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Unconditional Weak derandomization of weak algorithms Explicit versions of Yao s lemma Ronen Shaltiel, University of Haifa :

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Derandomization: The goal Main open problem: Show that BPP=P. (There is evidence that this is hard [IKW,KI]) More generally: Convert: randomized algorithm A(x,r) into: deterministic algorithm B(x) We d like to: 1. Preserve complexity: complexity(B) complexity(A) (known BPP EXP). 2. Preserve uniformity: transformation A B is explicit (known BPP P/poly). n bit long input m bit long coin tosses

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Strong derandomization is sometimes impossible Setting: Communication complexity. x=(x 1,x 2 ) where x 1,x 2 are shared between two players. Exist randomized algorithms A(x,r) (e.g. for Equality) with logarithmic communication complexity s.t. any deterministic algorithm B(x) requires linear communication. Impossible to derandomize while preserving complexity.

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(The easy direction of) Yao s Lemma: A straight forward averaging argument Given randomized algorithm that computes a function f with success 1-ρ on the worst case, namely: Given A:{0,1} n £ {0,1} m {0,1} s.t. x: Pr R Um [A(x,R)=f(x)] ¸ 1-ρ r 2 {0,1} m s.t. the deterministic algorithm B(x)=A(x,r) computes f well on average, namely: Pr X Un [B(X)=f(X)] ¸ 1-ρ Useful tool in bounding randomized algs. Can also be viewed as weak derandomization.

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Yao s lemma as weak derandomization Advantages Applies to any family of algorithms and any complexity measure. Communication complexity. Decision tree complexity. Circuit complexity classes. Construction: B(x)=A(x,r ) preserves complexity. e.g. if A has low communication complexity then B has low communication complexity. Drawbacks Weak derandomization is weak: deterministic alg B succeeds on most but not all inputs. Let s not be too picky. In some scenarios (e.g. communication complexity) strong derandomization is impossible. The argument doesn t give an explicit way to find r and produce B(x)=A(x,r ). Uniformity is not preserved: Even if A is uniform we only get that B(x)=A(x,r ) is nonuniform (B is a circuit).

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The goal: Explicit versions of Yao s Lemma Given randomized algorithm that computes a function f with success 1-ρ on the worst case, namely: Given A:{0,1} n £ {0,1} m {0,1} s.t. x: Pr R Um [A(x,R)=f(x)] ¸ 1-ρ Give explicit construction of a deterministic algorithm B(x) s.t.: B computes f well on average, namely: Pr X Un [B(X)=f(X)] ¸ 1-ρ Complexity is preserved: complexity(B) complexity(A). We refer to this as Explicit weak derandomization.

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Adelman s theorem ( BPP P/poly) follows from Yao s lemma Given randomized algorithm A that computes a function f with success 1-ρ on the worst case. 1. (amplification) amplify success prob. to 1-ρ for ρ=2 -(n+1) 2. (Yaos lemma) deterministic circuit B(x) such that: b := Pr X Un [B(X)f(X)]<ρ<2 -(n+1) b=0 B succeeds on all inputs. Corollary: Explicit version of Yaos lemma for general poly-time algorithms BPP=P. Reminder of talk: Explicit versions of Yaos lemma for weak algorithms: Communication games, Decision trees, Streaming algorithms, AC 0 algorithms.

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Related work: Extracting randomness from the input Idea [Goldreich and Wigderson]: Given a randomized alg A(x,r) s.t. |r| · |x| consider the deterministic alg: B(x)=A(x,x). Intuition: If input x is chosen at random then random coins r:=x is chosen at random. Problem: Input and coins are correlated. (e.g. consider A s.t. 8 input x, coin x is bad for x). GW: Does work if A has the additional property that whether or not a coin toss is good does not depend on the input. GW: It turns out that there are A s with this property.

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The role of extractors in [GW] In their paper Goldreich and Wigderson actually use: B(x)=maj seeds y A(x,E(x,y)) Where E(x,y) is a seeded extractor. Extractors are only used for deterministic amplification (that is to amplify success probability). Alternative view of the argument: 1. Set A (x,r)=maj seeds y A(x,E(r,y)) 2. Apply construction B(x)=A (x,x).

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Randomness extractors Daddy, how do computers get random bits? Do we have to tell that same old story again?

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Seeded Randomness Extractors: Definition and two flavors C is a class of distributions over n bit strings containing k bits of (min)-entropy. A deterministic (seedless) C- extractor is a function E such that for every XєC, E(X) is ε- close to uniform on m bits. A seeded extractor has an additional (short i.e. log n) independent random seed as input. For Seeded extractors C={all X with min-entropy ¸ k} source distribution from C Extractor seed random output Deterministic A distribution X has min-entropy k if x: Pr[X=x] 2 -k Two distributions are ε-close if the probability they assign to any event differs by at most ε.

