Penalized and Constrained Optimization: An Application to High-Dimensional Website Advertising

Firms are increasingly transitioning advertising budgets to Internet display campaigns, but this transition poses new challenges. These campaigns use numerous potential metrics for success (e.g., reach or click rate), and because each website represents a separate advertising opportunity, this is also an inherently high-dimensional problem. Further, advertisers often have constraints they wish to place on their campaign, such as targeting specific sub-populations or websites. These challenges require a method flexible enough to accommodate thousands of websites, as well as numerous metrics and campaign constraints. Motivated by this application, we consider the general constrained high-dimensional problem, where the parameters satisfy linear constraints. We develop the Penalized and Constrained optimization method (PaC) to compute the solution path for high-dimensional, linearly constrained criteria. PaC is extremely general; in addition to internet advertising, we show it encompasses many other potential applications, such as portfolio estimation, monotone curve estimation, and the generalized lasso. Computing the PaC coefficient path poses technical challenges, but we develop an efficient algorithm over a grid of tuning parameters. Through extensive simulations, we show PaC performs well. Finally, we apply PaC to a proprietary dataset in an exemplar Internet advertising case study and demonstrate its superiority over existing methods in this practical setting. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

企业正日益将广告预算转向互联网展示广告推广活动，但这一转型也带来了全新的挑战。此类推广活动采用多种衡量成功的指标（例如触达率或点击率），且由于每个网站均代表独立的广告投放机会，该问题本质上属于高维问题。此外，广告主通常会对推广活动设置各类约束条件，例如定向特定细分人群或特定网站。上述挑战要求所采用的方法具备足够灵活性，能够适配数千个网站，同时兼容多种指标与推广活动约束。基于这一应用场景，我们聚焦于参数满足线性约束的通用约束高维问题。我们提出了惩罚约束优化方法（Penalized and Constrained Optimization, PaC），用于求解高维线性约束准则下的解路径。PaC具备极强的通用性；除互联网广告场景外，我们证明其可覆盖诸多其他潜在应用，例如投资组合估计、单调曲线估计以及广义Lasso（generalized lasso）。求解PaC系数路径存在技术难点，但我们基于调优参数网格设计了高效算法。通过大量模拟实验，我们证明PaC表现优异。最后，我们将PaC应用于一项专有数据集开展典型互联网广告案例研究，并证实其在该实际场景中的性能优于现有方法。本文的补充材料（包含可复现研究的标准化材料说明）可通过在线补充资源获取。