A Dual Polynomial for OR

A Dual Polynomial for OR Robert ˇ Spalek Google, Inc. ∗ spalek@google.com Abstract W e reprov e that the approximate degree of the OR function on n bits is Ω( √ n ) . W e consider a linear program which is feasible if and only if there is an approximate polynomial for a given function, and apply the duality theory . The duality theory says that the primal program has no solution if and only if its dual has a solution. Therefore one can prove the nonexistence of an approximate polynomial by e xhibiting a dual solution, coined the dual polynomial . W e construct such a polynomial. 1 Intr oduction W e study the approximation of Boolean functions by real-v alued polynomials. This line of research was initiated by Minsky and Papert [MP68]. An n -bit Boolean function f is represented by a multiv ariate polynomial p ( x 1 , . . . , x n ) . Nisan and Szegedy [NS94] defined the approximate degree of a function f under the ` ∞ -norm, denoted g deg( f ) , as the smallest degree for which there e xists a polynomial that is close to the function pointwise. Sev eral complexity measures hav e been since shown to be lo wer -bounded in terms of g deg( f ) : circuit size [Bei93], or quantum query complexity [BBC + 01]. Consider the OR function on n bits. Nisan and Szegedy [NS94] sho wed that g deg(OR n ) = Θ( √ n ) , and Paturi [P at92] extended their bound to all symmetric functions. The existence of an approximate polynomial can be described by a linear program; let us coin it the primal program. Using the duality theory of linear programming, one can show the non-existence of an approximate polynomial for a function f by exhibiting a solution to its dual program, a so-called dual polynomial for f . Recently , se v eral papers hav e appeared that use dual polynomials to prove good communication complexity lo wer bounds: Sherstov [She07] and Shi and Zhu [SZ07] show two-party quantum communication lower bounds, and Lee and Shraibman [LS08] and Chattopadhyay and Ada [CA08] show multi-party randomized communication lo wer bounds in the number-on-the- forehead model. The basic idea of these papers is as follows. One defines a special pattern matrix (or tensor in the multi-party case) whose entries are values of a certain polynomial. The structure of the pattern matrix allo ws one to relate properties of the polynomial to properties of the matrix, such as its trace norm. The pattern matrix formed from the dual polynomial forms a witness to the lar ge trace norm of the matrix. The communication complexity is then lower -bounded in terms of the trace norm. None of these papers actually presents an explicit dual polynomial for any function; they only use its e xistence and some inequalities guaranteed by the duality principle from the kno wn bounds on the approximate degree. It is natural to ask what a dual polynomial looks like for the simplest functions. In this short note, we address this question and present an asymptotically optimal dual polynomial for the OR function. Our proof extends the ideas of Buhrman and Szegedy [BS03]. ∗ Most of the work conducted while at CWI, Amsterdam, in February 2003. 1 2 Pr eliminaries 2.1 Symmetric polynomials W e represent Boolean functions by polynomials in the F ourier basis, where +1 corresponds to the logical value 0 (false) and − 1 to the logical value 1 (true). In this basis, multiplication corresponds to the exclusi ve OR. W e say that f : {± 1 } n → {± 1 } is a symmetric function, if f ( x ) = f ( x σ ) for ev ery permutation σ ∈ S n and x ∈ {± 1 } n , where x σ denotes a σ -permuted version of x , with ( x σ ) i = x σ ( i ) . Let p : {± 1 } n → < be a polynomial in variables x 1 , . . . , x n . Since x 2 i = 1 , we can restrict ourselves to multilinear polynomials, where each variable appears with de gree at most 1. W e say that p has de gree d and pur e high de gr ee d 0 , if each term in p is a product of at most d and at least d 0 variables. W e say that p is an ε -appr oximation for a function f , if | p ( x ) − f ( x ) | ≤ ε for ev ery x ∈ {± 1 } n . If p is an ε -approximation of a symmetric function f , then there exists a symmetric polynomial p 0 with the same degree, pure high de gree, and approximation factor: p 0 ( x ) = 1 n ! P σ ∈ S n p ( x σ ) . Let [ n ] = { 0 , 1 , . . . , n } . Given a symmetric function f : {± 1 } n → {± 1 } , one can define a single-v ariate function F : [ n ] → {± 1 } such that f ( x ) = F ( | x | ) , where | x | = n − ( x 1 + ··· + x