Preserving Randomness for Adaptive Algorithms Adam R. Klivans - PowerPoint PPT Presentation

Analysis of warm-up steward 2 ε A ( µ ) B ( µ ) µ ◮ Imagine if the steward always returns A ( µ ) or B ( µ )... f 1 f A f B 2 2 f AA f AB f BA f BB 3 3 3 3 f AAA f AAB f ABA f ABB f BAA f BAB f BBA f BBB 4 4 4 4 4 4 4 4 ◮ Union bound: Pr [ X good for every function in tree ] ≥ 1 − 2 k δ 10 / 20

Analysis of warm-up steward 2 ε A ( µ ) B ( µ ) µ ◮ Imagine if the steward always returns A ( µ ) or B ( µ )... f 1 f A f B 2 2 f AA f AB f BA f BB 3 3 3 3 f AAA f AAB f ABA f ABB f BAA f BAB f BBA f BBB 4 4 4 4 4 4 4 4 ◮ Union bound: Pr [ X good for every function in tree ] ≥ 1 − 2 k δ ◮ If so, inductively, every f i is in the tree! 10 / 20

Main result ◮ Theorem : For all n , k , d , ε, δ, γ , there is an efficient one-query steward with 11 / 20

Main result ◮ Theorem : For all n , k , d , ε, δ, γ , there is an efficient one-query steward with ◮ Error ε ′ ≤ O ( ε d ) 11 / 20

Main result ◮ Theorem : For all n , k , d , ε, δ, γ , there is an efficient one-query steward with ◮ Error ε ′ ≤ O ( ε d ) ◮ Failure probability δ ′ ≤ k δ + γ 11 / 20

Main result ◮ Theorem : For all n , k , d , ε, δ, γ , there is an efficient one-query steward with ◮ Error ε ′ ≤ O ( ε d ) ◮ Failure probability δ ′ ≤ k δ + γ ◮ # random bits n + O ( k log( d + 1) + log k log(1 /γ )) 11 / 20

Main steward ◮ Pick random seed X , compute ( X 1 , . . . , X k ) = Gen( X ) 12 / 20

Main steward ◮ Pick random seed X , compute ( X 1 , . . . , X k ) = Gen( X ) ◮ For i = 1 to k : 12 / 20

Main steward ◮ Pick random seed X , compute ( X 1 , . . . , X k ) = Gen( X ) ◮ For i = 1 to k : ◮ Obtain W i = f i ( X i ) 12 / 20

Main steward ◮ Pick random seed X , compute ( X 1 , . . . , X k ) = Gen( X ) ◮ For i = 1 to k : ◮ Obtain W i = f i ( X i ) ◮ Shift and round W i to determine output Y i 12 / 20

Main steward ◮ Pick random seed X , compute ( X 1 , . . . , X k ) = Gen( X ) ◮ For i = 1 to k : ◮ Obtain W i = f i ( X i ) ◮ Shift and round W i to determine output Y i ◮ Ingredient 1: Gen: PRG for block decision trees 12 / 20

Main steward ◮ Pick random seed X , compute ( X 1 , . . . , X k ) = Gen( X ) ◮ For i = 1 to k : ◮ Obtain W i = f i ( X i ) ◮ Shift and round W i to determine output Y i ◮ Ingredient 1: Gen: PRG for block decision trees ◮ Ingredient 2: Deterministic shifting and rounding algorithm 12 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Shifting and rounding algorithm ( d + 1) · 2 ε W i 1 W i 2 W i 3 W i 4 W i 5 13 / 20

Analysis of shifting and rounding algorithm ◮ For W ∈ R d and ∆ ∈ [ d + 1], define R ∆ ( W ) ∈ R d by shifting W according to ∆, then rounding 14 / 20

Analysis of shifting and rounding algorithm ◮ For W ∈ R d and ∆ ∈ [ d + 1], define R ∆ ( W ) ∈ R d by shifting W according to ∆, then rounding ◮ By construction, Y i = R ∆ ( W i ) for some ∆ 14 / 20

Analysis of shifting and rounding algorithm ◮ For W ∈ R d and ∆ ∈ [ d + 1], define R ∆ ( W ) ∈ R d by shifting W according to ∆, then rounding ◮ By construction, Y i = R ∆ ( W i ) for some ∆ ◮ Imagine if Y i = R ∆ ( µ i ) for some ∆... 14 / 20

Analysis of shifting and rounding algorithm ◮ For W ∈ R d and ∆ ∈ [ d + 1], define R ∆ ( W ) ∈ R d by shifting W according to ∆, then rounding ◮ By construction, Y i = R ∆ ( W i ) for some ∆ ◮ Imagine if Y i = R ∆ ( µ i ) for some ∆... f 1 14 / 20

Analysis of shifting and rounding algorithm ◮ For W ∈ R d and ∆ ∈ [ d + 1], define R ∆ ( W ) ∈ R d by shifting W according to ∆, then rounding ◮ By construction, Y i = R ∆ ( W i ) for some ∆ ◮ Imagine if Y i = R ∆ ( µ i ) for some ∆... f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 14 / 20

