面向在线线性规划的低频求解算法
Infrequent Resolving Algorithm for Online Linear Programming
AI总结：
该研究针对在线线性规划两类现有算法的性能与计算成本权衡问题，提出低频求解LP的混合算法，实现了不同求解次数约束下的悔值保证，并通过实验验证了有效性。
摘要：
在线线性规划（Online Linear Programming，OLP）因在在线拍卖、网络收益管理、订单履约和广告投放等领域的广泛应用，受到了研究人员与从业者的高度关注。现有的OLP算法可分为两类：基于LP的算法和无LP的算法。前者通常能保证更优的性能，但需要求解大量线性规划问题，计算成本极高。相比之下，无LP算法仅需进行一阶计算，但性能表现较差。\n在本研究中，我们通过提出一种性能优异的算法来弥合这两类极端方法之间的差距：该算法仅在少数选定的时间点求解线性规划，在其余时间点执行一阶计算。具体而言，针对输入服从未知有限支撑分布的场景，所提算法能实现常数悔值（即使在难度较高的“退化”情形下也成立），而在整个时间跨度T内仅需求解O(log log T)次线性规划。此外，当我们仅被允许求解M次线性规划时，我们设计了对应的调度方案，使所提算法能保证接近O(T^((1/2)^(M-1)))的悔值。\n本研究凸显了在销售周期的起始和结束阶段进行重新求解的价值，并提出了一个新颖的框架，用于证明所提策略在不同低频求解调度方案下的性能保证。我们还开展了数值实验，验证了所提算法的有效性。
英文摘要：
Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications such as online auctions, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithms only require first-order computations but induce a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only $O(\log\log T)$ times over the time horizon $T$. Moreover, when we are allowed to solve LPs only $M$ times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly $O\left(T^{(1/2)^{M-1}}\right)$ regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms. The code is available at https://github.com/guokai-li/infrequent_resolving_olp.
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