A Primal-Dual Online Algorithm for Online Matching Problem in Dynamic Environments

Abstract

Recently, the online matching problem has attracted much attention due to its wide application on real-world decision-making scenarios. In stationary environments, by adopting the stochastic user arrival model, existing methods are proposed to learn dual optimal prices and are shown to achieve a fast regret bound. However, the stochastic model is no longer a proper assumption when the environment is changing, leading to an optimistic method that may suffer poor performance. In this paper, we study the online matching problem in dynamic environments in which the dual optimal prices are allowed to vary over time. We bound the dynamic regret of online matching problem by the sum of two quantities, including a regret of online max-min problem and a dynamic regret of online convex optimization (OCO) problem. Then we propose a novel online approach named Primal-Dual Online Algorithm (PDOA) to minimize both quantities. In particular, PDOA adopts the primal-dual framework by optimizing dual prices with the online gradient descent (OGD) algorithm to eliminate the online max-min problem's regret. Moreover, it maintains a set of OGD experts and combines them via an expert-tracking algorithm, which gives a sublinear dynamic regret bound for the OCO problem. We show that PDOA achieves an O(K sqrt{T(1+P_T)}) dynamic regret where K is the number of resources, T is the number of iterations and P_T is the path-length of any potential dual price sequence that reflects the dynamic environment. Finally, experiments on real applications exhibit the superiority of our approach.

Percentile Risk-Constrained Budget Pacing for Guaranteed Display Advertising in Online Optimization

“…However, the representative term is fixed without consideration of dynamic adjustment. Another research direction is to achieve budget pacing through feedback control, which can be further categorized into bid modification (Mehta et al 2007;Zhou et al 2021) and probabilistic throttling (Agarwal et al 2014;Xu et al 2015;Lee, Jalali, and Dasdan 2013). Bid modification influences the budget spending of an ad by adjusting its bidding price.…”

“…Mehta et al (Mehta et al 2007) modify the bid by multiplying it with a value that reflects the proportion of unused budget, and the impression would be allocated to the ad with the highest modified bid. Balseiro et al (Balseiro, Lu, and Mirrokni 2020) and Zhou et al (Zhou et al 2021) utilize a dual multiplier to form a virtual bid, which is consistently updated based on the variance between the actual and expected budget consumption. These methods adhere to the same principle of decreasing an ad's bid when the budget is being spent too rapidly.…”