Dual lower bounds for approximate degree and Markov–Bernstein inequalities

Abstract

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 error 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 [4,16,26,37,40]. The same lower bound was recently obtained independently by Sherstov using related techniques [33]. 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 ofŠpalek [42]. 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.

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“…First, only a handful of techniques are currently known for the construction of dual polynomials, especially for the case where ε = Θ(1). To date, dual polynomials have been constructed only for symmetric functions [46,19] and a handful of highly structured block-composed functions [20,19,40,41,42,43,39]. (A block-composed function F : {−1, 1} M·N → {−1, 1} is a function of the form of the form F = g( f (x 1 ), .…”

“…Specifically, our construction ensures that the analysis establishing Properties (1) and (2) becomes similar to the analyses of known dual polynomials for the OR function [46,19].…”

“…In more detail, our prior work [19] built on work of Špalek [46] to give a dual witness γ for the fact that deg ε (OR n ) = Ω( √ n) for any constant ε < 1; moreover, γ places non-zero weight only on sets H k , for values of k equal (up to scaling factors) to perfect squares. The pure high degree of γ is shown to be equal to (at least) the number of sets H k upon which γ places non-zero weight.…”

“…This "method of dual polynomials" was recently used to resolve the approximate degree of the AND-OR tree [40,19], closing a long line of incrementally larger lower bounds [44,7,23,41,30]. It has also been used to establish several "hardness amplification" results for approximate degree [48,20,42], and to prove new threshold degree lower bounds for several important classes of functions, including the intersection of two majorities [31,41] and AC 0 [42,43].…”

Weighted Sobolev spaces: Markov-type inequalities and duality

“…Some illustrative examples when T is either the derivative or the difference operator and you deal with some classical weights (Laguerre, Gegenbauer in the first case, Charlier, Meixner in the second one) are shown. Another recent application of Markov-Bernstein-type inequalities can be found in [5].…”