Upper and Lower Bounds on ε-approximate Degree of AND n and OR n Using Chebyshev Polynomials

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1 Upper an Lower Bouns on ε-approximate Degree of AND n an OR n Using Chebyshev Polynomials Mrinalkanti Ghosh, Rachit Nimavat December 11, Introuction The notion of approximate egree was first introuce an stuie in [6]. For any function f : {0, 1} n R on the boolean hypercube, ε-approximate egree of f, enote as eg ε f ), is efine as the minimum egree such that the following hols: there exists a n-variate polynomial g of egree such that for all x {0, 1} n, we have f x) gx) ε, i.e., everywhere on the hypercube, g is an approximation of f up to aitive error ε. Approximate egree, as a measure of complexity, has been wiely use in theoretical computer science; e.g., algorithm esign, circuit complexity, communication complexity, quantum query complexity an learning theory. Depening on how we compute error of approximation, ifferent notions of approximate egrees are also known in the literature. For any approximation g of f, let the error function e g : {0, 1} n R be efine as e g x) = f x) gx). In the approximate egree efine above, we look for polynomial g such that eg ε. One other notable norm through which approximate egree has been stuie is the l norm; e.g., in [5] the authors show upper bouns for approximate egrees in l norm for AC 0 computable functions. However, in this paper, we only restrict our attention to approximate egree efine through l norm. In approximation theory, Chebyshev polynomials are funamental objects of stuy ue to their extremal properties. Also, there are quite a few equivalent ways of efining them see [9] for a few of those efinitions). So, Chebyshev polynomials establish useful connections between various areas of mathematics. 1

2 We enote AND on n-bit boolean input as AND n an similarly OR n enote n-bit OR). In this paper we use Chebyshev polynomials to give elementary proofs of upper an lower bouns of approximate egrees of AND n an OR n functions. 1.1 Our Result For the upper bouns on approximate egree we prove: Theorem 1.1. For all ε > 0, eg ε AND n ) = O n log 1 /ε). We note here that our proof for this theorem was alreay known in folklore. But ieas in the construction of the approximating polynomial are use in the proof of lower boun as well. So, for the sake of completeness, an the reason state earlier, we inclue a proof of this theorem in Section 3. On the lower boun sie, we prove the following theorem: Theorem 1.. For any function m : N N such that mn) = o n) an any constant ε < 1, eg ε AND n ) = Ω mn) log 1 ε )). We note that the conclusion of the theorem above is obviously false for ε = 1. An here, we only concern ourselves with constant ε. Although, the n proof presente here continues to hol true for any ε such that ε /log1/ε) m n)/n. The proof of this appears in Section 4. The bouns prove here are known to suboptimal more on this in the next section. Our principal contribution, however, is an elementary proof. We note here that eg ε OR n ) = eg ε AND n ). Inee, if we have two polynomials P an Q with Px 1,..., x n ) = 1 Q1 x 1,..., 1 x n ) then P ε-approximates AND n iff Q ε-approximates OR n, an vice versa. So, as a corollary, we get upper an lower boun for OR n as well: Corollary 1.3. For ε > 0, eg ε OR n ) = O n log 1 /ε). Also, for any function m : N N such that mn) = o n) an 1 n < ε < 1, egε OR n ) = Ω mn) log 1 ε )). 1. Comparison With Previous Work The notion of approximate egree was first introuce in [6]. Then its relation to exact egree of function was extensively stuie in [7] the authors showe that approximate an exact egree are polynomially relate, for

