www.theoryofcomputing.org Approximating the AND-OR Tree

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f within 1=3 at every point. We prove that the function V n i=1 W n j=1 xi j, known as the AND-OR tree, has approximate degree W(n). This lower bound is tight and closes a line of research on the problem, the best previous bound being W(n 0:75 ). More generally, we prove that the function V m=1 W n j=1 xi j has approximate degree W( p mn); which is tight. The same lower bound was obtained independently by Bun and Thaler (2013) using related techniques.


Introduction
Over the past two decades, representations of Boolean functions by real polynomials have played an important role in theoretical computer science.The surveys [7,31,10,32,1] provide a fairly comprehensive overview of this body of work.Several kinds of representation [24,23,5,7,25] have been studied, depending on the intended application.For our purposes, a real polynomial p represents a Boolean function f : {0, 1} n → {0, 1} if ALEXANDER A. SHERSTOV 1/3 is called the approximate degree of f , denoted deg ( f ).The constant 1/3 is chosen for aesthetic reasons and can be replaced by any other in (0, 1/2) without affecting the theory in any way.
The formal study of the approximate degree began in 1969 with the seminal work of Minsky and Papert [23], who famously proved that the parity function in n variables cannot be approximated by a polynomial of degree less than n.Since then, the approximate degree has been used to solve a vast array of problems in complexity theory and algorithm design.The earliest use of the approximate degree was to prove circuit lower bounds and oracle separations of complexity classes [27,41,5,20,21,35].Over the past decade, the approximate degree has been used many times to prove tight lower bounds on quantum query complexity, e. g., [6,9,2,18].The approximate degree has enabled remarkable progress [12,28,11,36,29,32] in communication complexity, with complete resolutions of difficult open problems.The results listed up to this point are of negative character, i. e., they are lower bounds in relevant computational models.More recently, the approximate degree has found important algorithmic applications.In computational learning theory, the approximate degree has been used to obtain the fastest known algorithms for PAC-learning DNF formulas [42,19] and read-once formulas [4] and the fastest known algorithm for agnostically learning disjunctions [17].Another well-known use of the approximate degree is an algorithm for approximating the inclusion-exclusion formula based on its initial terms [22,16,33,43].
These applications motivate the study of the approximate degree as a complexity measure in its own right.As one would expect, methods of approximation theory have been instrumental in determining the approximate degree for specific Boolean functions of interest [8,25,40,2,3,33,39].In addition, quantum query algorithms have been used to prove upper bounds on the approximate degree [15,43,4,30], and duality-based methods have yielded lower bounds [26,34,38].Nevertheless, our understanding of this complexity measure remains fragmented, with few general results available [25,39].
The limitations of known techniques are nicely illustrated by the so-called AND-OR tree, Despite its seeming simplicity, it has been a frustrating function to analyze.Its approximate degree has been studied for the past 19 years [25,40,15,3,34] and has been recently re-posed as an open problem by Aaronson [1].Table 1 gives a quantitative summary of this line of research.The best lower and upper bounds prior to this paper were Ω(n 0.75 ) and O(n), respectively.Our contribution is to close this gap by improving the lower bound to Ω(n).We obtain the following more general result.
This lower bound is tight for all m and n, by the results of Høyer et al. [15].

