Allocation of dynamically-arriving (i.e., online) divisible resources among a set of offline agents is a fundamental problem, with applications to online marketplaces, scheduling, portfolio selection, signal processing, and many other areas. The water-filling algorithm, which allocates an incoming resource to maximize the minimum load of compatible agents, is ubiquitous in many of these applications whenever the underlying objectives prefer more balanced solutions; however, the analysis and guarantees differ across settings. We provide a justification for the widespread use of water-filling by showing that it is a universally minimax optimal policy in a strong sense. Formally, our main result implies that water-filling is minimax optimal for a large class of objectives -- including both Schur-concave maximization and Schur-convex minimization -- under $α$-regret and competitive ratio measures. This optimality holds for every fixed tuple of agents and resource counts. Remarkably, water-filling achieves these guarantees as a myopic policy, remaining entirely agnostic to the objective function, agent count, and resource availability. Our techniques notably depart from the popular primal-dual analysis of online algorithms, and instead develop a novel way to apply the theory of majorization in online settings to achieve universality guarantees.

翻译：动态到达（即在线）的可分割资源在一组离线智能体之间的分配是一个基础问题，广泛应用于在线市场、调度、投资组合选择、信号处理等多个领域。注水算法将流入资源分配给兼容智能体以最大化其最小负荷，当底层目标偏好更均衡的解时，该算法在众多应用中无处不在；然而，不同场景下的分析和保证各不相同。我们为注水算法的广泛使用提供了理论依据，表明它在强意义上是一种普遍极小极大最优策略。形式上，我们的主要结果表明，对于一大类目标函数——包括舒尔凹最大化与舒尔凸最小化——在$α$-遗憾和竞争比度量下，注水算法是极小极大最优的。这一最优性对每个固定的智能体元组和资源数量均成立。值得注意的是，注水算法作为短视策略实现了这些保证，完全无需了解目标函数、智能体数量和资源可用性。我们的方法显著偏离了流行的在线算法原始-对偶分析，而是开创性地将优超理论应用于在线场景，从而获得普适性保证。