We introduce and study the problem of dueling optimization with a monotone adversary, which is a generalization of (noiseless) dueling convex optimization. The goal is to design an online algorithm to find a minimizer $\mathbf{x}^{*}$ for a function $f\colon X \to \mathbb{R}$, where $X \subseteq \mathbb{R}^d$. In each round, the algorithm submits a pair of guesses, i.e., $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$, and the adversary responds with any point in the space that is at least as good as both guesses. The cost of each query is the suboptimality of the worse of the two guesses; i.e., ${\max} \left( f(\mathbf{x}^{(1)}), f(\mathbf{x}^{(2)}) \right) - f(\mathbf{x}^{*})$. The goal is to minimize the number of iterations required to find an $\varepsilon$-optimal point and to minimize the total cost (regret) of the guesses over many rounds. Our main result is an efficient randomized algorithm for several natural choices of the function $f$ and set $X$ that incurs cost $O(d)$ and iteration complexity $O(d\log(1/\varepsilon)^2)$. Moreover, our dependence on $d$ is asymptotically optimal, as we show examples in which any randomized algorithm for this problem must incur $\Omega(d)$ cost and iteration complexity.

翻译：我们引入并研究了具有单调对手的对决优化问题，这是（无噪声）对决凸优化的一般化形式。目标是设计一种在线算法，以找到函数$f\colon X \to \mathbb{R}$的最小化点$\mathbf{x}^{*}$，其中$X \subseteq \mathbb{R}^d$。在每一轮中，算法提交一对猜测，即$\mathbf{x}^{(1)}$和$\mathbf{x}^{(2)}$，而对手则响应空间中任何一个不比这两个猜测中更差的点。每次查询的成本是两个猜测中较差者的次优性，即${\max} \left( f(\mathbf{x}^{(1)}), f(\mathbf{x}^{(2)}) \right) - f(\mathbf{x}^{*})$。目标是最小化找到$\varepsilon$-最优点所需的迭代次数，并最小化多轮猜测的总成本（遗憾）。我们的主要结果是针对函数$f$和集合$X$的几种自然选择提出了一种高效的随机化算法，该算法产生的成本为$O(d)$，迭代复杂度为$O(d\log(1/\varepsilon)^2)$。此外，我们对$d$的依赖性是渐近最优的，因为我们通过实例表明，任何针对该问题的随机化算法都必须承受$\Omega(d)$的成本和迭代复杂度。