Proof overview
The problem of approximating a given function f pointwise to within error ε by polynomials of degree at most d can be viewed as a search for a point in the intersection of two convex sets, namely, the Nisan and Szegedy [25] Ω( Ω(n 0.66... ) Ambainis [3] Ω(n 0.75 ) Sherstov [34] Ω(n) This paper Table 1: Approximate degree of the AND-OR tree.
ε-neighborhood of f and the set of polynomials of degree at most d.As a result, the nonexistence of an approximating polynomial for f is equivalent to the existence of a so-called dual polynomial for f , whose defining properties are orthogonality to degree-d polynomials and large inner product with f .Geometrically, the dual polynomial is a separating hyperplane for the two convex sets in question.
Our proof is quite short (barely longer than a page).We view f (x) = m i=1 n j=1 x i j as the componentwise composition of the functions AND m and OR n .We use the dual polynomial for OR n to prove the existence of an operator L with the following properties: (ii) L decreases the degree of the function to which it is applied by a factor of Ω( √ n); (iii) L f ≈ AND m pointwise.
The existence of L directly implies our main result.Indeed, for any polynomial p that approximates f pointwise, the polynomial Lp has degree Ω( √ n) times smaller and approximates AND m pointwise; since the latter approximation task is known [25] to require degree Ω( √ m), the claimed lower bound of Ω( √ mn) on the degree of p follows.What makes the construction of L possible is the following very special property of any dual polynomial for OR n : it maintains the same sign on OR −1 n (0) and has almost half of its 1 norm there.We call such dual polynomials one-sided.This property was proved several years ago by Gavinsky and the author in [14], where it was used to obtain lower bounds for nondeterministic and Merlin-Arthur communication protocols.

Independent work by Bun and Thaler
In an upcoming paper, Bun and Thaler [13] independently prove an Ω( √ mn) lower bound on the approximate degree of f (x) = m i=1 n j=1 x i j .The proof in [13] and ours are both based on the fact that OR n has a one-sided dual polynomial.The two papers differ in how they use this fact to prove an Ω( √ mn) lower bound on the approximate degree.The treatment in this paper is a combination of the dual view (one-sided dual polynomial for OR n ) and the primal view (construction of an approximating polynomial for AND m ).The treatment in [13] is a refinement of [34] and uses exclusively the dual view (construction of a dual polynomial for f using dual polynomials for AND m and OR n ).In our opinion, the proof in this paper has the advantage of being shorter and simpler.On the other hand, the approach in [13] has the advantage of giving an explicit dual polynomial for f , which is of interest because explicit dual polynomials have found several uses in communication complexity [32].

Preliminaries
For a function f : X → R on a finite set X, we let f ∞ = max x∈X | f (x)|.The total degree of a multivariate real polynomial p : R n → R is denoted deg p.We use the terms degree and total degree interchangeably in this paper.For a function f : X → R on a finite set X ⊂ R n , the ε-approximate degree deg ε ( f ) of f is defined as the least degree of a real polynomial p with f − p ∞ ≤ ε.Throughout this paper, we will work with the ε-approximate degree for a small constant ε > 0. For Boolean functions f : X → {0, 1}, the choice of constant 0 < ε < 1/2 affects the quantity deg ε ( f ) by at most a constant factor: where c = c(ε) and C = C(ε) are positive constants.By convention, one studies ε = 1/3 as the canonical case and reserves for it the special symbol deg [36,38] of the approximate degree is as follows.We adopt the usual definitions of the Boolean functions AND n , OR n : {0, 1} n → {0, 1}.Their approximate degree was determined by Nisan and Szegedy [25].
Theorem 2.2 (Nisan and Szegedy).The functions AND n and OR n obey By combining the above two theorems, Gavinsky and the author [14, Thm.5.1] obtained the following result, which plays a key role in this paper.
For the sake of completeness, we include the proof.
Proof of Claim 3.3.By the linearity of L, it suffices to consider factored polynomials p of the form p(x) = ∏ m i=1 p i (x i,1 , x i,2 , . . ., x i,n ).In this case we have the convenient formula By (2.4)  Using the pattern matrix method [36], one can immediately translate the main result of this paper into lower bounds on communication complexity.For example, it follows that the two-party communication problem f (x, y) = m i=1 n j=1 (x i j ∧ y i j ) has bounded-error quantum complexity Ω( √ mn), regardless of prior entanglement.We refer the interested reader to [36,38,37] for further details and applications.

Fact 2 . 1 .
Let f : X → R be given, for a finite set X ⊂ R n .Then deg ε ( f ) ≥ d if and only if there exists a function ψ : X → R such that ∑ x∈X |ψ(x)| = 1 , ∑ x∈X ψ(x) f (x) > ε , and ∑ x∈X ψ(x)p(x) = 0 for every polynomial p of degree less than d.