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1. Approximating the noise sensitivity of a monotone Boolean function

   Rubinfeld, R.; Vasilyan, A. (January 2019, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019)

   The noise sensitivity of a Boolean function f: {0,1}^n - > {0,1} is one of its fundamental properties. For noise parameter delta, the noise sensitivity is denoted as NS_{delta}[f]. This quantity is defined as follows: First, pick x = (x_1,...,x_n) uniformly at random from {0,1}^n, then pick z by flipping each x_i independently with probability delta. NS_{delta}[f] is defined to equal Pr [f(x) != f(z)]. Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure. The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating NS_{delta}[f] given an oracle access to the function f. We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [O'Donnell, 2014]). Specifically, we study the algorithmic problem of approximating NS_{delta}[f] for monotone f, given the promise that NS_{delta}[f] >= 1/n^{C} for constant C, and for delta in the range 1/n <= delta <= 1/2. For such f and delta, we give a randomized algorithm performing O((min(1,sqrt{n} delta log^{1.5} n))/(NS_{delta}[f]) poly (1/epsilon)) queries and approximating NS_{delta}[f] to within a multiplicative factor of (1 +/- epsilon). Given the same constraints on f and delta, we also prove a lower bound of Omega((min(1,sqrt{n} delta))/(NS_{delta}[f] * n^{xi})) on the query complexity of any algorithm that approximates NS_{delta}[f] to within any constant factor, where xi can be any positive constant. Thus, our algorithm's query complexity is close to optimal in terms of its dependence on n. We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias. 

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2. The Polynomial Method Strikes Back: Tight Quantum Query Bounds via Dual Polynomials

   https://doi.org/10.1145/3188745.3188784  

   Bun, Mark; Kothari, Robin; Thaler, Justin (October 2020, Theory of computing)

   null (Ed.)

   The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. k-Distinctness: For any constant k, the approximate degree and quantum query complexity of the k-distinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of k-distinctness is O(n3/4−1/(2k+2−4)). Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n]→[n] is Ω~(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. k-Junta testing: A tight Ω~(k1/2) lower bound for k-junta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance from uniform: A tight Ω~(n1/2) lower bound for approximating the statistical distance of a distribution from uniform, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). Shannon entropy: A tight Ω~(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). Surjectivity: The approximate degree of the surjectivity function is Ω~(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of O~(n3/4) due to Sherstov (STOC 2018), which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for surjectivity introduces new techniques for approximating Boolean functions by low-degree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017). 

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3. The query complexity of certification

   https://doi.org/10.1145/3519935.3519993  

   Blanc, G.; Koch, C.; Lange, J.; Tan, L. (January 2022, 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 22))

   We study the problem of certification: given queries to a function f : {0,1}n → {0,1} with certificate complexity ≤ k and an input x⋆, output a size-k certificate for f’s value on x⋆. For monotone functions, a classic local search algorithm of Angluin accomplishes this task with n queries, which we show is optimal for local search algorithms. Our main result is a new algorithm for certifying monotone functions with O(k8 logn) queries, which comes close to matching the information-theoretic lower bound of Ω(k logn). The design and analysis of our algorithm are based on a new connection to threshold phenomena in monotone functions. We further prove exponential-in-k lower bounds when f is non-monotone, and when f is monotone but the algorithm is only given random examples of f. These lower bounds show that assumptions on the structure of f and query access to it are both necessary for the polynomial dependence on k that we achieve. 

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4. Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an $$~O(n\sqrt{d})$$ Monotonicity Tester

   Hadley Black, Deeparnab Chakrabarty (June 2023, Proceedings of the annual ACM Symposium on Theory of Computing)

   The problem of testing monotonicity for Boolean functions on the hypergrid, $$f:[n]^d \to \{0,1\}$$ is a classic topic in property testing. When $n=2$, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making $$\otilde(\eps^{-2}\sqrt{d})$$ queries. Up to polylog $$d$$ and $$\eps$$ factors, this bound matches the $$\widetilde{\Omega}(\sqrt{d})$$-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any $n > 2$, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a $$\otilde(d^{5/6})$$-query upper bound (SODA 2020), quite far from the $$\sqrt{d}$$ bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant $$n$$, up to $$\poly(\eps^{-1}\log d)$$ factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making $$\otilde(\eps^{-2}n\sqrt{d})$$ queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid $[n]^d$. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube. 

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5. A $$d^{1/2+o(1)}$$ Monotonicity Tester for Boolean Functions on $$d$$-Dimensional Hypergrids

   Hadley Black, Deeparnab Chakrabarty (November 2023, Annual Symposium on Foundations of Computer Science)

   Monotonicity testing of Boolean functions on the hypergrid, $$f:[n]^d \to \{0,1\}$$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $$n$$, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity $$\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$$. This complexity is independent of $$n$$, but has a suboptimal dependence on $$d$$. Recently, [Braverman-Khot-Kindler-Minzer, ITCS 2023] and [Black-Chakrabarty-Seshadhri, STOC 2023] describe $$\widetilde{O}(\varepsilon^{-2} n^3\sqrt{d})$$ and $$\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$$-query testers, respectively. These testers have an almost optimal dependence on $$d$$, but a suboptimal polynomial dependence on $$n$$. \smallskip In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity $$O(\varepsilon^{-2} d^{1/2 + o(1)})$$, \emph{independent} of $$n$$. Up to the $$d^{o(1)}$$-factors, our result resolves the non-adaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of $$n$$ yields a non-adaptive, one-sided $$O(\varepsilon^{-2} d^{1/2 + o(1)})$$-query monotonicity tester for Boolean functions $$f:\mathbb{R}^d \to \{0,1\}$$ associated with an arbitrary product measure. 

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