n ) 2 is the Hamming weight of x , i.e., the number of minuses in x . Analogously , follo wing [MP68], gi v en a symmetric multilinear polynomial p : {± 1 } n → < , one can define a single-variate polynomial P : [ n ] → < of the same degree such that P ( k ) = p ( − 1 , . . . , − 1 | {z } k , +1 , . . . , +1 | {z } n − k ) for all k ∈ [ n ] , p = P  n − ( x 1 + · · · + x n ) 2  mod ( x 2 1 − 1) , . . . , mod ( x 2 n − 1) . Note that the pure high degree of p does not correspond to the smallest degree of a k -term in P ( k ) . When we talk about the pure high de gree of a single-v ariate polynomial, we mean the pure high degree of its corresponding multilinear polynomial. Let p, q : {± 1 } n → < . Define a scalar pr oduct as p · q = P x ∈{± 1 } n p ( x ) q ( x ) . This induces a scalar product P · Q = P n i =0  n i  P ( i ) Q ( i ) on the space of symmetric polynomials. Similarly , the ` 1 -norm k p k 1 = P x ∈{± 1 } n | p ( x ) | induces an ` 1 -norm k P k 1 = P n i =0  n i  | P ( i ) | . Let p : {± 1 } n → < be a multilinear polynomial of degree d and pure high degree d 0 , and consider q ( x ) = p ( x ) · ( x 1 · · · x n ) mo d ( x 2 i − 1) . In the functional interpretation, q ( x ) equals p ( x ) multiplied by the parity of x . Thanks to the term cancellation x 2 i = 1 , each term in q corresponds to the complement of a term in p , and therefore q has degree n − d 0 and pure high degree n − d . Now , assume that p (and thus also q ) are symmetric, and consider their corresponding single-v ariate polynomials P, Q . Then Q ( k ) = P ( k ) · ( − 1) k , and the de gree of P corresponds to n minus the pure high de gree of Q and vice versa. 2.2 Linear program f or polynomial approximation Theorem 1. A total Boolean function f : {± 1 } n → {± 1 } has ε -appr oximate degr ee at least d if and only if there exists a polynomial b : {± 1 } n → < with pur e high degr ee d such that k b k 1 b · f < 1 ε . Pr oof. f can be ε -approximated by a polynomial of degree d − 1 is equi valent to the feasibility of the follo wing primal linear program. Consider the F ourier basis on the space of multilinear polynomials: { χ S } S ⊆{ 1 ,...,n } , where χ S ( x ) = Q i ∈ S x i . Let F = { χ S ( x ) } x,S denote the F ourier transform o ver Z n 2 , index ed by {± 1 } n and S ⊆ { 1 , . . . , n } , and let a denote a v ector of Fourier coef ficients. F a ≥ f − ε F a ≤ f + ε a S = 0 for | S | ≥ d 2 The primal program is unfeasible if and only if its dual is feasible. The dual program is as follows. ( b + − b − ) · f > ( b + + b − ) · ε ( b + − b − ) F = c b + , b − ≥ 0 c S = 0 for | S | < d ⇐ ⇒ b · f > | b | · ε bF = c c S = 0 for | S | < d W e can assume that b + and b − of the optimal solution are disjoint, i.e., b + ( x ) b − ( x ) = 0 for each x , otherwise we could lower the right-hand side of the first inequality by subtracting the same constant min( b + ( x ) , b − ( x )) > 0 from both b + ( x ) and b − ( x ) , and the remaining expressions would stay unchanged. Let b = b + − b − and | b | = b + + b − . The constraints bF = c and c S = 0 for | S | < d say that b has pure high degree d . The dual is feasible if and only if there exists such a b with b · f > | b | · ε = ε k b k 1 . Note that if f is symmetric, then it suffices to look for a dual polynomial b in the space of symmetric polynomials. Let us reformulate the condition in the language of single-variate polynomials. Corollary 2. A total symmetric Boolean function F : [ n ] → {± 1 } has ε -appr oximate de gr ee at least d if and only if ther e e xists a polynomial B : [ n ] → < with pure high de gree d such that k B k 1 B · F < 1 ε . 3 Dual polynomial f or OR First, we define a certain lo w-degree polynomial P and sho w that its norm k P k 1 is not too large compared to its value P (0) . This polynomial will be crucial for defining the dual polynomial for OR. The design of our polynomial comes from extending the ideas of Buhrman and Sze gedy [BS03]. Definition 3. Let m = b √ n c and let S = { i 2 : i ∈ [ m ] } ∪ { 2 } denote the set containing the integ er squar es up to n and the number 2 . Define a polynomial P ( x ) = 2( − 1) n − m − 1 m ! 2 n ! · Y i ∈ [ n ] −S ( x − i ) . The multiplicativ e factor of P is chosen such that P (0) = 1 . The degree of P is n − m − 1 . Lemma 4. F or every pair of inte gers k, m with k ≤ m , m ! 2 ( m + k )!( m − k )! ≤ 1 . Pr oof. The term is a product of numbers that are all smaller than 1: m ! 2 ( m + k )!