Budget pacing is a popular service that has been offered by major internet advertising platforms since their inception. Budget pacing systems seek to optimize advertiser returns subject to budget constraints by smoothly spending advertiser budgets. In the past few years, autobidding products that provide real-time bidding as a service to advertisers have seen a prominent rise in adoption. A popular autobidding strategy is value maximization subject to return-on-spend (ROS) constraints. For historical/business reasons, the systems that govern these two services, namely budget pacing and ROS pacing, are not always a unified and coordinated entity that optimizes a global objective subject to both constraints. The purpose of this work is to theoretically and empirically compare algorithms with different degrees of coordination between these two pacing systems. In particular, we compare (a) a fully-decoupled sequential algorithm that first constructs the advertiser's ROS-pacing bid and then lowers that bid for budget pacing; (b) a minimally-coupled min-pacing algorithm that runs these two services independently, obtains the bid multipliers from both of them and applies the minimum of the two multipliers as the effective multiplier; and (c) a fully-coupled dual-based algorithm that optimally combines the dual variables from both the systems. Our main contribution is to theoretically analyze the min-pacing algorithm and show that it attains similar guarantees to the fully-coupled canonical dual-based algorithm. On the other hand, we show that the sequential algorithm, even though appealing by virtue of being fully decoupled, could badly violate the constraints. We validate our theoretical findings empirically by showing that the min-pacing algorithm performs almost as well as the canonical dual-based algorithm on a semi-synthetic dataset based on a large online advertising platform's data.

翻译：预算节奏是一项自大型互联网广告平台问世以来便提供的流行服务。预算节奏系统旨在通过平滑支出广告主预算，在满足预算约束的前提下优化广告主回报。过去几年中，为广告主提供实时竞价作为服务的自动竞价产品采用率显著上升。一种流行的自动竞价策略是在满足投资回报率（ROS）约束下实现价值最大化。由于历史或业务原因，管理这两项服务的系统（即预算节奏与ROS节奏）并非始终是一个统一协调的实体，能够同时优化满足两种约束的全局目标。本文旨在从理论和实证角度比较不同协调程度的算法。具体而言，我们比较了：(a) 完全解耦的序贯算法，该算法首先构建广告主的ROS节奏出价，随后为该出价降低预算节奏；(b) 最小耦合的最小值节奏算法，该算法独立运行这两项服务，从两者获取出价乘数，并将两者中的最小值作为有效乘数；(c) 完全耦合的基于对偶的算法，该算法最优地结合了两个系统的对偶变量。我们的主要贡献在于从理论上分析了最小值节奏算法，并证明其能获得与完全耦合的标准基于对偶的算法相似的保障。另一方面，我们证明尽管序贯算法因完全解耦而具有吸引力，但其可能严重违反约束。我们通过实证验证了理论发现，表明基于大型在线广告平台数据的半合成数据集上，最小值节奏算法的表现几乎与标准基于对偶的算法相当。