The approximate degree of a boolean function , denoted , is the least degree of a polynomial over the reals that pointwise approximates . We focus on constant factor approximations: that is, we require that for all , whenever , and whenever , for some constants . Note that the choices of and are arbitrary, in the sense that varying them changes the approximate degree by at most a constant multiplicative factor.
Lower and upper bounds on approximate degree have found many natural applications in computer science, including lower bounds on (quantum) query complexity, learning theory, and communication complexity (see e.g. [She13] for a brief history). Indeed, researchers have studied representations of boolean functions by polynomials since as early as the 1960s [MPB69].
Many of the earliest lower and upper bounds on approximate degree were proved using symmetrization, a technique that involves transforming a (typically symmetric) multivariate polynomial into a univariate polynomial. Then, one can typically appeal to classical results in approximation theory to analyze the univariate polynomial. As an example, in the early 90s, Paturi [Pat92] used symmetrization arguments to tightly characterize the approximate degree of all symmetric boolean functions (i.e. functions that only depend on the Hamming weight of the input). However, symmetrization has limited power in proving lower bounds. Symmetrization is typically only useful for studying highly symmetric functions. Additionally, symmetrization is inherently a lossy technique, as a univariate polynomial can only capture part of the behavior of a multivariate polynomial.
A more recently introduced technique, variously called the method of dual witnesses or dual polynomials, has recently become the method of choice for analyzing more complicated functions. This method involves formulating approximate degree as a linear program, then exhibiting a solution to the dual linear program to prove a lower bound on approximate degree. The dual witness method has the advantage that it can theoretically prove tight lower bounds for every boolean function, but finding explicit dual witnesses can be difficult.
One of the major successes of the dual witness method is in resolving the approximate degree of the two-level - tree: the function . While an upper bound on was proved in 2003 via a quantum algorithm for the problem [HMDW03], the first tight lower bound on the approximate degree of the two-level - tree was proved in 2013 using dual witnesses, discovered independently by Sherstov [She13] and Bun and Thaler [BT13]. This resolved a question that had been open for nearly two decades, closing a line of successively tighter lower bounds (see Table 1).
|Symmetrization||Nisan and Szegedy [NS92]|
|Dual witnesses||Sherstov [She09]|
|Dual witnesses||Bun and Thaler [BT13]; Sherstov [She13]|
|Other||Ben-David, Bouland, Garg, and Kothari [BDBGK18]|
For many years, the two-level - tree was viewed as a symbol of the limitations of the (at the time) known techniques for proving lower bounds using symmetrization. Indeed, the inabilty to prove a lower bound on the quantum query complexity of the two-level - tree via approximate degree was the original motivation for the development of the quantum adversary method [Amb00]. To this date, essentially all tight lower bounds on rely on the dual formulation of approximate degree in some capacity111Recently, Ben-David, Bouland, Garg, and Kothari [BDBGK18] proved that for any total boolean function . By a reduction involving polynomials derived from quantum algorithms, they show that . They then appeal to the known result [She11] that . Though they never construct an explicit dual witness, the lower bound on relies on dual witnesses in an essential way..
In this work, we show that nevertheless, one can prove a (nearly) tight lower bound using symmetrization arguments alone. Specifically, we show that , which is tight up to the log factor. Our proof relies on a technique due to Aaronson, Kothari, Kretschmer, and Thaler [AKKT19]: we use Laurent polynomials (polynomials that can have positive and negative exponents) and an associated symmetrization (Lemma 2) that reduces bivariate polynomials to univariate polynomials.
In fact, our proof outline is very similar to the lower bound in [AKKT19] on the one-sided approximate degree of . At a high level, our proof begins with what is essentially a lower bound on the degree of a “robust”, partially symmetrized polynomial that approximates . In particular, we show a generalization of the following:
Theorem 4 (Informal).
Suppose that is a polynomial with the property that for all :
If for some , then .
If for all , then .
To give intuition, the variables roughly correspond to the Hamming weight of the inputs to each gate. Indeed, any polynomial that satisfies the statement of the theorem can be turned into one that approximates by letting equal the sum of the inputs to the th gate. However, the polynomial is also required to be “robust” in the sense the polynomial must behave similarly when is not an integer.
The proof of Theorem 4 works as follows: we group the variables into pairs and apply the Laurent polynomial symmetrization to each pair. We argue that this has the effect of “switching” the role of and , in the sense that the resulting polynomial (in variables) looks like a partially symmetrized polynomial that approximates , which has approximate degree .
We then show (Theorem 5) that starting with a polynomial that approximates , we can “robustly symmetrize” to construct a polynomial of the same degree that behaves like the one in the statement of Theorem 4, at the cost of a factor in the lower bound on the degree of the polynomial. This polynomial is obtained by applying the “erase-all-subscripts” symmetrization (Lemma 1) to the variables corresponding to each gate, producing a polynomial in variables. This immediately implies (Corollary 6) that any polynomial that approximates has degree .
We use to denote the set . We will need the following two symmetrization lemmas, which were both introduced (in these forms) in [AKKT19]:
Lemma 1 (Erase-all-subscripts symmetrization [Akkt19]).
