We prove a simple, nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that ˜ deg(AND m ˆ OR n ) = ˜ Ω(√ mn ). We prove this lower bound via reduction to the OR function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [6, 10, 21]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson et al. [2].
We prove a simple, nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that ˜ deg(AND m ˆ OR n ) = ˜ Ω(√ mn ). We prove this lower bound via reduction to the OR function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [6, 10, 21]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson et al. [2].
The ε-approximate degree of a Boolean function f:{−1,1}n→{−1,1} is the minimum degree of a real polynomial that approximates f to within ε in the ℓ∞ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the ε-approximate degree of the two-level AND-OR tree for any constant ε>0. We show that this quantity is Θ(n), closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov using related techniques. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.
We prove that the OR function on {-1,1\} n can be pointwise approximated with error ε by a polynomial of degree O ( k ) and weight 2 O ( n log (1/ε)/k) , for any k ≥ √ n log (1/ε). This result is tight for any k ≤ (1-Ω (1)) n . Previous results were either not tight or had ε = Ω (1). In general, we obtain a tight approximate degree-weight result for any symmetric function. Building on this, we also obtain an approximate degree-weight result for bounded-width CNF. For these two classes no such result was known. We prove that the OR function on {−1,1}n can be pointwise approximated with error ϵ by a polynomial of degree O(k) and weight 2O(nlog(1/ϵ)/k) , for any k≥nlog(1/ϵ) . This result is tight for any k≤(1−Ω(1))n . Previous results were either not tight or had ϵ=Ω(1) . In general, we obtain a tight approximate degree-weight result for any symmetric function. Building on this, we also obtain an approximate degree-weight result for bounded-width CNF . For these two classes no such result was known. One motivation for such results comes from the study of indistinguishability. Two distributions P , Q over n -bit strings are (k,δ) -indistinguishable if their projections on any k bits have statistical distance at most δ . The above approximations give values of (k,δ) that suffice to fool OR , symmetric functions, and bounded-width CNF , and the first result is tight for all k while the second result is tight for k≤(1−Ω(1))n . We also show that any two (k,δ) -indistinguishable distributions are O(nk/2δ) -close to two distributions that are (k,0) -indistinguishable, improving the previous bound of O(n)kδ . Finally, we present proofs of some known approximate degree lower bounds in the language of indistinguishability, which we find more intuitive.
The approximate degree of a Boolean function f : {-1, 1}n→ {-1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n · polylog(n)) variables with approximate degree at least D = Ω(n1/3· d2/3). In particular, if d = n1-Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a constant-depth polynomial-size Boolean circuit, then so is F. By recursively applying our transformation, for any constant δ > 0 we exhibit an AC0function of approximate degree Ω(n1-δ). This improves over the best previous lower bound of Ω(n2/3) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: · For any constant δ > 0, an Ω(n1-δ) lower bound on the quantum communication complexity of a function in AC0. · A Boolean function f with approximate degree at least C(f)2-o(1), where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent. · Improved secret sharing schemes with reconstruction procedures in AC0.
A Nearly Optimal Lower Bound on the Approximate Degree of AC0
The approximate degree of a Boolean function f:{−1,1}n→{−1,1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n⋅polylog(n)) variables with approximate degree at least D=Ω(n1/3⋅d2/3). In particular, if d=n1−Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a polynomial-size Boolean circuit of constant depth, then so is F. By recursively applying our transformation, for any constant δ>0 we exhibit an AC0 function of approximate degree Ω(n1−δ). This improves over the best previous lower bound of Ω(n2/3) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant δ>0, an Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC0. * A Boolean function f with approximate degree at least C(f)2−o(1), where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC0.
The approximate degree of a Boolean function f captures how well f can be approximated pointwise by low-degree polynomials. This monograph surveys what is known about approximate degree and illustrates its applications in theoretical computer science. A particular focus of the survey is a method of proving lower bounds via objects called dual polynomials. These represent a reformulation of approximate degree using linear programming duality. We discuss in detail a recent, powerful technique for constructing dual polynomials, called “dual block composition”.