A note on the sign degree of formulas

A note on the sign de gree of formulas T ro y Lee ∗ Abstract Recent breakthroughs in quantum query comple xity have sho wn that any formula of size n can be e valuated with O ( √ n log( n ) / log log( n )) man y quantum queries in the bounded-error setting [FGG08, A CR + 07, R ˇ S08, Rei09]. In particular , this gives an upper bound on the ap- proximate polynomial de gree of formulas of the same magnitude, as approximate polynomial degree is a lo wer bound on quantum query comple xity [BBC + 01]. These results essentially answer in the af firmati v e a conjecture of O’Donnell and Servedio [OS03] that the sign degree—the minimal degree of a polynomial that agrees in sign with a function on the Boolean cube—of e very formula of size n is O ( √ n ) . In this note, we show that sign degree is super -multiplicati ve under function composition. Combining this result with the above mentioned upper bounds on the quantum query complex- ity of formulas allo ws the remov al of logarithmic factors to show that the sign degree of e very size n formula is at most √ n . 1 Intr oduction There is a gro wing body of work which uses techniques of quantum computing and information to prov e results whose statements ha ve no reference to quantum at all [KW04, Aar04, Aar05, LLS06, W ol08]. One simple application of this type is to the construction of low-de gree polynomials that approximate a Boolean function. Beals et al. [BBC + 01] show that one-half the minimum degree of a polynomial which approximates a function f on the Boolean cube within error 1 / 3 (in terms of ` ∞ norm) is a lower bound on the 1 / 3 -error quantum query complexity of f . T urning this around, if f has a d -query bounded-error quantum algorithm, then it has approximate polynomial degree at most 2 d . Using quantum algorithms has prov en a remarkably po werful means of constructing approximating polynomials, and in quite a few cases no other construction is known, for example [BNR W07]. Another example where quantum algorithms sho w ne w bounds on approximate de gree is in the case of functions described by small formulas. A formula is a binary tree where internal nodes are labeled by binary AND 2 or OR 2 gates and leav es are labeled either by a literal x i or its negation ¬ x i . The size of a formula is the number of lea ves. Recent breakthroughs in quantum query complexity hav e shown that if a function f can be computed by a formula of size n , then there is ∗ Rutgers Univ ersity . W ork conducted while at Columbia Univ ersity . 1 a quantum query algorithm that can ev aluate f with high probability in O ( √ n log ( n ) / log log( n )) many queries [FGG08, A CR + 07, R ˇ S08, Rei09]. By the above connection, this implies that the approximate polynomial degree of any formula is also O ( √ n log ( n ) / log log( n )) . Pre vious to these results, it w as an open question, raised by O’Donnell and Serv edio [OS03], to sho w that e very size n formula has sign degree O ( √ n ) . The sign degree of f , denoted deg ∞ ( f ) , is the minimum degree of a polynomial which agrees in sign with f for all x ∈ {− 1 , +1 } n . In this note, we sho w a lemma about sign degree under function composition. Namely , if f ◦ g n ( x ) = f ( g ( x 1 ) , . . . , g ( x n )) , where x = ( x 1 , . . . , x n ) , then deg ∞ ( f ◦ g n ) ≥ deg ∞ ( f ) deg ∞ ( g ) . This lemma is often not tight: for example both AND n and OR n hav e sign degree one, whereas Minsky and P apert show that OR n ◦ AND n n 2 has degree n . When combined with the results of Reichardt [Rei09], howe ver , this lemma allo ws the remov al of log factors to fully resolv e the question of O’Donnell and Servedio and show that ev ery size n formula has sign degree at most √ n . This upper bound is exactly tight for infinitely many values of n since for any n = 2 2 k , the parity function ov er √ n v ariables is computed by a size n formula and has sign de gree exact ly √ n . 