3 arbitrary functions. In [7], the authors also showe that eg 1/3 OR n ) = Θ n) which was later extene in [8] to give tight bouns for eg 1/3 for symmetric functions. The lower boun in [8] uses the Bernstein-Markov inequality an oes not reaily generalize to the case of eg ε. An explicit proof of lower boun on eg 1/3 OR n ) using LP-uality is also shown in [11]. In [4] the authors establishe almost tight up to polylog n) factor) upper an lower boun for eg ε OR n ). Although the authors in [4] on t explicitly consier eg ε OR n ), the result can be recovere easily. The authors in [4] use Chebyshev polynomials for the upper boun an LPuality to show their lower boun. In [1] a tight boun of the form eg ε OR n ) = Θ n log 1 /ε) was obtaine. The authors in [1] uses quantum algorithms an their connections to approximate egree to obtain their upper boun. Their lower boun proof is morally similar to our proof of lower boun given here. For the lower boun, the authors in [1] uses an inequality ue to Coppersmith an Rivlin [] which establishes an upper boun on the absolute value of a polynomial given that the polynomial is boune on some given equispace set of points. Our lower boun proof, instea, employs Taylor s approximation to use the bouneness of the polynomial on the equispace input points. The result of [4] was later generalize to give tight estimates for eg ε f ) for any symmetric functions in [10, 3]. Preliminaries.1 Notation In the following we use bol-face) x to enote the both finite an infinite) tuple x 1, x,..., x n,...) an x i to enote the i-th element of the tuple x.. Multivariate Polynomials an Symmetrization A multivariate) monomial is of the form x t = n i=1 xt i i, where t i {0} N for i [n]. For the monomial x t, we efine the egree of the monomial to be n i=1 t i. For a multivariate polynomial Px 1,..., x n ) = t {0,1,...} [n] c t x t we ll be concerne with the case c t R), we efine the egp) = sup t : ct =0 egxt ). The summation in the efinition of the polynomial efine here is only a formal summation. But we ll only be concerne with the polynomials where there are finitely many non-zero c t an hence the sup in the efinition egree is always finite for us. 3

4 For a multivariate polynomial Px 1,..., x n ), we efine: P sym x 1,..., x n ) = 1 n! σ Sym n Pσx)) 1, σx))..., σ n x)) n ) It is clear that P sym is a symmetric polynomial, i.e., P sym x) = P sym σx)) for all σ Sym n. Moreover, it is well known that there exists an univariate polynomial py) such that p n i=1 x i ) = P sym x). for all x {0, 1} n, with egp) = egp). applying the following observations: A simple proof follows by It is enough to show it for homogeneous polynomials P sym ) sym = P sym if n i=1 x i = k the contribution of each monomial x t of egree to symmetrization can be expresse as a egree- polynomial in k..3 Chebyshev Polynomials In this section we state some relevant facts about Chebyshev polynomials more etaile account of Chebyshev polynomials an their use in approximation of functions an algorithm esign can be foun in [9] an the references therein. First, we efine the Chebyshev polynomials: Definition.1. The -th Chebyshev polynomial T x) is efine inuctively on as: T +1 x) = x T x) T 1 x). where T 0 x) = 1 an T 1 x) = x. The -th Chebyshev polynomial T has exactly roots in the interval [ 1, 1]. We ll escribe these roots an an alternate representation of T x) in the interval [ 1, 1] next: Claim.. T cos θ) = cos θ). An hence, in the interval [ 1, 1], T x) = cos arccosx)). A proof of this can be foun in [9]. 4

5 Proof: [9] We prove this by inuction on. For the base cases: T 0 x) = cos0) = 1 an T 1 cos θ) = cosθ). For the inuction step, let us assume T cosθ)) = cos θ) an T 1 cosθ)) = cos 1) θ). Now, by the inuctive efinition: T +1 cosθ)) = cosθ)t cosθ)) T 1 cosθ)) = cosθ) cosθ) cos 1)θ) = cosθ) cosθ) cosθ) cosθ) sinθ) sinθ) = cosθ) cosθ) sinθ) sinθ) = cos + 1)θ). This completes the inuction step. { ) } As a corollary, we see that T x) = 0 for x jπ cos : j [], j )) jπ an, T cos = 1) j for j []. Now we state an extremal property of Chebyshev Polynomials: Fact.3. Let px) is an arbitrary egree- polynomial such that for all x [ 1, 1], px) 1. Then for all y 1, py) T y). A proof of this fact can be foun in [9] in fact the proof of Theorem 1. is inspire by the proof given in [9]. Finally, we give one more alternate representation of T : Fact.4. The polynomial T x) = x+ x 1) +x x 1) for x R. An inuctive proof of this fact can be foun in [9]. representation is vali even in the interval [ 1, 1]. Note that this 3 Upper boun of Approximate Degree of AND n In this section we prove the following upper-boun: Theorem 1.1. For all ε > 0, eg ε AND n ) = O n log 1 /ε). 3.1 Overview The following proof of the upper boun of approximate egree of AND n is one using Chebyshev polynomials. We note that in orer to approximate AND n, it is enough to fin a low-egree univariate) polynomial q, the supremum of which, in the interval [0, 1 1 /n], is as small as possible, an 5

