Approximating the noise sensitivity of a monotone Boolean function

The noise sensitivity of a Boolean function f : {0, 1}ⁿ → {0, 1} is one of its fundamental properties. For noise parameter δ, the noise sensitivity is denoted as NS_δ[f]. This quantity is defined as follows: First, pick x = (x1, . . ., xn) uniformly at random from {0, 1}ⁿ, then pick z by flipping each xi independently with probability δ. NS_δ[f] is defined to equal Pr[f(x) =6 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_δ[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 [21]). Specifically, we study the algorithmic problem of approximating NS_δ[f] for monotone f, given the promise that NS_δ[f] ≥ 1/n^C for constant C, and for δ in the range 1/n ≤ δ ≤ 1/2. For such f and δ, we give a randomized algorithm performing O (Formula presented.) queries and approximating NS_δ[f] to within a multiplicative factor of (1 ± ε). Given the same constraints on f and δ, we also prove a lower bound of Ω (Formula presented.) on the query complexity of any algorithm that approximates NS_δ[f] to within any constant factor, where ξ 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.