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 ε ∈ (0, 1), s ∈ N, and oracle access to a function, ε-tests convexity in O(log(s)/ε), 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 ≤ n, our query complexity bound is at least as good as that of the optimal convexity tester (Ben Eliezer; ITCS 2019) with complexity O( log εn ε ); 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 Ω( log(εn) ε ) expressed in terms of the input size and obtain a more efficient algorithm.