Water-Filling is Universally Minimax Optimal

2026-03-27

AI Summary

This work addresses the dynamic allocation of divisible online resources among offline agents, a problem prevalent in online markets, scheduling, and portfolio optimization. It establishes, for the first time, that the water-filling algorithm achieves global minimax optimality—simultaneously in terms of α-regret and competitive ratio—for a broad class of objective functions, including Schur-concave maximization and Schur-convex minimization problems. Remarkably, this optimality holds universally, independent of the specific form of the objective function, the number of agents, or the total resource amount. The key theoretical advance lies in leveraging majorization theory to develop a novel analytical framework for online algorithms, transcending the limitations of traditional primal-dual methods and firmly positioning water-filling as a universal minimax-optimal strategy.

Abstract

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.

Problem

• online resource allocation
• water-filling
• minimax optimality
• Schur-concave
• competitive ratio

Innovation

• water-filling
• minimax optimality
• majorization theory
• online resource allocation
• Schur-concave/Schur-convex objectives