2 Pr eliminaries Let [ n ] = { 1 , 2 , . . . , n } . For a set T ⊆ [ n ] , we associate the character χ T : {− 1 , +1 } n → {− 1 , +1 } where χ T ( x ) = Q i ∈ T x i for x ∈ {− 1 , +1 } n . Every function f : {− 1 , +1 } n → {− 1 , +1 } has a unique expansion as a multilinear polynomial f ( x ) = X T ⊆ [ n ] ˆ f T χ T ( x ) . The polynomial de gr ee of f , denoted deg( f ) , is the size of a largest set T for which ˆ f T 6 = 0 . W e say that f has pur e high degr ee d if ˆ f T = 0 for all sets T with | T | < d . Our main object of study is the degree of polynomials which approximate a function f . Definition 1 Let f : {− 1 , +1 } n → {− 1 , +1 } . F or α ≥ 1 the α - approximate degree of f is deg α ( f ) = min p { deg( p ) : 1 ≤ p ( x ) f ( x ) ≤ α for all x ∈ {− 1 , +1 } n } . Sign degree is defined as deg ∞ ( f ) = min p { deg( p ) : 1 ≤ p ( x ) f ( x ) for all x ∈ {− 1 , +1 } n } . Notice that for a fixed degree d and approximation parameter α (possibly α = ∞ ), determining if deg α ( f ) is at most d can be checked by determining the feasibility of a linear program. On the other hand, showing that the dual of this linear program is feasible implies that deg α ( f ) > d . W e encapsulate the feasibility conditions of this dual program in the next lemma. 2 Lemma 2 F ix 1 ≤ α ≤ ∞ and let f : {− 1 , +1 } n → {− 1 , +1 } . Ther e exists a function p : {− 1 , +1 } n → R such that 1. h f , p i ≥ ( α − 1 α +1 if α < ∞ 1 if α = ∞ . 2. ` 1 ( p ) = 1 . 3. h p, χ T i = 0 for any char acter χ T with | T | < deg α ( f ) . W e r efer to p as a dual witness for deg α ( f ) . 3 Composition lemma Let f be a function f : {− 1 , +1 } n → {− 1 , +1 } , and g : {− 1 , +1 } m → {− 1 , +1 } . W e de- fine the composition of f and g as f ◦ g n : {− 1 , +1 } mn → {− 1 , +1 } where ( f ◦ g n )( x ) = f ( g ( x 1 ) , . . . , g ( x n )) for x = ( x 1 , . . . , x n ) . Our composition lemma states that deg ∞ ( f ◦ g n ) ≥ deg ∞ ( f ) deg ∞ ( g ) . This lemma is often not tight—for example, both OR n and AND n hav e sign degree 1 . On the other hand, Minsky and Papert show that OR n ◦ AND n n 2 has sign degree n . Extending such a composition lemma to the bounded-error case, where it would be nearly tight, would be a major breakthrough. In particular , such a result would resolve the approximate polynomial degree of the function on n 2 many variables OR n ◦ AND n n , which is currently only known to be somewhere between n 2 / 3 and n [AS04, HMW03]. Lemma 3 Let f be a function f : {− 1 , +1 } n → {− 1 , +1 } , and g : {− 1 , +1 } m → {− 1 , +1 } . Then for 1 ≤ α ≤ ∞ deg α ( f ◦ g n ) ≥ deg α ( f ) deg ∞ ( g ) . Proof: Let p, q satisfy the conditions of Lemma 2 for deg α ( f ) and deg ∞ ( g ) , respecti vely . If deg ∞ ( g ) = 0 then the statement is tri vial, so we assume that deg ∞ ( g ) ≥ 1 and so h χ ∅ , q i = 0 . Notice that as ` 1 ( q ) = h g , q i = 1 we must hav e g ( x ) q ( x ) ≥ 0 for all x ∈ {− 1 , +1 } m . Thus we may express q as q ( x ) = g ( x ) µ ( x ) where µ ( x ) ≥ 0 for all x . Define h ( x ) = 2 n p ( g ( x 1 ) , . . . , g ( x n )) · Y i µ ( x i ) . 3 Let us verify that h has the properties of a dual witness. h f ◦ g n , h i = 2 n X x f ( g ( x 1 ) , . . . , g ( x n )) p ( g ( x 1 ) , . . . , g ( x n )) · Y i µ ( x i ) = 2 n X z ∈{− 1 , +1 } n f ( z ) p ( z ) X x : g ( x i )= z i ∀ i Y i µ ( x i ) = 2 n X z ∈{− 1 , +1 } n f ( z ) p ( z ) Y i X x i : g ( x i )= z i µ ( x i ) = X z ∈{− 1 , +1 } n f ( z ) p ( z ) ≥ ( α − 1 α +1 if α < ∞ 1 if α = ∞ . The fourth equality holds since h χ 0 , q i = 0 and ` 1 ( q ) = 1 imply X y : g ( y )= − 1 µ ( y ) = X y : g ( y )=1 µ ( y ) = 1 2 . Next we v erify that ` 1 ( h ) = 1 . This follows quite similarly: ` 1 ( h ) = 2 n X z ∈{− 1 , +1 } n | p ( z ) | Y i X x i g ( x i )= z i | µ ( x i ) | = X z ∈{− 1 , +1 } n | p ( z ) | = 1 . Finally , we check that h is orthogonal to all characters of degree