Strong Direct Sum for Randomized Query Complexity Eric Blais Joshua Brody University of Waterloo Swarthmore College Conference on Computational Complexity New Brunswick, New Jersey July 18, 2019
Outline • Introduction • Strong Direct Sum • Query Resistance • Separation Theorem • Open Problems
Direct Sum Theorems Does computing f(x) on k copies scale with k?
Direct Sum Theorems Does computing f(x) on k copies scale with k? Direct Sum Theorem: Computing k copies of f requires k times the resources Direct Product Theorem: Success prob. of computing k copies of f with << k resources is 2 - Ω (k)
Direct Sum Theorems Does computing f(x) on k copies scale with k? Direct Sum Theorem: Computing k copies of f requires k times the resources Direct Product Theorem: Success prob. of computing k copies of f with << k resources is 2 - Ω (k) Strong Direct Sum: computing k copies of f w/error ε requires >> k times the resources
Our Main Results Strong Direct Sum for average query complexity: For any f and any k, computing f k satisfies: R ε (f k ) = ϴ (kR ε /k (f)) Separation Theorem: for all ε > 2 -n^1/3 , there is total function f : {0,1} N → {0,1} such that R ε (f) = ϴ (R(f)log(1/ ε )) Corollary: There is f such that R ε (f k ) = ϴ (klog(k)R ε (f))
Query Complexity aka Decision Tree Complexity x 1 x 3 0 x 8 1 0 abort
Query Complexity aka Decision Tree Complexity Decision Tree for f: {0,1} n → {0,1}: x 1 • internal nodes labeled w/input bits x i x 3 0 • leaves labeled w/output or ABORT • cost(T,x): depth of T on input x x 8 1 Randomized DT: distribution A on decision trees • cost( A ) = max T,x cost(T,x) 0 abort • acost(A) = max x E T~A [cost(T, x)]
Query Complexity aka Decision Tree Complexity Decision Tree for f: {0,1} n → {0,1}: x 1 • internal nodes labeled w/input bits x i x 3 0 • leaves labeled w/output or ABORT • cost(T,x): depth of T on input x x 8 1 Randomized DT: distribution A on decision trees • cost( A ) = max T,x cost(T,x) 0 abort • acost(A) = max x E T~A [cost(T, x)] μ Distributional QC : min E x [cost(T,x)] s.t. Pr[abort] ≤ 훿 and Pr[error] ≤ ε D 훿 , ε (f) Randomized QC : minimum cost of randomized algorithm s.t. R 훿 , ε (f) Pr[abort] ≤ 훿 and Pr[error] ≤ ε Average case Randomized QC : R ε (f) minimum acost of randomized algorithm s.t. Pr[error] ≤ ε
Basic Results μ μ Minimax Lemma: max D 2 훿 ,2 ε (f) max D 훿 /2, ε /2 (f) ≤ R 훿 , ε (f) ≤ μ μ Error Reduction: R (f) ≤ O(log(t)R 1/2, 1/3 (f) ) O(1/t), O(1/t) Average QC vs Aborts: 훿 R 훿 , ε (f) R 훿 ,(1- 훿 ) ε (f) /(1- 훿 ) ≤ R ε (f) ≤
Basic Results Average QC vs Aborts: 훿 R 훿 , ε (f) R 훿 ,(1- 훿 ) ε (f) /(1- 훿 ) ≤ R ε (f) ≤ First inequality: Algorithm A : ε -error, acost(A)= q Second inequality: Algorithm B’ : (1- 훿 ) ε -error, 훿 -abort , q queries.
Basic Results Average QC vs Aborts: 훿 R 훿 , ε (f) R 훿 ,(1- 훿 ) ε (f) /(1- 훿 ) ≤ R ε (f) ≤ First inequality: Algorithm B(x) { emulate A(x) Algorithm A : ε -error, abort if > q/ 훿 queries acost(A)= q } Second inequality: Algorithm A’(x) { repeat: Algorithm B’ : (1- 훿 ) ε -error, emulate B’(x) 훿 -abort , q queries. until no aborts }
Previous Work Information Complexity: [MWY13, MWY15] • strong direct sum for information complexity w/aborts + error • applications for streaming/sketching algorithms Direct Product Theorem: [Drucker 12] • direct product theorems for randomized query complexity Separation Theorems: [GPW15, ABBLSS17] • query complexity separations based on pointer functions • polynomial separation R 0 (f) vs R ε (f) Direct Sum Theorems: [Jain Klauck Santha 10]: R ε (f k ) ≥ 훿 2 k R ε /(1- 훿 )+ 훿 (f) • [Ben-David Kothari 18]: R ε (f k ) ≥ kR ε (f) •
Our Results μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = Ω (kD 1/5 , 40 ε /k (f)) Separation Theorem: There is f : {0,1} N → {0,1} such that for all ε > 2 -N^1/3 , we have R = Ω (R 1/3 (f)log(1/ ε )) 훿 , ε (f) Corollary: There is f such that R 1/3 (f k ) = Ω (klog(k)R ε (f))
Our Results μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = Ω (kD 1/5 , 40 ε /k (f)) Separation Theorem: There is f : {0,1} N → {0,1} such that for all ε > 2 -N^1/3 , we have R = Ω (R 1/3 (f)log(1/ ε )) 훿 , ε (f) Corollary: There is f such that R 1/3 (f k ) = Ω (klog(k)R ε (f)) proof: R 1/3 (f k ) ≥ R 0,1/3 (f k ) = Ω (kR 1/5 ,40/3k (f)) = Ω (klog(k)R 1/3 (f))
Our Results μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = Ω (kD 1/5 , 40 ε /k (f)) Separation Theorem: There is f : {0,1} N → {0,1} such that for all ε > 2 -N^1/3 , we have R = Ω (R 1/3 (f)log(1/ ε )) 훿 , ε (f) Corollary: There is f such that R 1/3 (f k ) = Ω (klog(k)R ε (f)) proof: R 1/3 (f k ) ≥ R 0,1/3 (f k ) = Ω (kR 1/5 ,40/3k (f)) = Ω (klog(k)R 1/3 (f)) Key Technical result: Query-resistant codes: probabilistic encoding G: Σ → {0,1} N such that N/3 bits of G(x) needed to learn anything about x
Outline • Introduction • Strong Direct Sum • Query Resistance • Separation Theorem • Open Problems
μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 ,40 ε /k (f)) Let A be an ε -error algorithm for f k with q queries. Goal: ( ε /k)-error algorithm B for f with q/k queries. Let y = (y 1 ,…, y k ) . Embed(y,i,x) := y , w/i-th coord replaced by x .
μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 ,40 ε /k (f)) Let A be an ε -error algorithm for f k with q queries. Goal: ( ε /k)-error algorithm B for f with q/k queries. Let y = (y 1 ,…, y k ) . Embed(y,i,x) := y , w/i-th coord replaced by x . Algorithm B(x) { carefully select y,i emulate A(EMBED(y,i,x)) abort if problems found }
μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 ,40 ε /k (f)) Let A be an ε -error algorithm for f k with q queries. Goal: ( ε /k)-error algorithm B for f with q/k queries. Let y = (y 1 ,…, y k ) . Embed(y,i,x) := y , w/i-th coord replaced by x . Algorithm B(x) { carefully select y,i emulate A(EMBED(y,i,x)) abort if problems found } Intuition: success on typical coordinate ≥ 1- 10 ε /k else overall success < (1- 10 ε /k) k < 1- ε
μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 , 40 ε /k (f)) k 1- ε ≤ Pr[A(Y) = f k (Y)] = ∏ Pr[A(Y) i = f k (Y) i | A(Y) <i = f k (Y) <i ] Y~ μ k Y~ μ k i=1
μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 , 40 ε /k (f)) k 1- ε ≤ Pr[A(Y) = f k (Y)] = ∏ Pr[A(Y) i = f k (Y) i | A(Y) <i = f k (Y) <i ] Y~ μ k Y~ μ k i=1 Want: i such that (1) conditional error very low: Pr[A err. on i-th coord. | correct on < i] ≤ 10 ε /k (2) Expected # queries on i-th coord not too high: E[queries on i-th coord.] ≤ 3q/k
μ k μ Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 , 40 ε /k (f)) k 1- ε ≤ Pr[A(Y) = f k (Y)] = ∏ Pr[A(Y) i = f k (Y) i | A(Y) <i = f k (Y) <i ] Y~ μ k Y~ μ k i=1 Want: i such that (1) conditional error very low: Pr[A err. on i-th coord. | correct on < i] ≤ 10 ε /k (2) Expected # queries on i-th coord not too high: E[queries on i-th coord.] ≤ 3q/k Fact: at least 2k/3 coords. satisfy (1) Fact: at least 2k/3 coords. satisfy (2) ⟹ There is i* satisfying (1) and (2). Y* := Embed(Y,i*,x).
μ μ k Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 , 40 ε /k (f)) This i* satisfies: 1. E Y~ μ k [ Pr x~ μ [A(Y*) <i* ≠ f k (Y*) <i* ] ] ≤ ε 2. E Y~ μ k [Pr x~ μ [A(Y*) i* ≠ f k (Y*) i* | A(Y*) <i* = f k (Y*) <i* ] ≤ 10 ε /k 3. E Y~ μ k [ E X [q i* (Y*)] ] ≤ 3q/k
μ μ k Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 , 40 ε /k (f)) This i* satisfies: 1. E Y~ μ k [ Pr x~ μ [A(Y*) <i* ≠ f k (Y*) <i* ] ] ≤ ε 2. E Y~ μ k [Pr x~ μ [A(Y*) i* ≠ f k (Y*) i* | A(Y*) <i* = f k (Y*) <i* ] ≤ 10 ε /k 3. E Y~ μ k [ E X [q i* (Y*)] ] ≤ 3q/k Markov Inequality: there is y* such that 1. Pr x~ μ [A(Y*) <i* ≠ f k (Y*) <i* ] ≤ 4 ε 2. Pr x~ μ [A(Y*) i* ≠ f k (Y*) i* | A(Y*) <i* = f k (Y*) <i* ] ≤ 40 ε /k 3. E X [q i* (Y*)] ≤ 12q/k
μ μ k Strong Direct Sum Theorem: D 0, ε (f k ) = 훀 (kD 1/5 , 40 ε /k (f)) This i* satisfies: 1. E Y~ μ k [ Pr x~ μ [A(Y*) <i* ≠ f k (Y*) <i* ] ] ≤ ε 2. E Y~ μ k [Pr x~ μ [A(Y*) i* ≠ f k (Y*) i* | A(Y*) <i* = f k (Y*) <i* ] ≤ 10 ε /k 3. E Y~ μ k [ E X [q i* (Y*)] ] ≤ 3q/k Markov Inequality: there is y* such that 1. Pr x~ μ [A(Y*) <i* ≠ f k (Y*) <i* ] ≤ 4 ε 2. Pr x~ μ [A(Y*) i* ≠ f k (Y*) i* | A(Y*) <i* = f k (Y*) <i* ] ≤ 40 ε /k 3. E X [q i* (Y*)] ≤ 12q/k Algorithm B(x) { z := EMBED(y*,i*,x) emulate A(z) abort if q i* (z) > 120q/k abort if A(z) <i* ≠ f k (z) <i* }
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