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 $O(\log² n)$. 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 $\Omega((\frac{n}{d})<sup>{\frac{d}{2}})$.