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Zimand: explicit version of Yao s lemma for decision trees Zimand defines and constructs a stronger variant of seeded extractors E(x,y) called exposure resilient extractors. He considers: B(x)=maj seeds y A(x,E(x,y)) Thm: [Zimand07] If A is a randomized decision tree with q queries that tosses q random coins then: B succeeds on most inputs. (a (1-ρ)-fraction). B can be implemented by a deterministic decision tree with q O(1) queries. Zimand states his result a bit differently. We improve to O(q)

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Our results Develop a general technique to prove explicit versions of Yao s Lemma (that is weak derandomization results). Use deterministic (seedless) extractors that is B(x)=A(x,E(x)) where E is a seedless extractor. The technique applies to any class of algorithms with |r| · |x|. Can sometimes handle |r|>|x| using PRGs. More precisely: Every class of randomized algorithm defines a class C of distributions. An explicit construction of an extractor for C immediately gives an explicit version of Yao s Lemma (as long as |r| · |x|).

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Explicit version of Yao s lemma for communication games Thm: If A is a randomized (public coin) communication game with communication complexity q that tosses m

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Explicit weak derandomization results ExtractorsAlgorithms Extractors for bit-fixing sources [KZ03,GRS04,R07] Decision trees (Improved alternative proof of Zimand s result). 2-source extractors [CG88,Bou06] Communication games Construct from 2-source extractors. Inspired by [KM06,KRVZ06]. Streaming algorithms (can handle |r| ¸ |x|). We construct inspired by PRGs for AC 0. [N,NW] AC 0 (constant depth) (can handle |r| ¸ |x|). We construct using low-end hardness assumptions. Poly-time algorithms (can handle |r| ¸ |x|).

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Constant depth algorithms Consider randomized algorithms A(x,r) that are computable by uniform families of poly-size constant depth circuits. [NW,K] : Strong derandomization in quasi-poly time. Namely, there is a uniform family of quasi-poly-size circuits that succeed on all inputs. Our result: Weak derandomization in poly-time. Namely, there is a uniform family of poly-size circuits that succeed on most inputs. (can also preserve constant depth). High level idea: Reduce # of random coins of A from n c to (log n) O(1) using a PRG. (Based on the hardness of the parity function [H,RS]) Extract random coins from input x using an extractor for sources recognizable by AC 0 circuits. Construct extractors using the hardness of the parity function and ideas from [NW,TV].

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High level overview of the proof To be concrete we consider communication games

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Preparations Thm: If A is a randomized communication game with communication complexity q that tosses m random coins then set B(x)=A(x,E(x)) where E is a 2-source extractor. B succeeds on most inputs. A (1-ρ)-fraction. B can be implemented by a deterministic communication game with communication complexity O(m+q). Define independent random variables X,R by: X Un, R Um. We have that: x: Pr[A(x,R)=f(x)] ¸ 1-ρ It follows that: a := Pr[A(X,R)=f(X)] ¸ 1-ρ We need to show:b := Pr[A(X,E(X))=f(X)] ¸ 1-ρ – (2ρ+2 -2m ). The plan is to show that b ¸ a – (2ρ + 2 -2m ).

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High level intuition At the end of protocol all inputs in a rectangle answer the same way. Consider the entropy in the variable X|rectangle = (X|Q r (X)=v). Independent of answer. Idea: extract the randomness from this entropy. Doesn t make sense: rectangle is defined only after random coins r are fixed. For every choice of random coins r the game A(,r) is deterministic w/complexity q. It divides the set of strings x of length n into 2 q rectangles. Let Q r (x) denote the rectangle of x. x1x1 x2x2 Q r (x 1,x 2 ) Rectangle = 2-source

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Averaging over random coins and rectangles a = Pr[A(X,R)=f(X)] = Σ r Pr[A(X,R)=f(X) R=r] = Σ r Σ v Pr[A(X,R)=f(X) R=r Q r (X)=v] = Σ r Σ v Pr[A(X,r)= f(X) R=r Q r (X)=v] = Σ r Σ v Pr[Q r (X)=v]Pr[R=r|Q r (X)=v]Pr[A(X,r)=f(X)|R=rQ r (X)=v] For every choice of random coins r the game A(,r) is deterministic w/complexity q. It divides the set of strings x of length n into 2 q rectangles. Let Q r (x) denote the rectangle of x. x1x1 x2x2 Q r (x 1,x 2 )

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Averaging over random coins and rectangles b = Pr[A(X,E(X))=f(X)] = Σ r Pr[A(X, E(X))=f(X) E(X)=r] = Σ r Σ v Pr[A(X,E(X))=f(X) E(X)=r Q r (X)=v] = Σ r Σ v Pr[A(X,r)= f(X) E(X)=r Q r (X)=v] = Σ r Σ v Pr[Q r (X)=v]Pr[E(X)=r|Q r (X)=v] Pr[A(X,r)=f(X)|E(X)=rQ r (X)=v] For every choice of random coins r the game A(,r) is deterministic w/complexity q. It divides the set of strings x of length n into 2 q rectangles. Let Q r (x) denote the rectangle of x. x1x1 x2x2 Q r (x 1,x 2 )

