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 ; 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 expressed in terms of the input size and obtain a more efficient algorithm.
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 ; 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 expressed in terms of the input size and obtain a more efficient algorithm.
We study the query complexity of testing quantum juntas. Given an unknown n-qubit unitary or quantum channel, the goal is to decide whether it acts nontrivially on at most k qubits or is ε-far from every such operation. For unitaries, we consider the forward-only model, where the tester has access to U, but not to U† or controlled-U. We give tight bounds for both unitary and channel junta testing. For sufficiently large k, n≥2k, and a broad range of ε, both problems have query complexity Θ(k/(ε2logk)). Our upper bounds are achieved by simple nonadaptive testers using product inputs and single-qubit measurements. For junta channels, this closes the gap by Bao and Yao (COLT 23, TPAMI 2025) between an O(k) upper bound and a Ω(k) lower bound at constant ε, establishing the optimal Θ(k/logk) query complexity. At the heart of both bounds is a strong connection between quantum junta testing and classical support-size testing. For the upper bounds, we reduce quantum junta testing to a generalized support-size problem for distributions over random subsets, and develop a new tester with improved sample complexity. For the lower bounds, we proceed in the reverse direction, reducing classical support-size testing to quantum junta testing through a phase-masked hard ensemble and showing that forward quantum queries can be exactly classicalized on average. Together, these reductions identify support-size testing as the classical core of quantum junta testing and yield matching lower bounds. At constant ε, they also give a separation between Θ(k/logk) forward queries and O(k) queries with inverse access.
Functions are a fundamental object in mathematics, with countless applications to different fields, and are usually classified based on certain properties, given their domains and images. An important property of a real-valued function is its convexity, which plays a very crucial role in many areas, such as thermodynamics and geometry. Motivated by recent advances in quantum computation as well as the quest for quantum advantage, we give a quantum algorithm for testing convexity of polynomial functions, which appears frequently in multiple contexts, such as optimization, machine learning, physics, etc. We show that quantum computers can reveal the convexity property superpolynomially faster than classical computers with respect to number of variables. As a corollary, we provide a significant improvement and extension on quantum Newton's method constructed in earlier work of Rebentrost et al [New J. Phys. \textbf{21} 073023 (2019)]. We further discuss our algorithm in a broader context, such as potential application in the study of geometric structure of manifold, testing training landscape of variational quantum algorithm and also gradient descent/Newton's method for optimization.
Given oracle access to a real-valued function on the n-dimensional Boolean cube, how many queries does it take to estimate the squared Euclidean distance to its closest linear function within ε? Our main result is that O(log³(1/ε) ⋅ 1/ε²) queries suffice. Not only is the query complexity independent of n but it is optimal up to the polylogarithmic factor. Our estimator evaluates f on pairs correlated by noise rates chosen to cancel out the low-degree contributions to f while leaving the linear part intact. The query complexity is optimized when the noise rates are multiples of Chebyshev nodes. In contrast, we show that the dependence on n is unavoidable in two closely related settings. For estimation from random samples, Θ(√n/ε + 1/ε²) samples are necessary and sufficient. For agnostically learning a linear approximation with ε mean-square regret under the uniform distribution, Ω(n/√ε) nonadaptively chosen queries are necessary, while O(n/ε) random samples are known to be sufficient (Linial, Mansour, and Nisan). Our upper bounds apply to functions with bounded 4-norm. Our lower bounds apply even to ± 1-valued functions.
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:ℝⁿ → ℝ and its (sub)gradient. Our goal is to find an ε-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ε)²) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n. In this paper we reprove the randomized lower bound of Ω((GR/ε)²) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ε) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ε)²) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.