Online allocation is a broad class of problems where items arriving online have to be allocated to agents who have a fixed utility/cost for each assigned item so to maximize/minimize some objective. This framework captures a broad range of fundamental problems such as the Santa Claus problem (maximizing minimum utility), Nash welfare maximization (maximizing geometric mean of utilities), makespan minimization (minimizing maximum cost), minimization of $\ell_p$-norms, and so on. We focus on divisible items (i.e., fractional allocations) in this paper. Even for divisible items, these problems are characterized by strong super-constant lower bounds in the classical worst-case online model. In this paper, we study online allocations in the {\em learning-augmented} setting, i.e., where the algorithm has access to some additional (machine-learned) information about the problem instance. We introduce a {\em general} algorithmic framework for learning-augmented online allocation that produces nearly optimal solutions for this broad range of maximization and minimization objectives using only a single learned parameter for every agent. As corollaries of our general framework, we improve prior results of Lattanzi et al. (SODA 2020) and Li and Xian (ICML 2021) for learning-augmented makespan minimization, and obtain the first learning-augmented nearly-optimal algorithms for the other objectives such as Santa Claus, Nash welfare, $\ell_p$-minimization, etc. We also give tight bounds on the resilience of our algorithms to errors in the learned parameters, and study the learnability of these parameters.

翻译：在线分配是一类广泛的问题，其中在线到达的物品需要分配给具有固定效用/成本的智能体，以最大化/最小化某个目标。该框架涵盖了诸多基础性问题，如圣诞老人问题（最大化最小效用）、纳什福利最大化（最大化效用的几何平均值）、完工时间最小化（最小化最大成本）、$\ell_p$-范数最小化等。本文重点关注可分割物品（即分数分配）。即使对于可分割物品，这些问题在经典最坏情况在线模型中也存在强超常数下界。本文研究了学习增强型在线分配问题，即算法可获取关于问题实例的额外（机器学习）信息。我们提出了一种**通用**的学习增强型在线分配算法框架，该框架仅需为每个智能体学习单一参数，即可针对上述广泛的极大化与极小化目标生成接近最优的解。作为该通用框架的推论，我们改进了Lattanzi等人（SODA 2020）以及Li和Xian（ICML 2021）关于学习增强型完工时间最小化的先前结果，并为圣诞老人问题、纳什福利问题、$\ell_p$-范数最小化等其他目标首次提出了学习增强型近似最优算法。我们还给出了算法对于学习参数误差的紧致鲁棒性边界，并研究了这些参数的可学习性。