6 q1) = 1. On the other han, Fact.3 states that T is the fastest growing polynomial in interval [1, )) which is boune in absolute value by 1 in the interval [ 1, 1]. This suggests using a single Chebyshev polynomial of appropriate egree possibly compose with an affine transformation to match-up the intervals an points in question). 3. Proof of Theorem 1.1 Proof: Consier the polynomial gx) = c T n n 1 x 1) where an ef c > 0 are to be etermine later. For x A n = { j /n : j {0, 1,..., n 1}, we have that x n /n 1 1 [ 1, 1]. Hence for x A n, gx) c. Recall from Fact.4) that T x) = x+ x 1) +x x 1). Hence, g1) =c T 1 + ) n 1 ) = c 1 + n n n 1) ) + c 1 + n 1 4 n n 1) = c Θ 1 + / n) + 1 / n) ) =c Θe n ) We set c = ε an choose = Θ n log 1 /ε), so that g1) = 1. So, hx 1,..., x n ) = g n i=1 x i/n) is a ε-approximation to AND n with egree Θ n log 1 /ε), an we get the result: eg ε OR n ) = O n log 1 /ε). 4 Lower Boun of Approximate Degree of AND n In this section we prove the following lower-boun theorem: Theorem 1.. For any function m : N N such that mn) = o n) an any constant ε < 1, eg ε AND n ) = Ω mn) log 1 ε )). The proof of the theorem goes through metho of contraiction. Assuming there exists a low-egree polynomial representation of AND n, we fin two polynomials g 1 an g with egree at most. We then emonstrate that g 1 g has + 1 roots a contraiction. The etaile proof is as follows: 6

7 Proof: Consier the univariate polynomial h efine by ) xi ef h = P sym x 1,..., x n ) 1) n We know that egh) = egp). Fix any function m : N N such that mn) = o n). Let a be the affine transformation efine as ax) = n 1 n x + 1) an so, a 1 x) = n n 1 x 1. Further, we let = mn) log1 /ε), an as we observe: c ef = T 1 + ) ) e mn) n log 1 ε ) 1 o1) = 1 n 1 ε ε. We observe that the last inequality gets stronger as n gets larger. The estimation of T 1 + /n 1)) one here is similar to the one one in the proof in Theorem 1.1. For sake of contraiction, let us assume egp) <. Let g 1 x) = hax)) an g x) = h1) c T x). We note that g n 1) = ) h1) = g 1 + n 1. Now for j {0,..., n 1}, a 1 j /n) = 1 + j n 1. Hence for j {0,..., n 1}, g n 1) j = h j/n) ε by combining Eq. 1) with the fact that P ) is an ε-approximation to AND n. Recall that from Claim.) in the interval x [ 1, 1], the following equality hols: T x) = ) T cosarccosx))) = cos arccosx)). So, for k N an z k = cos k π, we have T x) = 1. An, for all large enough n, g z k ) h1) /c 1 ε)/c ε as h1) 1 ε an ε < 1. Next prove the following claim: Claim 4.1. There exists ε 0 such that for all large enough n an small enough ε with ε < ε 0, there exists a sequence y 1 > y >... > y 1 such that, for any k [ 1] we have: g 1 y k ) ε, g y k ) > ε an, sgng y k )) = 1) k. ) Proof: We note here that the intene points y k s are in fact a 1 j n for some j [n 1]. Recall that z k = cosk π ) are the extremal inputs for g an z 1 > z >... > z 1. In this proof, for each k, we choose smallest jk [n 1] such that z k a 1 j k/n) an set y k = a 1 j k/n). By construction, g 1 y k ) ε. To obtain the require properties of g y k ), we analyze the Taylor series expansion of T x) near x = z k. We observe that ) by the Taylor series expansion of f x) = cos arccosx)) near z k = cos kπ : f z k + δ) f z k ) f z k ) δ + f z k ) 7 δ + Oδ 3 ).

