A SENSITIVITY ESTIMATE FOR BOOLEAN FUNCTIONS

A Boolean response to a random binary input of length n can be modeled as a {0, 1}-valued function upsilon defined on a discrete probability space OMEGA of all subsets of a finite set of size n. An omega is-an-element-of OMEGA represents the locations of 1's in the input. For a particular j(th) location, 1 less-than-or-equal-to j less-than-or-equal-to n, we assume that 1 appears with probability rho(j) independently of other locations. Then, for rhoBAR = (rho1, ..., rho(n)), we will investigate rho(rhoBAR)(upsilon = 1) as a function of rhoBAR. Using the sharp version of the Khinchin inequality, we give an upper estimate for the 12 norm of the gradient of P(rhoBAR)(upsilon = 1) evaluated at rhoBAR = (1/2, ..., 1/2) (cf. (5) below). For monotone functions, the estimate applies also to vector of influences of Boolean functions. We also provide a handy expansion of P(.) (upsilon = 1) based on a Fourier expansion of upsilon (cf. (4) below). Numerical analysis of the bounds leads to the conjecture about the sharp bound that depends on cardinality of the underlying set; the sharp version of the Khinchin inequality is also conjectured.