[Paper Review] Adaptive Boolean Monotonicity Testing in Total Influence Time

Deeparnab Chakrabarty, C. Seshadhri|arXiv (Cornell University)|Jan 9, 2018

Formal Methods in Verification7 references3 citations

TL;DR

This paper presents an adaptive, one-sided monotonicity tester for Boolean functions that runs in time proportional to the total influence $\mathbf{I}(f)$, achieving $O(\mathbf{I}(f) \cdot \mathrm{poly}(\log n)/\varepsilon^4)$ query complexity. By leveraging random walks on the undirected hypercube and analyzing sticky edges, the tester efficiently detects violations of monotonicity, beating the $\widetilde{O}(\sqrt{n}/\varepsilon^2)$ non-adaptive bound for low-influence functions.

ABSTRACT

The problem of testing monotonicity of a Boolean function $f:\{0,1\}^n o \{0,1\}$ has received much attention recently. Denoting the proximity parameter by $\varepsilon$, the best tester is the non-adaptive $\widetilde{O}(\sqrt{n}/\varepsilon^2)$ tester of Khot-Minzer-Safra (FOCS 2015). Let $I(f)$ denote the total influence of $f$. We give an adaptive tester whose running time is $I(f)poly(\varepsilon^{-1}\log n)$.

Motivation & Objective

• To resolve whether adaptivity can provably improve monotonicity testing for Boolean functions with low total influence.
• To design a one-sided adaptive tester whose query complexity depends on the total influence $\mathbf{I}(f)$ rather than $n$.
• To demonstrate that adaptivity provides a super-polynomial speedup over non-adaptive testers for functions with $\mathbf{I}(f) \ll \sqrt{n}$.
• To provide a tight analysis of random walks on the undirected hypercube to detect violating edges in monotonicity.

Proposed method

• The tester performs random walks of varying lengths $\ell$ on the undirected hypercube $H_n$, selecting $\ell$ uniformly at random from powers of two.
• It checks for influential edges along the walk and uses binary search to locate the first violating edge where $f(x)=1$, $f(y)=0$, and $x \prec y$.
• The analysis relies on identifying $\ell$-sticky edges—edges that are the only influential edge in a walk segment—whose probability of occurrence is bounded below using conditional independence.
• The key technical tool is a directed isoperimetry lemma from KMS, which guarantees the existence of a subgraph with $\sigma^2 d = \Theta(\varepsilon^2 / \log^4 n)$, ensuring a sufficient number of violating edges when $f$ is $\varepsilon$-far from monotone.
• The rejection probability is lower-bounded by summing over all $\ell$-sticky edges in a walk, using the fact that $\Pr[\mathcal{E}_{u,v}] \geq \ell / (4n 2^n)$ for each such edge.
• The algorithm selects $\ell$ uniformly from $\log n$ candidates, ensuring a constant fraction of the total rejection probability is captured with probability $1/\log n$.

Experimental results

Research questions

• RQ1 Can adaptivity lead to a super-polynomial speedup in monotonicity testing for Boolean functions with low total influence?
• RQ2 Is there a natural function class—specifically, those with $\mathbf{I}(f) \ll \sqrt{n}$—for which adaptive testers outperform the best non-adaptive testers?
• RQ3 Can random walks on the undirected hypercube be used to detect violating edges in a way that leverages adaptivity to achieve sub-$\sqrt{n}$ query complexity?
• RQ4 Does the existence of a subgraph with $\sigma^2 d = \Theta(\varepsilon^2 / \log^4 n)$ imply a robust lower bound on the probability of detecting violations via adaptive walks?

Key findings

• The proposed adaptive tester achieves a query complexity of $O(\mathbf{I}(f) \cdot \mathrm{poly}(\log n)/\varepsilon^4)$, which is sub-$\widetilde{O}(\sqrt{n}/\varepsilon^2)$ for functions with $\mathbf{I}(f) \ll \sqrt{n}$.
• For functions with total influence $\mathbf{I}(f) \leq 6\sqrt{n}$, the tester's rejection probability is at least $\Omega(\varepsilon^2 / (\mathbf{I}(f) \log^5 n))$, ensuring high confidence in the test.
• When $\sigma < 100/\sqrt{n}$, the tester still achieves $\Omega(\varepsilon^4 / (\mathbf{I}(f) \log^8 n))$ rejection probability by sampling random edges, confirming robustness in low-influence regimes.
• The analysis shows that $\ell^*$-sticky edges contribute significantly to rejection probability, with $|F_{\ell^*}| \geq \sigma d 2^n / 2$, ensuring sufficient edge coverage.
• The tester's performance is analyzed via a decomposition of the walk into segments where each influential edge is the first such edge, enabling a clean union bound over disjoint events.
• The final rejection probability is bounded below by $\Omega(\varepsilon^2 / (\mathbf{I}(f) \log^5 n))$, proving correctness and efficiency for the class of functions with bounded influence.