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1. Smooth Convex Optimization Using Sub-Zeroth-Order Oracles

   Karabag, Mustafa ; Neary, Cyrus ; and Topcu, Ufuk ( January 2021 , Proceedings of the AAAI Conference on Artificial Intelligence)

   null (Ed.)

   We consider the problem of minimizing a smooth, Lipschitz, convex function over a compact, convex set using sub-zerothorder oracles: an oracle that outputs the sign of the directional derivative for a given point and a given direction, an oracle that compares the function values for a given pair of points, and an oracle that outputs a noisy function value for a given point. We show that the sample complexity of optimization using these oracles is polynomial in the relevant parameters. The optimization algorithm that we provide for the comparator oracle is the first algorithm with a known rate of convergence that is polynomial in the number of dimensions. We also give an algorithm for the noisy-value oracle that incurs sublinear regret in the number of queries and polynomial regret in the number of dimensions. 

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2. Procrastinated Tree Search: Black-Box Optimization with Delayed, Noisy, and Multi-Fidelity Feedback

   https://doi.org/10.1609/aaai.v36i9.21280  

   Wang, Junxiong ; Basu, Debabrota ; Trummer, Immanuel ( June 2022 , Proceedings of the AAAI Conference on Artificial Intelligence)

   In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), called ProCrastinated Tree Search (PCTS), that flexibly accommodates a delay and noise-tolerant bandit algorithm. We provide a generic proof technique to quantify regret of PCTS under delayed, noisy, and multi-fidelity feedbacks. Specifically, we derive regret bounds of PCTS enabled with delayed-UCB1 (DUCB1) and delayed-UCB-V (DUCBV) algorithms. Given a horizon T, PCTS retains the regret bound of non-delayed HOO for expected delay of O(log T), and worsens by T^((1-α)/(d+2)) for expected delays of O(T^(1-α)) for α ∈ (0,1]. We experimentally validate on multiple synthetic functions and hyperparameter tuning problems that PCTS outperforms the state-of-the-art black-box optimization methods for feedbacks with different noise levels, delays, and fidelity. 

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3. Active Learning a Convex Body in Low Dimensions

   Har-Peled, Sariel ; Jones, Mitchell ; Saladi, Rahul ( July 2020 , Leibniz international proceedings in informatics)

   Czumaj, Artur ; Dawar, Anuj ; Merelli, Emanuela (Ed.)

   Consider a set P ⊆ ℝ^d of n points, and a convex body C provided via a separation oracle. The task at hand is to decide for each point of P if it is in C using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using O(⬡_P log n) queries, where ⬡_P is the largest subset of points of P in convex position. In 2D, we provide an algorithm which efficiently generates these adaptive queries. Furthermore, we show that in two dimensions one can solve this problem using O(⊚(P,C) log² n) oracle queries, where ⊚(P,C) is a lower bound on the minimum number of queries that any algorithm for this specific instance requires. Finally, we consider other variations on the problem, such as using the fewest number of queries to decide if C contains all points of P. As an application of the above, we show that the discrete geometric median of a point set P in ℝ² can be computed in O(n log² n (log n log log n + ⬡(P))) expected time. 

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4. An Optimal Learning Algorithm for Online Unconstrained Submodular Maximization

   Roughgarden, Tim ; Wang, Joshua R. ( July 2018 , 31st Annual Conference on Learning Theory (COLT))

   This paper considers a basic problem at the interface of submodular optimization and online learning. In the online unconstrained submodular maximization problem, there is a universe [n] = {1, 2, . . . , n} and a sequence of T nonnegative (not necessarily monotone) submodular functions arrive over time. The goal is to design a computationally efficient online algorithm, which chooses a subset of [n] at each time step as a function only of the past, such that the accumulated value of the chosen subsets is as close as possible to the maximum total value of a fixed subset in hindsight. Our main result is a polynomial time no-1/2-regret algorithm for this problem, meaning that for every sequence of nonnegative submodular functions, the algorithm’s expected total value is at least 1/2 times that of the best subset in hindsight, up to an error term sublinear in T. The factor of 1/2 cannot be improved upon by any polynomial-time online algorithm when the submodular functions are presented as value oracles. 

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5. A unified approach for maximizing continuous DR-submodular functions

   Mohammad Pedramfar, Christopher John ( December 2023 , Advances in Neural Information Processing Systems)

   This paper presents a unified approach for maximizing continuous DRsubmodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in three cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining four cases. Notably, our approach for the stochastic function valuebased oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions. 

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