The Total Influence (Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function f : {0, 1} n → {0, 1}, which we denote by I[f ]. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of (1 ± ) by performing O √ n I[f ] poly(1/ ) queries. We also prove a lower bound of Ω √ n log n·I[f ] on the query complexity of any

more »

... -factor approximation algorithm for this problem (which holds for I[f ] = Ω(1)), hence showing that our algorithm is almost optimal in terms of its dependence on n. For general functions we give a lower bound of Ω n I[f ] , which matches the complexity of a simple sampling algorithm.