Analysis of shifting and rounding algorithm ◮ For W ∈ R d and ∆ ∈ [ d + 1], define R ∆ ( W ) ∈ R d by shifting W according to ∆, then rounding ◮ By construction, Y i = R ∆ ( W i ) for some ∆ ◮ Imagine if Y i = R ∆ ( µ i ) for some ∆... f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 f R 2 ( µ 1 ) , R 1 ( µ 2 ) f R 2 ( µ 1 ) , R 2 ( µ 2 ) f R 2 ( µ 1 ) , R 3 ( µ 2 ) ⊥ 3 3 3 14 / 20

Certification tree f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 f R 2 ( µ 1 ) , R 1 ( µ 2 ) f R 2 ( µ 1 ) , R 2 ( µ 2 ) f R 2 ( µ 1 ) , R 3 ( µ 2 ) ⊥ 3 3 3 15 / 20

Certification tree f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 f R 2 ( µ 1 ) , R 1 ( µ 2 ) f R 2 ( µ 1 ) , R 2 ( µ 2 ) f R 2 ( µ 1 ) , R 3 ( µ 2 ) ⊥ 3 3 3 ◮ A sequence ( X 1 , . . . , X k ) of query points determines: 15 / 20

Certification tree f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 f R 2 ( µ 1 ) , R 1 ( µ 2 ) f R 2 ( µ 1 ) , R 2 ( µ 2 ) f R 2 ( µ 1 ) , R 3 ( µ 2 ) ⊥ 3 3 3 ◮ A sequence ( X 1 , . . . , X k ) of query points determines: ◮ A transcript ( f 1 , Y 1 , f 2 , Y 2 , . . . , f k , Y k ) 15 / 20

Certification tree f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 f R 2 ( µ 1 ) , R 1 ( µ 2 ) f R 2 ( µ 1 ) , R 2 ( µ 2 ) f R 2 ( µ 1 ) , R 3 ( µ 2 ) ⊥ 3 3 3 ◮ A sequence ( X 1 , . . . , X k ) of query points determines: ◮ A transcript ( f 1 , Y 1 , f 2 , Y 2 , . . . , f k , Y k ) ◮ A path P through tree 15 / 20

Certification tree f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 f R 2 ( µ 1 ) , R 1 ( µ 2 ) f R 2 ( µ 1 ) , R 2 ( µ 2 ) f R 2 ( µ 1 ) , R 3 ( µ 2 ) ⊥ 3 3 3 ◮ A sequence ( X 1 , . . . , X k ) of query points determines: ◮ A transcript ( f 1 , Y 1 , f 2 , Y 2 , . . . , f k , Y k ) ◮ A path P through tree ◮ If we pick X 1 , . . . , X k independently and u.a.r., ( X 1 ,..., X k ) [ P has a ⊥ node] ≤ k δ Pr 15 / 20

Certification tree f 1 f R 1 ( µ 1 ) f R 2 ( µ 1 ) f R 3 ( µ 1 ) ⊥ 2 2 2 f R 2 ( µ 1 ) , R 1 ( µ 2 ) f R 2 ( µ 1 ) , R 2 ( µ 2 ) f R 2 ( µ 1 ) , R 3 ( µ 2 ) ⊥ 3 3 3 ◮ A sequence ( X 1 , . . . , X k ) of query points determines: ◮ A transcript ( f 1 , Y 1 , f 2 , Y 2 , . . . , f k , Y k ) ◮ A path P through tree ◮ If we pick X 1 , . . . , X k independently and u.a.r., ( X 1 ,..., X k ) [ P has a ⊥ node] ≤ k δ Pr ◮ (Certification) No ⊥ nodes in P = ⇒ every Y i has error O ( ε d ) 15 / 20

Block decision trees ◮ ( k , n , q ) block decision tree: Full q -ary tree of height k 16 / 20

Block decision trees ◮ ( k , n , q ) block decision tree: Full q -ary tree of height k v v a v b v c v aa v ab v ac v ba v bb v bc v ca v cb v cc 16 / 20

Block decision trees ◮ ( k , n , q ) block decision tree: Full q -ary tree of height k ◮ Each internal node v s has a function v s : { 0 , 1 } n → [ q ] v v a v b v c v aa v ab v ac v ba v bb v bc v ca v cb v cc 16 / 20

Block decision trees ◮ ( k , n , q ) block decision tree: Full q -ary tree of height k ◮ Each internal node v s has a function v s : { 0 , 1 } n → [ q ] v 00 , 11 10 01 v a v b v c v aa v ab v ac v ba v bb v bc v ca v cb v cc 16 / 20

Block decision trees ◮ ( k , n , q ) block decision tree: Full q -ary tree of height k ◮ Each internal node v s has a function v s : { 0 , 1 } n → [ q ] ◮ Tree reads nk bits and outputs a leaf v 00 , 11 10 01 v a v b v c v aa v ab v ac v ba v bb v bc v ca v cb v cc 16 / 20

PRG for block decision trees ◮ Theorem : There is an efficient γ -PRG for block decision trees with seed length n + O ( k log q + log k log(1 /γ )) 17 / 20

PRG for block decision trees ◮ Theorem : There is an efficient γ -PRG for block decision trees with seed length n + O ( k log q + log k log(1 /γ )) ◮ Proof idea: Modify parameters of INW generator 17 / 20

PRG for block decision trees ◮ Theorem : There is an efficient γ -PRG for block decision trees with seed length n + O ( k log q + log k log(1 /γ )) ◮ Proof idea: Modify parameters of INW generator ◮ This generator fools the certification tree 17 / 20