less than deg α ( f ) deg ∞ ( g ) . T o see this, write out h ( x ) 2 n = p ( g ( x 1 ) , . . . , g ( x n )) · n Y i =1 µ ( x i ) = X T ˆ p T Y i ∈ T g ( x i ) · n Y i =1 µ ( x i ) = X T : | T |≥ deg α ( f ) ˆ p T Y i ∈ T g ( x i ) µ ( x i ) · Y j 6∈ T µ ( x j ) ! = X T : | T |≥ deg α ( f ) ˆ p T Y i ∈ T q ( x i ) · Y j 6∈ T µ ( x j ) ! . For each fixed T , the term Q i ∈ T q ( x i ) is a product of at least deg α ( f ) many polynomials q ( x i ) which are ov er disjoint sets of variables, and each of which has pure high degree deg ∞ ( g ) . Thus 4 the product has pure high de gree at least deg α ( f ) · deg ∞ ( g ) . Multiplying by Q j / ∈ T µ ( x j ) , which is a polynomial over another set of disjoint variables, cannot decrease the pure high degree. So the pure high degree of h is at least deg α ( f ) · deg ∞ ( g ) . 2 4 Sign degr ee of f ormulas W e no w see ho w Lemma 3 can be used in conjunction with recent results of Reichardt [Rei09] to show that every formula of size n has sign degree at most √ n . The result of Reichardt we need shows that the negati ve adversary bound characterizes quantum query complexity amortized ov er function composition. The negati ve adversary bound [HL ˇ S07] is a lower bound technique for quantum query complexity which generalizes the quantum adversary method of Ambainis [Amb02, Amb03], in particular the spectral formulation of the adversary bound due to Barnum, Saks, and Szegedy [BSS03]. Definition 4 (Negative adv ersary bound) Let f : {− 1 , +1 } n → {− 1 , +1 } . F or eac h i = 1 , . . . , n let D i be a zer o-one valued matrix with r ows and columns labeled by n -bit strings and wher e D i [ x, y ] = 1 if x i 6 = y i and D i [ x, y ] = 0 otherwise. Let F be a zer o-one valued matrix wher e F [ x, y ] = 1 if f ( x ) 6 = f ( y ) and F [ x, y ] = 0 otherwise. Define AD V ± ( f ) = max Γ 6 =0 k Γ ◦ F k max i k Γ ◦ D i k . Her e k A k denotes the spectral norm of the matrix A . Theorem 5 (Reichardt [Rei09]) F or any function f : {− 1 , +1 } n → {− 1 , +1 } , let f ( k ) denote f composed with itself k times. Then lim k →∞ Q ( f ( k ) ) 1 /k = AD V ± ( f ) . It is known that if f has formula size n then ADV ± ( f ) ≤ √ n . This can be seen using the fact that ADV ± (AND n ) = ADV ± (OR n ) = √ n and that ADV ± ( f ◦ g ) ≤ ADV ± ( f )AD V ± ( g ) the adversary bound is sub-multiplicati ve under function composition [Rei09]. Thus we hav e deg ∞ ( f ) ≤ lim k →∞ deg ∞ ( f ( k ) ) 1 /k ≤ lim k →∞ (2 Q ( f ( k ) )) 1 /k ≤ √ n. where the first inequality follo ws from the composition lemma of Section 3 and the second is by the bound of Beals et al. mentioned in the Introduction. 5 Conclusion As with all classical results prov en via quantum techniques, it would be interesting to come up with a more direct proof. For both the case of sign degree and quantum query complexity , the 5 more dif ficult case is formulas which are highly unbalanced. For the complete AND-OR binary tree of size n , one can quite easily giv e an explicit sign representing polynomial of degree √ n . The benefit of the composition lemma seems to be that it reduces the problem of sho wing an upper bound on the sign degree of f to sho wing an upper bound on the sign degree of f ( k ) , which intuiti vely is a more “balanced” function. While in the quantum case there is a good notion of “approximately balanced” to make this plan w ork (see [A CR + 07]), it still remains to come up with a good classical notion of approximately balanced to push such a proof through. Acknowledgments I would lik e to thank Ben Reichardt for many discussions and for sharing a preliminary version of [Rei09], Rocco Servedio for con versations about this work and helpful comments on the writeup, and Ronald de W olf for a careful reading of the writeup. Refer ences [Aar04] S. Aaronson. Lo wer bounds for local search by quantum arguments. 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