A Unified Lower Bound on the Noisy Query Complexity of Boolean Functions

2026-06-09

AI Summary

This work investigates lower bounds on the query complexity of Boolean functions in the noisy query model, where each query is flipped independently with a fixed probability. By analyzing degree statistics of specific subgraphs of the Boolean hypercube and integrating tools from Boolean function analysis, hypercube graph theory, and probabilistic methods, the authors introduce the first general lower bound framework that surpasses the trivial random-query baseline. This framework not only unifies and simplifies existing results but also resolves an open problem concerning the relationship between total influence $I(f)$ and noisy query complexity $N_p(f)$, establishing that $N_p(f) = \Omega(I(f) \log I(f))$. Furthermore, it yields tight bounds for several new functions and recovers nearly all known lower bounds up to constant factors.

Abstract

We study the query complexity of Boolean functions $f: \{0, 1\}^n \rightarrow \{0, 1\}$ in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability $p \in (0, 1/2)$; repeated queries are allowed. The noisy query complexity $\mathsf{N}_p(f)$ of a function $f$ is defined as the minimum expected number of queries needed to compute $f(x)$ with error probability at most $1/3$, for the worst case input $x$. We prove a general lower bound on $\mathsf{N}_p(f)$ based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that $\mathsf{N}_p(f)$ is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that $\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))$, where $\mathsf{I}(f)$ denotes the total influence of $f$. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.

Problem

• noisy query complexity
• Boolean functions
• query complexity lower bound
• total influence
• Boolean hypercube

Innovation

• noisy query complexity
• Boolean functions
• lower bound
• total influence
• hypercube subgraphs