Min-max optimization arises in many domains such as game theory, adversarial machine learning, etc. For these problems, gradient-based methods are well understood and enjoy strong guarantees. However, in the absence of convexity or concavity, existing approaches study convergence to an approximate saddle point or first-order stationary points, which may be arbitrarily far from global optima. In this work, we present an algorithmic framework for computing the global minimax value in convex--non-concave and non-convex--concave min-max optimization. For convex--non-concave min-max problems, we use a reformulation that transforms the problem into a non-concave--convex max-min optimization problem with suitably defined feasible sets and objective function. This reformulation can be viewed as an extension of Sion's minimax theorem to the convex--non-concave setting. We then introduce EXOTIC -- an Exact, Optimistic, Tree-based algorithm for solving the reformulated max-min problem. EXOTIC combines an iterative convex optimization solver for the inner minimization with an optimistic hierarchical tree search for the outer maximization, inspired by StroquOOL~\cite{bartlett2019simple}. Unlike StroquOOL, which assumes stochastic zero-mean noisy evaluations, EXOTIC handles deterministic, biased, and budget-dependent evaluation errors arising from finite-time solutions of the inner convex subproblems. We establish an upper bound on its optimality gap. The same framework also applies to non-convex--concave min-max optimization. Empirically, EXOTIC outperforms gradient-based methods on popular benchmarks from the literature. Finally, we demonstrate the utility of EXOTIC by computing security strategies in multi-player games with three or more players -- a computationally challenging task that, to our knowledge, no prior method solves exactly.

翻译：极小极大优化出现在博弈论、对抗性机器学习等多个领域。对于这些问题，基于梯度的方法已被充分理解并享有强理论保证。然而，在缺乏凸性或凹性的情况下，现有方法研究收敛到近似鞍点或一阶稳定点，但这些点可能任意远离全局最优值。本文提出了一种计算凸-非凹和非凸-凹极小极大优化中全局极小极大值的算法框架。对于凸-非凹极小极大问题，我们通过一种重构方法，将问题转化为定义在适当可行集和目标函数上的非凹-凸极大极小优化问题。该重构可视为Sion极小极大定理在凸-非凹情形下的推广。接着，我们引入EXOTIC——一种用于求解重构后极大极小问题的精确、乐观、基于树的算法。EXOTIC将内层最小化的迭代凸优化求解器与外层最大化的乐观分层树搜索相结合，后者受StroQOOL~\cite{bartlett2019simple}启发。与假设随机零均值噪声评估的StroQOOL不同，EXOTIC处理由内层凸子问题有限时间求解产生的确定性、有偏且预算依赖的评估误差。我们建立了其最优性间隙的上界。同一框架也适用于非凸-凹极小极大优化。实验上，EXOTIC在文献中的常用基准测试上优于基于梯度的方法。最后，我们通过计算包含三个或更多玩家的多人博弈中的安全策略，展示了EXOTIC的实用性——这是一个计算上极具挑战性的任务，据我们所知，此前尚无方法能精确求解。