Mark Bun

Dual lower bounds for approximate degree and Markov-Bernstein inequalities

The e-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 e in the ? ∞ 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 e-approximate degree of the two-level AND-OR tree for any constant e 0 . We show that this quantity is ? ( n ) , closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov (Theory Comput. 2013) 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 Spalek (2008). 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.

Dual lower bounds for approximate degree and markov-bernstein inequalities

The e-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 e 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 e-approximate degree of the two-level AND-OR tree for any constant e>0. We show that this quantity is

Hardness Amplification and the Approximate Degree of Constant-Depth Circuits

\Theta(\sqrt{n})

Simultaneous Private Learning of Multiple Concepts

, closing a line of incrementally larger lower bounds [3,11,21,30,32]. The same lower bound was recently obtained independently by Sherstov using related techniques [25]. 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 Spalek [34]. 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.

Make up your mind: the price of online queries in differential privacy

We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to within constant error in a certain one-sided sense, then an OR of disjoint copies of that circuit cannot be pointwise approximated even with very high error. As our main application, we show that for every sequence of degrees \(d(n)\), there is an explicit depth-three circuit \(F: \{-1,1\}^n \rightarrow \{-1,1\}\) of polynomial-size such that any degree-\(d\) polynomial cannot pointwise approximate \(F\) to error better than \(1-\exp (-\tilde{\Omega }(nd^{-3/2}))\). As a consequence of our main result, we obtain an \(\exp (-\tilde{\Omega }(n^{2/5}))\) upper bound on the the discrepancy of a function in AC\(^{\text{0 }}\), and an \(\exp (\tilde{\Omega }(n^{2/5}))\) lower bound on the threshold weight of AC\(^{\text{0 }}\), improving over the previous best results of \(\exp (-\Omega (n^{1/3}))\) and \(\exp (\Omega (n^{1/3}))\) respectively.

A Nearly Optimal Lower Bound on the Approximate Degree of AC^0

We investigate the {\em direct-sum} problem in the context of differentially private PAC learning: What is the sample complexity of solving k learning tasks simultaneously under differential privacy, and how does this cost compare to that of solving k learning tasks without privacy? In our setting, an individual example consists of a domain element x labeled by k unknown concepts (c1,...,ck). The goal of a multi-learner is to output k hypotheses (h1,...,hk) that generalize the input examples. Without concern for privacy, the sample complexity needed to simultaneously learn

Dual Polynomials for Collision and Element Distinctness.

k

The polynomial method strikes back: tight quantum query bounds via dual polynomials

concepts is essentially the same as needed for learning a single concept. Under differential privacy, the basic strategy of learning each hypothesis independently yields sample complexity that grows polynomially with k. For some concept classes, we give multi-learners that require fewer samples than the basic strategy. Unfortunately, however, we also give lower bounds showing that even for very simple concept classes, the sample cost of private multi-learning must grow polynomially in k.

Separating Computational and Statistical Differential Privacy in the Client-Server Model

We consider the problem of answering queries about a sensitive dataset subject to differential privacy. The queries may be chosen adversarially from a larger set Q of allowable queries in one of three ways, which we list in order from easiest to hardest to answer: Offline: The queries are chosen all at once and the differentially private mechanism answers the queries in a single batch. Online: The queries are chosen all at once, but the mechanism only receives the queries in a streaming fashion and must answer each query before seeing the next query. Adaptive: The queries are chosen one at a time and the mechanism must answer each query before the next query is chosen. In particular, each query may depend on the answers given to previous queries. Many differentially private mechanisms are just as efficient in the adaptive model as they are in the offline model. Meanwhile, most lower bounds for differential privacy hold in the offline setting. This suggests that the three models may be equivalent. We prove that these models are all, in fact, distinct. Specifically, we show that there is a family of statistical queries such that exponentially more queries from this family can be answered in the offline model than in the online model. We also exhibit a family of search queries such that exponentially more queries from this family can be answered in the online model than in the adaptive model. We also investigate whether such separations might hold for simple queries like threshold queries over the real line.

Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness.

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(n^{1/3} d^{2/3}). In particular, if d = n^{1-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 AC° function of approximate degree Ω(n^{1-δ}). This improves over the best previous lower bound of Ω(n^{2/3}) 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(n^{1-δ}) lower bound on the quantum communication complexity of a function in AC°.• 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 AC°.