Every decision tree has an influential variable

TL;DR

This paper establishes a fundamental inequality relating the variance, influence, and probability of variables in decision trees, leading to new lower bounds on the depth and influence for various decision tree models and applications.

Contribution

It proves a general inequality connecting variance, influence, and read probability in decision trees, extending influence lower bounds to randomized and non-boolean decision trees.

Findings

Proves variance-influence inequality for decision trees under arbitrary product measures.

Derives lower bounds on decision tree depth based on variable influence.

Provides a simplified proof of lower bounds on randomized query complexity for monotone graph properties.

Abstract

We prove that for any decision tree calculating a boolean function $f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n \delta_i \Inf_i(f), \] where $\delta_i$ is the probability that the $i$th input variable is read and $\Inf_i(f)$ is the influence of the $i$th variable on $f$. The variance, influence and probability are taken with respect to an arbitrary product measure on $\{-1,1\}^n$. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced boolean function with a decision tree of depth $d$ has a variable with influence at least $\frac{1}{d}$. The only previous nontrivial lower bound known was $\Omega(d 2^{-d})$. Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with non-boolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least $\Omega(v^{4/3}/p^{1/3})$, where $v$ is the number of vertices and $p \leq \half$ is the critical threshold probability. This supersedes the milestone $\Omega(v^{4/3})$ bound of Hajnal and is sometimes superior to the best known lower bounds of Chakrabarti-Khot and Friedgut-Kahn-Wigderson.