Testing convexity of functions over finite domains

Abstract

We establish new upper and lower bounds on the number of queries required to test convexity of functions over various discrete domains. 1. We provide a simplified version of the non-adaptive convexity tester on the line. We re-prove the upper bound $O(\frac{\log(\epsilon n)}{\epsilon})$ in the usual uniform model, and prove an $O(\frac{\log n}{\epsilon})$ upper bound in the distribution-free setting. 2. We show a tight lower bound of $\Omega(\frac{\log(\epsilon n)}{\epsilon})$ queries for testing convexity of functions $f: [n] \rightarrow \mathbb{R}$ on the line. This lower bound applies to both adaptive and non-adaptive algorithms, and matches the upper bound from item 1, showing that adaptivity does not help in this setting. 3. Moving to higher dimensions, we consider the case of a stripe $[3] \times [n]$. We construct an \emph{adaptive} tester for convexity of functions $f\colon [3] \times [n] \to \mathbb R$ with query complexity $O(\log^2 n)$. We also show that any \emph{non-adaptive} tester must use $\Omega(\sqrt{n})$ queries in this setting. Thus, adaptivity yields an exponential improvement for this problem. 4. For functions $f\colon [n]^d \to \mathbb R$ over domains of dimension $d \geq 2$, we show a non-adaptive query lower bound $\Omega((\frac{n}{d})^{\frac{d}{2}})$.