8 Now, an, f z k ) = sin arccosz k)) 1 z k f z k ) = f z k ) 1 z k = sink π) 1 z k + z k sink π) = 1 x ) 3/ = 0 sin k π ). So, for k [ 1], f z k ) 4 since for this values of k, sin π k π /) π. ef Hence for δ δ 0 = ε/, f z k + δ) f z k ) ε < ε. For our choice of, δ 0 = ε ε = g n) log1/ε) = ω ε log1/ε) 1 n ) as mn) = o n)). Hence for large enough n, δ 0 > 4 n 1. We recall a 1 j /n) = 1 + j n 1. Hence, we get that for each k [ 1], there exists j such that a 1 j n ) [z k, z k δ 0 ]. Let j k be the minimal such j an we set y k = a 1 j kn ). By the approximation shown above, g y k ) > ε an sgng y k )) = sgng z k )) = 1) k. We efine y = 1 = a 1 0) an y 0 = 1 = a 1 n 1 n ). We note g y 0 ) = g y ) = h1) c ε > ε. Now from the Claim 4.1 an the efinition above, g y k ) ε, g y k ) an g y k 1 ) has opposite signs for all k []. Since g 1, g are continuous functions, for each k [] there exists r k [y k 1, y k ] such that g 1 r k ) = g r k ) as g 1 x) g x) has opposite signs at x = y k 1, y k for k []. Hence {r k : k []} {1 + n 1 } are the roots of the polynomial g 1x) g x). But egg ) < an hence egg 1 x) g x)) =. So we have foun + 1 roots of a egree- polynomial a contraiction. 5 Acknowlegement The authors wish to thank Mahur Tulsiani for consierable simplification of the proof of the upper boun of the approximate egree. The authors also thank Haris Angleakis for his help in correcting various errors an typos an for his suggestions to improve the presentation of the results. References [1] Harry Buhrman, Richar Cleve, Ronal e Wolf, an Christof Zalka. Bouns for small-error an zero-error quantum algorithms. In Procee- 8

9 ings of the 40th Annual Symposium on Founations of Computer Science, FOCS 99, pages 358, Washington, DC, USA, IEEE Computer Society. [] Don Coppersmith an T. J. Rivlin. The growth of polynomials boune at equally space points. SIAM J. Math. Anal., 34): , July 199. [3] Ronal e Wolf. A note on quantum algorithms an the minimal egree of ɛ-error polynomials for symmetric functions. Quantum Info. Comput., 810): , November 008. [4] Jeff Kahn, Nathan Linial, an Alex Samoronitsky. Inclusionexclusion: Exact an approximate. Combinatorica, 164): , [5] Nathan Linial, Yishay Mansour, an Noam Nisan. Constant epth circuits, fourier transform, an learnability. J. ACM, 403):607 60, July [6] Marvin Minsky an Seymour Papert. Perceptrons: An Introuction to Computational Geometry. MIT Press, Cambrige, MA, USA, [7] Noam Nisan an Mario Szegey. On the egree of boolean functions as real polynomials. In Proceeings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC 9, pages , New York, NY, USA, 199. ACM. [8] Ramamohan Paturi. On the egree of polynomials that approximate symmetric boolean functions preliminary version). In Proceeings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC 9, pages , New York, NY, USA, 199. ACM. [9] Sushant Sacheva an Nisheeth K. Vishnoi. Faster algorithms via approximation theory. Foun. Trens Theor. Comput. Sci., 9):15 10, March 014. [10] Alexaner A. Sherstov. Approximate inclusion-exclusion for arbitrary symmetric functions. computational complexity, 18):19 47, 009. [11] R. Spalek. A Dual Polynomial for OR. ArXiv e-prints, March