On the maximal L1 influence of real-valued boolean functions

TL;DR

This paper establishes a lower bound on the maximum L1 influence of well-behaved real-valued boolean functions under p-biased distributions, linking influence to variance and logarithmic factors.

Contribution

It introduces a new lower bound on the maximal L1 influence for sequences of real-valued boolean functions under p-biased measures.

Findings

Maximal L1 influence is at least proportional to variance times log(n)/n.

The bound applies to well-behaved functions, including bounded and non-constant.

The result holds for any p in (0,1).

Abstract

We show that any sequence of well-behaved (e.g. bounded and non-constant) real-valued functions of $n$ boolean variables $\{f_n\}$ admits a sequence of coordinates whose $L^1$ influence under the $p$-biased distribution, for any $p\in(0,1)$, is $\Omega(\text{var}(f_n) \frac{\ln n}{n})$.