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.
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Kane, D.M. The Gaussian Surface Area and Noise Sensitivity of Degree-d Polynomial Threshold Functions. comput. complex. 20, 389–412 (2011). https://doi.org/10.1007/s00037-011-0012-6
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DOI: https://doi.org/10.1007/s00037-011-0012-6