Parameterized Convexity Testing

In this work, we develop new insights into the fundamental problem of convexity testing of real-valued functions over the domain $[n]$. Specifically, we present a nonadaptive algorithm that, given inputs $\eps \in (0,1), s \in $\mathbb{N}$$,andoracleaccesstoafunction,$\eps$-testsconvexityin$O(\log (s)/\eps)$,where$s$ is an upper bound on the number of distinct discrete derivatives of the function. We also show that this bound is tight. Since $s \leq n$, our query complexity bound is at least as good as that of the optimal convexity tester (Ben Eliezer; ITCS 2019) with complexity $O(\frac{\log \eps n}{\eps})$;ourboundisstrictlybetterwhen$s = o(n)$. The main contribution of our work is to appropriately parameterize the complexity of convexity testing to circumvent the worst-case lower bound (Belovs et al.; SODA 2020) of $\mathrm{\Omega }(\frac{\log (\text{eps}n)}{\text{eps}})$ expressed in terms of the input size and obtain a more efficient algorithm.