Polynomial bounds for decoupling, with applications
February 3, 2016

Let f(x_1, ..., x_n) be an n-variate real multilinear polynomial of degree at most k. For its one-block decoupled version f~, we show tail-bound comparisons of the form

Pr[ |f~(y, z)| > C_k t] <= D_k Pr[ |f(x)| > t ]

Our constants C_k, D_k are significantly better than those known for ''full decoupling''. For example, when x, y, z are independent Gaussians we obtain C_k = D_k = O(k); when x, y, z are {+1, -1} random variables we obtain C_k = O(k^2), D_k = k^{O(k)}. By contrast, for full decoupling only C_k = D_k = k^{O(k)} is known in these settings.

We describe consequences of these results for query complexity (related to conjectures of Aaronson and Ambainis) and for analysis of Boolean functions (including an optimal sharpening of the DFKO~Inequality).


Yu Zhao is a PhD candidate at Carnegie Mellon University advised by Ryan O'Donnell. Previously, he completed M.S. in Carnegie Mellon University and Bachelor in Computer Science in Tsinghua University. His research interests include complexity and analysis of boolean functions.