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 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.
Monotone Boolean functions
noise sensitivity
influence
Theory of computation~Design and analysis of algorithms
Theory of computation
52:1-52:17
RANDOM
Full version: https://arxiv.org/abs/1904.06745
We are grateful to the anonymous referees, Daniel Grier and MIT EECS Communication Lab for helpful comments and suggestions.
Ronitt
Rubinfeld
Ronitt Rubinfeld
CSAIL at MIT, Cambridge, MA, USA
Blavatnik School of Computer Science at Tel Aviv University, Israel
https://people.csail.mit.edu/ronitt/
NSF grants CCF-1650733, CCF-1733808, IIS-1741137 and CCF-1740751
Arsen
Vasilyan
Arsen Vasilyan
CSAIL at MIT, Cambridge, MA, USA
NSF grant IIS-1741137, EECS SuperUROP program, the MIT Summer UROP program and the DeFlorez Endowment Fund
10.4230/LIPIcs.APPROX-RANDOM.2019.52
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Ronitt Rubinfeld and Arsen Vasilyan
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