Parameterized Convexity Testing

TL;DR

This paper introduces a parameterized, nonadaptive algorithm for convexity testing of functions over discrete domains, achieving improved query complexity bounds by leveraging the number of distinct derivatives, and proves these bounds are tight.

Contribution

It presents a new parameterized convexity testing algorithm that outperforms previous methods when the number of distinct derivatives is small, and establishes the optimality of these bounds.

Findings

Query complexity of $O(rac{ ext{log}(s)}{ ext{eps}})$ for functions with $s$ distinct derivatives.

Bound is tight, matching lower bounds for the problem.

Improves upon previous convexity testing bounds when $s = o(n)$.

Abstract

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 $\eps \in (0,1), s \in \mathbb{N}$, and oracle access to a function, $\eps$-tests convexity in $O(\log (s)/\eps)$, 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 $O(\frac{\log \eps n}{\eps})$; our bound is strictly better when $s = o(n)$. 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 $\Omega(\frac{\log (\eps n)}{\eps})$ expressed in terms of the input size and obtain a more efficient algorithm.