Infrequent Resolving Algorithm for Online Linear Programming

Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications, such as online auction, network revenue management and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former one typically guarantees better performance, even offering a constant regret, but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithm only requires first-order computations but induces a worse performance, lacking a constant regret bound. In this work, we study the case where the inputs are drawn from an unknown finite-support distribution, and bridge the gap between these two extremes by proposing an algorithm that achieves a constant regret 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 propose an algorithm that can guarantee an $O\left(T^{(1/2+\epsilon)^{M-1}}\right)$ regret. Furthermore, when the arrival probabilities are known at the beginning, our algorithm can guarantee a constant regret by solving LPs $O(\log\log T)$ times, and an $O\left(T^{(1/2+\epsilon)^{M}}\right)$ regret by solving LPs only $M$ times. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms.