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The sensitivity of a Boolean function f is the maximum over all inputs x, ofthe number of sensitive coordinates of x. The well-known sensitivity conjectureof Nisan (see also Nisan and Szegedy) states that every sensitivity-s Booleanfunction can be computed by a polynomial over the reals of degree poly(s). Thebest known upper bounds on degree, however, are exponential rather thanpolynomial in s.  Our main result is an approximate version of the conjecture: every Booleanfunction with sensitivity s can be epsilon-approximated (in L_2) by apolynomial whose degree is O(s log(1/epsilon)). This is the first improvementon the folklore bound of s/epsilon. Further, we show that improving the boundto O(s^c log(1/epsilon)^d)$ for any d < 1 and any c > 0 will imply thesensitivity conjecture. Thus our result is essentially the best one can hopefor without proving the conjecture.  We postulate a robust analogue of the sensitivity conjecture: if most inputsto a Boolean function f have low sensitivity, then most of the Fourier mass off is concentrated on small subsets, and present an approach towards provingthis conjecture.

Degree and Sensitivity: tails of two distributions