ a 0 then f ◦ g admits a regular LP certificate witnessing adeg ( f ◦ g ) ≥ a 1 adeg ( ¬ f ◦ g ) ≥ a 0 G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 9 / 12

Regular certificates for M AJ 3 – 1 / 2 – 1 / 2 0 5 / 6 5 / 6 5 / 6 1 / 6 1 / 6 1 / 6 1 1 1 + = – 5 / 6 – 5 / 6 – 5 / 6 0 0 0 – 5 / 6 – 5 / 6 – 5 / 6 0 0 0 ˆ Ψ 1 + ˆ Ψ 1 Ψ 1 Ψ 1 M AJ ◦ 2 M AJ ◦ 3 M AJ ◦ 4 M AJ 3 3 3 3 # dual variables: 3 5 9 17 2.581... 2 2.596... 3 lower bound: 2.5 Open! G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 10 / 12

Subsequent application Eight-author paper: Anshu, Belovs, Ben-David, G¨ o¨ os, Jain, Kothari, Lee, and Santha [ECCC’16] ∃ total F : BPP cc ( F ) ≥ ˜ Ω ( BQP cc ( F ) 2.5 ) 1 BPP cc ( F ) ≥ log 2 − o ( 1 ) χ ( F ) ∃ total F : 2 G¨ o¨ os and Jayram A composition theorem for conical juntas 29th May 2016 11 / 12