Polynomial bounds for decoupling, with applications

TL;DR

This paper establishes improved polynomial bounds for one-block decoupling of multilinear polynomials, leading to tighter tail bounds and applications in query complexity and Boolean function analysis.

Contribution

It provides the first polynomial bounds for one-block decoupling with better constants than full decoupling, enhancing tail-bound comparisons for specific distributions.

Findings

C_k, D_k constants are significantly improved over full decoupling.

Results apply to Gaussian and Rademacher variables with polynomial bounds.

Implications for query complexity and Boolean function analysis.

Abstract

Let f(x) = f(x_1, ..., x_n) = \sum_{|S| <= k} a_S \prod_{i \in S} x_i be an n-variate real multilinear polynomial of degree at most k, where S \subseteq [n] = {1, 2, ..., n}. For its "one-block decoupled" version, f~(y,z) = \sum_{|S| <= k} a_S \sum_{i \in S} y_i \prod_{j \in S\i} z_j, 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, Rademacher 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).