Let be a real multilinear polynomial, and for any real number , let denote the distribution over wherein each coordinate is selected independently to be with probability
with probability. Then there exists a real polynomial with such that for all :
Then the lemma follows from linearity of expectation, because is assumed to be multilinear. ∎
Lemma 2 (Laurent polynomial symmetrization [Akkt19]).
Let be a real polynomial that is symmetric (i.e. for all ). Then for any real , there exists a real polynomial with such that for all real , .
Write as a Laurent polynomial (a polynomial in and ). Because is symmetric, we have that . This implies that the coefficients of the and terms of are equal for all , as otherwise would not be identically zero. Write for some coefficients , where . Then, it suffices to show that can be expressed as a real polynomial of degree in for all .
We prove by induction on . The case is a constant polynomial. For , observe that , where is some real Laurent polynomial of degree satisfying . In particular, we have that , where the right side of the equation is a real polynomial of degree in (by the induction assumption). ∎
We remark that Lemma 2 can also be viewed as a consequence of the fundamental theorem of symmetric polynomials222Indeed, our proof even mirrors the standard proof of the fundamental theorem of symmetric polynomials., which states that every symmetric polynomial in variables can be expressed uniquely as a polynomial in the elementary symmetric polynomials in variables. In 2 variables, the elementary symmetric polynomials are and . So, if , then restricting to the set and writing in terms of corresponds to taking , which is just a polynomial in . (Of course, one would also have to show that this transformation can be applied in a degree-preserving way, as the proof of Lemma 2 does).
Note also that Lemma 1 and Lemma 2 can be applied to polynomials with additional variables, because of the isomorphism between the polynomial rings and . For example, the Laurent polynomial symmetrization can be applied more generally to any polynomial that is symmetric in and by rewriting as a sum of the form:
where and are sets of polynomials and the s are all symmetric. Then, symmetrizing each according to Lemma 2 gives a polynomial .
Finally, we note the tight characterization of the approximate degree of and :
Lemma 3 ([Pat92]).
3 Main Result
We begin with the following theorem, which essentially lower bounds the degree of a robust, partially symmetrized polynomial that approximates . For convenience, a diagram of Theorem 4 is shown in Figure 1.
Let be arbitrary constants, and let such that also holds. Suppose that is a polynomial with the property that for all :
If for some , then .
If for all , then .
Assume without loss of generality that is symmetric in . Apply the Laurent polynomial symmetrization (Lemma 2) to each adjacent pair of variables to obtain a polynomial in corresponding variables with . We think of as corresponding to the restriction . Then we observe that for all :
when for some , as corresponds to either or , which corresponds to either or , respectively. (This is where we need the assumption , as otherwise this case never holds).
when for all , as corresponds , which corresponds to .
Perform an affine shift of with to obtain . The reason for this choice is for convenience, so that the cutoffs for the inequalities above become 0 and 1, respectively. It is easiest to see this by using the identity . Let . Then we observe that for all :
when for some , as corresponds to .
when for all , as corresponds .
Notice that approximates a partially symmetrized function. Now, we “un-symmetrize” . Let . Then approximates over the variables . Since we know that (Lemma 3), and since this construction satisfies we conclude that . ∎
Next, we show that a polynomial that approximates can be “robustly symmetrized” one like in the statement of Theorem 4 with and .
Let be a polynomial in variables that -approximates , where . Specifically, we assume that on a -instance, and on a -instance. Then there exists a polynomial with such that for all :
If for some , then .
If for all , then .
Assume without loss of generality that is multilinear (because over ), so that we can apply the erase-all-subscripts symmetrization (Lemma 1) separately to the inputs of each gate. This erases all of the “” subscripts, giving a polynomial with such that:
where denotes the distribution over
where each coordinate is selected from an independent Bernoulli distribution with probability.
Suppose for all . Then we have that:
On the other hand, suppose for all , and that for some . Then we have that:
by similar inequalities.
Since in general we can bound by
we can conclude that for all :
when for some .
when for all .
Putting these two theorems together gives:
Though our lower bound on is not tight, it suggests a natural way get a tighter lower bound via the same method: either tighten the “robust symmetrization” argument in Theorem 5 to eliminate the log factor (i.e. construct a polynomial of the same degree with and ), or show that Theorem 4 can be tightened by a factor. We conjecture that Theorem 4 is tight, and that Theorem 5 can be improved.
We remark that nevertheless, it is an open problem to exhibit polynomials of minimal degree that satisfy the statement of Theorem 4 with and constant, even in the case where ! We highlight this below:
What is the minimum degree of a polynomial with the property that for all :
If for some , then .
If for all , then .
In particular, do such polynomials exist of degree ?
Clearly, degree polynomials are necessary, by the lower bound on (Lemma 3). Additionally, degree polynomials are sufficient. We apply error reduction polynomials of degree to each input, mapping and to and , respectively. Then we sum the error-reduced inputs and feed the sum into an appropriately shifted Chebyshev polynomial of degree , giving a polynomial of degree overall.