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b = Pr[A(X,E(X))=f(X)] Proof (continued) Σ r Σ v Pr[Q r (X)=v]Pr[E(X)=r|Q r (X)=v]Pr[A(X,r)=f(X)|E(X)=rQ r (X)=v] Σ r Σ v Pr[Q r (X)=v]Pr[R=r |Q r (X)=v]Pr[A(X,r)=f(X)|R=r Q r (X)=v] a = Pr[A(X,R)=f(X)]

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Problem: It could be that A(,r) does well on rectangle but poorly on {E(X)=r} Note: A(,r) is constant over rectangle. b = Pr[A(X,E(X))=f(X)] Proof (continued) a = Pr[A(X,R)=f(X)] v v Σ r Σ v Pr[Q r (X)=v]Pr[R=r |Q r (X)=v]Pr[A(X,r)=f(X)|R=r Q r (X)=v] Σ r Σ v Pr[Q r (X)=v]Pr[E(X)=r|Q r (X)=v]Pr[A(X,r)=f(X)|E(X)=rQ r (X)=v] 2 -m R is uniform and independent of X 2 -m E is a 2-source extractor and {Q r (X)=v} is a rectangle Q r (x 1,x 2 )=v x1x1 x2x2 {E(X)=r} Would be fine if f was also constant over rectangle

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Modifying the argument We have that: Pr[A(X,R)=f(x)] ¸ 1-ρ By Yaos lemma deterministic game F w/complexity q Pr[F(X)=f(X)] ¸ 1-ρ Consider randomized algorithm A(x,r) which Simulates A(x,r) Simulates F(x) Let Q r (x) denote the rectangle of A and note that: A(,r) is constant on rectangle {Q r (X)=v}. F(x) is constant on rectangle {Q r (X)=v}.

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Problem: It could be that A(,r) does well on rectangle but poorly on {E(X)=r} Note: A(,r) is constant over rectangle. b = Pr[A(X,E(X))=f(X)] Proof (continued) a = Pr[A(X,R)=f(X)] v v Σ r Σ v Pr[Q r (X)=v]Pr[R=r |Q r (X)=v]Pr[A(X,r)=f(X)|R=r Q r (X)=v] Σ r Σ v Pr[Q r (X)=v]Pr[E(X)=r|Q r (X)=v]Pr[A(X,r)=f(X)|E(X)=rQ r (X)=v] 2 -m R is uniform and independent of X 2 -m E is a 2-source extractor and {Q r (X)=v} is a rectangle Q r (x 1,x 2 )=v x1x1 x2x2 {E(X)=r} Would be fine if f was also constant over rectangle

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Problem: It could be that A(,r) does well on rectangle but poorly on {E(X)=r} Note: A(,r) is constant over rectangle. b = Pr[A(X,E(X))=F(X)] Proof (replace f F) a = Pr[A(X,R)=F(X)] v v Σ r Σ v Pr[Q r (X)=v]Pr[R=r |Q r (X)=v]Pr[A(X,r)=F(X)|R=r Q r (X)=v] Σ r Σ v Pr[Q r (X)=v]Pr[E(X)=r|Q r (X)=v]Pr[A(X,r)=F(X)|E(X)=rQ r (X)=v] 2 -m R is uniform and independent of X 2 -m E is a 2-source extractor and {Qr(X)=v} is a rectangle Q r (x 1,x 2 )=v x1x1 x2x2 {E(X)=r} We have that F is constant over rectangle! |a -a| · ρ |b -b| · ρ

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Finishing up Thm: If A is a randomized communication game with communication complexity q that tosses m random coins then set B(x)=A(x,E(x)) where E is a 2-source extractor. B succeeds on most inputs. A (1-ρ)-fraction. B can be implemented by a deterministic communication game with communication complexity O(m+q). 2-source extractors cannot be computed by communication games. However, we need extractors for relatively large rectangles. Namely 2-source extractors for min-entropy n-(m+q). Each of the two parties can send the first 3(m+q) bits of his input. The sent strings have entropy rate at least ½. Run explicit 2-source extractor on substrings. q.e.d. ???

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Generalizing the argument Consider e.g. randomized decision trees A(x,r). Define Q r (x) to be the leaf the decision tree A(,r) reaches when reading x. Simply repeat argument noting that {Q r (X)=v} is a bit-fixing source. More generally, for any class of randomized algorithms we can set Q r (x)=A(x,r) Can do the argument if we can explicitly construct extractors for distributions that are uniform over {Q r (X)=v} = {A(X,r)=v}. Loosely speaking, need extractors for sources recognizable by functions of the form A(,r). There is a generic way to construct them from a function that cannot be approximated by functions of the form A(,r).

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Conclusion and open problem Loosely speaking: Whenever we have a function that is hard on average against a nonuniform version of a computational model we get an explicit version of Yao s lemma (that is explicit weak derandomization) for the model. Can handle AC0 using the hardness of parity. Gives a conditional weak derandomization for general poly-time algorithms. Assumption is incomparable to [NW,GW]. Open problems: Other ways to handle |r| > |x|. Distributions that aren t uniform.

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That s it …

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