We consider a fair resource allocation problem in the no-regret setting against an unrestricted adversary. The objective is to allocate resources equitably among several agents in an online fashion so that the difference of the aggregate $\alpha$-fair utilities of the agents between an optimal static clairvoyant allocation and that of the online policy grows sub-linearly with time. The problem is challenging due to the non-additive nature of the $\alpha$-fairness function. Previously, it was shown that no online policy can exist for this problem with a sublinear standard regret. In this paper, we propose an efficient online resource allocation policy, called Online Proportional Fair (OPF), that achieves $c_\alpha$-approximate sublinear regret with the approximation factor $c_\alpha=(1-\alpha)^{-(1-\alpha)}\leq 1.445,$ for $0\leq \alpha < 1$. The upper bound to the $c_\alpha$-regret for this problem exhibits a surprising phase transition phenomenon. The regret bound changes from a power-law to a constant at the critical exponent $\alpha=\frac{1}{2}.$ As a corollary, our result also resolves an open problem raised by Even-Dar et al. [2009] on designing an efficient no-regret policy for the online job scheduling problem in certain parameter regimes. The proof of our results introduces new algorithmic and analytical techniques, including greedy estimation of the future gradients for non-additive global reward functions and bootstrapping adaptive regret bounds, which may be of independent interest.

翻译：我们考虑在无遗憾设定下应对无限制对手的公平资源分配问题。目标是以在线方式在多个智能体之间公平分配资源，使得最优静态先见分配与在线策略在各智能体聚合$\alpha$-公平效用之差随时间呈次线性增长。由于$\alpha$-公平函数的非加性特性，该问题具有挑战性。此前研究表明，对于此问题不存在具有次线性标准遗憾的在线策略。本文提出一种高效的在线资源分配策略——在线比例公平（OPF），该策略实现了$c_\alpha$近似次线性遗憾，近似因子$c_\alpha=(1-\alpha)^{-(1-\alpha)}\leq 1.445$，适用于$0\leq \alpha < 1$。此问题$c_\alpha$-遗憾的上界呈现出令人惊讶的相变现象：遗憾边界在临界指数$\alpha=\frac{1}{2}$处从幂律形式转变为常数形式。作为推论，我们的结果还解决了Even-Dar等人[2009]提出的关于在特定参数区间内设计在线作业调度问题高效无遗憾策略的开放问题。本文证明引入了新的算法与分析技术，包括对非加性全局奖励函数进行未来梯度的贪心估计，以及引导自适应遗憾边界，这些技术可能具有独立的研究价值。