More nontrivially, polynomials of degree are sufficient for some constant , where denotes the iterated logarithm. Such polynomials can be derived from a quantum algorithm for search on bounded-error inputs described in the introduction of [HMDW03], using the well known connection between quantum query algorithms and approximating polynomials [BBC01]. The algorithm involves partitioning the input into blocks of size . Search on a single block can be done in queries using the trivial error reduction procedure in combination with Grover search. Then, using an additional queries, one can reduce the error probability in the case where for all . Overall, the search on the block accepts with constant probability if for some , and with probability if for all . One then applies this quantum algorithm recursively.
Note that the main algorithm of Høyer, Mosca, and de Wolf [HMDW03] does not give degree polynomials for creftypecap 7, as the polynomial derived from that quantum algorithm requires each input to be in either or . In contrast, in case 1 of creftypecap 7, there may be some inputs in . Likewise, Sherstov’s result on making polynomials robust to noise [She12] does not resolve the question, because it also only makes polynomials robust on inputs in .
Beyond the results that we (re-)proved, we wonder if the techniques introduced will find applications elsewhere. For example, we observed a connection between Lemma 2 and the fundamental theorem of symmetric polynomials. Is the Laurent polynomial symmetrization a special case of a more general class of symmetrizations that involve analyzing symmetric polynomials in the basis of elementary symmetric polynomials? If so, do any of these other types of symmetrizations have applications in proving new lower bounds on approximate degree?
- [AKKT19] Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum lower bounds for approximate counting via Laurent polynomials, 2019. URL: https://arxiv.org/abs/1904.08914, arXiv:arXiv:1904.08914.
Quantum lower bounds by quantum arguments.
Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, STOC ’00, pages 636–643, New York, NY, USA, 2000. ACM. URL: http://doi.acm.org/10.1145/335305.335394, doi:10.1145/335305.335394.
- [Amb05] Andris Ambainis. Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory of Computing, 1(3):37–46, 2005. URL: http://www.theoryofcomputing.org/articles/v001a003, doi:10.4086/toc.2005.v001a003.
- [BBC01] Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778–797, July 2001. URL: http://doi.acm.org/10.1145/502090.502097, doi:10.1145/502090.502097.
- [BDBGK18] Shalev Ben-David, Adam Bouland, Ankit Garg, and Robin Kothari. Classical lower bounds from quantum upper bounds. In Proceedings of the 2018 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’18, 2018. URL: https://doi.org/10.1109/FOCS.2009.18.
- [BT13] Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and markov-bernstein inequalities. In Proceedings of the 40th International Conference on Automata, Languages, and Programming - Volume Part I, ICALP’13, pages 303–314, Berlin, Heidelberg, 2013. Springer-Verlag. URL: http://dx.doi.org/10.1007/978-3-642-39206-1_26, doi:10.1007/978-3-642-39206-1_26.
- [HMDW03] Peter Høyer, Michele Mosca, and Ronald De Wolf. Quantum search on bounded-error inputs. In Proceedings of the 30th International Conference on Automata, Languages and Programming, ICALP’03, pages 291–299, Berlin, Heidelberg, 2003. Springer-Verlag. URL: http://dl.acm.org/citation.cfm?id=1759210.1759241.
- [MPB69] M. Minsky, S.A. Papert, and L. Bottou. Perceptrons: An Introduction to Computational Geometry. MIT Press, 1969.
- [NS92] Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. In Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC ’92, pages 462–467, New York, NY, USA, 1992. ACM. URL: http://doi.acm.org/10.1145/129712.129757, doi:10.1145/129712.129757.
- [Pat92] Ramamohan Paturi. On the degree of polynomials that approximate symmetric boolean functions. In Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC ’92, pages 468–474, New York, NY, USA, 1992. ACM. URL: http://doi.acm.org/10.1145/129712.129758, doi:10.1145/129712.129758.
- [She09] Alexander A. Sherstov. The intersection of two halfspaces has high threshold degree. In Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’09, pages 343–362, Washington, DC, USA, 2009. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2009.18, doi:10.1109/FOCS.2009.18.
- [She11] Alexander A. Sherstov. Strong direct product theorems for quantum communication and query complexity. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC ’11, pages 41–50, New York, NY, USA, 2011. ACM. URL: http://doi.acm.org/10.1145/1993636.1993643, doi:10.1145/1993636.1993643.
- [She12] Alexander A. Sherstov. Making polynomials robust to noise. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC ’12, pages 747–758, New York, NY, USA, 2012. ACM. URL: http://doi.acm.org/10.1145/2213977.2214044, doi:10.1145/2213977.2214044.
- [She13] Alexander A. Sherstov. Approximating the and-or tree. Theory of Computing, 9(20):653–663, 2013. URL: http://www.theoryofcomputing.org/articles/v009a020, doi:10.4086/toc.2013.v009a020.
- [Shi02] Yaoyun Shi. Approximating linear restrictions of boolean functions, 2002. URL: https://web.eecs.umich.edu/~shiyy/mypapers/linear02-j.ps.