( m − k )! = m ( m − 1) . . . ( m − k + 1) ( m + k )( m + k − 1) . . . ( m + 1) = k Y i =1  1 − k m + i  ≤ 1 Lemma 5.  n 2  | P (2) | ≤ 12 and  n k 2  | P ( k 2 ) | ≤ 8 k 2 for every k = 1 , 2 , . . . , m . Pr oof. First, we substitute x = 2 into | P ( x ) | and re write the product over i ∈ [ n ] − S as the ratio of two products, one ov er i ∈ [ n ] − { 0 , 1 , 2 } and one over i ∈ S − { 0 , 1 , 2 } . W e then pull the j = 2 term out of the product in the denominator , use | j 2 − 2 | < j 2 − 4 , and apply Lemma 4. | P (2) | = 2 m ! 2 n ! ( n − 2)! 2 Q m j =3 | 2 − j 2 | < m ! 2 n ! ( n − 2)! Q m j =3 ( j 2 − 4) = m ! 2 n ! ( n − 2)! Q m j =3 ( j + 2)( j − 2) = 1 n ( n − 1) m ! 2 ( m +2)! 4! ( m − 2)! ≤ 4! n ( n − 1) = 12  n 2  . 3 Second, we substitute x = k 2 to | P ( x ) | and rewrite the product over i ∈ [ n ] − S as the ratio of two products, one ov er i ∈ [ n ] − { k 2 } and one o ver i ∈ S − { k 2 } . The term i = k 2 does not appear in any of products, because it is 0 . | P ( k 2 ) | = 2 m ! 2 n ! · Q i ∈ [ n ] i 6 = k 2 | k 2 − i | | k 2 − 2 | · Q j ∈ [ m ] j 6 = k ( k + j ) | k − j | = 2 m ! 2 n ! · k 2 !( n − k 2 )! ( k + m )! 2 k · ( k − 1)! · k !( m − k )! · 1 | k 2 − 2 | = 4 · k 2 !( n − k 2 )! n ! · m ! 2 ( m + k )!( m − k )! · 1 | k 2 − 2 | Apply Lemma 4 and | k 2 − 2 | ≥ k 2 / 2 , which holds for all integers k ≥ 1 . ≤ 4  n k 2  · 1 | k 2 − 2 | ≤ 4  n k 2  · 1 k 2 / 2 ≤ 8  n k 2  k 2 . Note that if we did not include the number 2 into S , in Definition 3, then the upper bound on | P ( k 2 ) | would be much weaker , without the factor of 1 /k 2 . Now we sho w that a constant fraction of the norm of P comes from the term P (0) = 1 . Theorem 6. k P k 1 < 27 . Pr oof. First, use the fact that P ( i ) = 0 for i ∈ [ n ] − S , non-square integers i other than 2. k P k 1 = n X i =0  n i  | P ( i ) | = X i ∈S  n i  | P ( i ) | = P (0) +  n 2  P (2) + m X k =1  n k 2  | P ( k 2 ) | Now , use P (0) = 1 , Lemma 5, and P k 1 k 2 = π 2 6 . ≤ 13 + 8 m X k =1 1 k 2 < 13 + 8 π 2 6 < 27 . Finally , we are ready to present the dual polynomial for OR. Theorem 7. The 1 14 -appr oximate de gree of OR on n bits is at least √ n . Pr oof. Consider the polynomial Q ( k ) = ( − 1) k P ( k ) , that is P from Definition 3 multiplied by parity . W e show that Q is a good dual polynomial for OR. First, the pure high degree of Q is n − ( n − m − 1) = m + 1 > √ n . Second, we compute the ratio from Corollary 2. Since OR(0) = 1 and OR( k ) = − 1 for k ≥ 1 , Q · OR = 2 Q (0) − Q · 1 = 2 Q (0) , because Q has no constant coef ficient. Now , we use Theorem 6 to upper-bound the numerator and conclude k Q k 1 Q · OR = k P k 1 2 P (0) < 27 2 < 14 . 4 4 Open pr oblems The approximate degree of the t -threshold function on n bits is Θ( p t ( n − t )) [Pat92]. It would be interesting to find an explicit dual polynomial for the threshold function. A good candidate may be Q ( k ) = ( − 1) k P ( k ) with P ( x ) = Y i ∈ [ n ] − T ( x − t − i ) , where T is a set of integers that can be written as k 2 − ` 2 , where k ∈ [ b √ n − t c ] and ` ∈ [ b √ t c ] . Note that | T | = Θ( p t ( n − t )) . The approximate degree of the two-le v el AND-OR tree on n bits (with all gates of fan-in √ n ) is only known to lie between O ( √ n ) and Ω( 3 √ n ) . Both bounds have been obtained through quantum algorithms, as follows. Consider a T -query quantum algorithm. Its acceptance probability on input x can be expressed as a 2 T -degree polynomial p in the variables x 1 , . . . , x n [BBC + 01]. If the algorithm computes a function f with bounded error, then p approximates f . Therefore quantum algorithms gi ve approximate polynomials, and approximate degree lower bounds give quantum query lower bounds. For the two-le vel AND-OR tree, the upper bound is via a quantum search algorithm on noisy inputs [HMW03] and the lo wer bound is via a reduction from the element distinctness problem [AS04]. Can one compute the approximate degree of the AND-OR tree by sho wing a good dual polynomial? Acknowledgments W e thank Harry Buhrman and Mario Szegedy for starting the project, coming up with the crucial ideas, and many fruitful discussions. W e also thank Ronald de W olf for fruitful discussions, and T ro y Lee for proofreading. Refer ences [AS04] S. Aaronson and Y . Shi. Quantum lower bounds for the collision and the element distinctness problems. Journal of the A CM , 51(4):595–605, 2004. [BBC + 01] R. Beals, H. Buhrman, R. Clev e, M. Mosca, and R. de W olf. 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