The Gaussian Surface Area and Noise Sensitivity of Degree-d Polynomial Threshold Functions

Abstract

We prove asymptotically optimal bounds on the Gaussian noise sensitivity and Gaussian surface area of degree-d polynomial threshold functions. In particular, we show that for f a degree-d polynomial threshold function that the Gaussian noise sensitivity of f with parameter $${\epsilon}$$ is at most $${\frac{d\arcsin\left(\sqrt{2\epsilon-\epsilon^2}\right)}{\pi}}$$ . This bound translates into an optimal bound on the Gaussian surface area of such functions, namely that the Gaussian surface area is at most $${\frac{d}{\sqrt{2\pi}}}$$ . Finally, we note that the later result implies bounds on the runtime of agnostic learning algorithms for polynomial threshold functions.