Budget-management systems are one of the key components of modern auction markets. Internet advertising platforms typically offer advertisers the possibility to pace the rate at which their budget is depleted, through budget-pacing mechanisms. We focus on multiplicative pacing mechanisms in an online setting in which a bidder is repeatedly confronted with a series of advertising opportunities. After collecting bids, each item is then allocated through a single-item, second-price auction. If there were no budgetary constraints, bidding truthfully would be an optimal choice for the advertiser. However, since their budget is limited, the advertiser may want to shade their bid downwards in order to preserve their budget for future opportunities, and to spread expenditures evenly over time. The literature on online pacing problems mostly focuses on the setting in which the bidder optimizes an additive separable objective, such as the total click-through rate or the revenue of the allocation. In many settings, however, bidders may also care about other objectives which oftentimes are non-separable, and therefore not amenable to traditional online learning techniques. Building on recent work, we study the frequent case in which advertisers seek to reach a certain distribution of impressions over a target population of users. We introduce a novel regularizer to achieve this desideratum, and show how to integrate it into an online mirror descent scheme attaining the optimal order of sub-linear regret compared to the optimal allocation in hindsight when inputs are drawn independently, from an unknown distribution. Moreover, we show that our approach can easily be incorporated in standard existing pacing systems that are not usually built for this objective. The effectiveness of our algorithm in internet advertising applications is confirmed by numerical experiments on real-world data.
The Complexity of Pacing for Second-Price Auctions
预算约束在在线广告拍卖中是普遍存在的。为了管理这些限制并平滑拍卖中的支出，投标人（或代表他们的平台）通常会使用起搏：每个投标人都会被分配一个介于0和1之间的起搏乘数，她对每个项目的出价都会按起搏乘数成倍地缩小。这自然会产生一个博弈，在这个博弈中，每个投标人都有策略地选择一个乘数。在这个博弈中，均衡的适当概念被称为节奏均衡。在这项工作中，我们表明，找到一个近似的节奏均衡的问题是PPAD完全的第二价格拍卖。这解决了Conitzer et al. [2021]的一个未决问题。作为我们的硬度结果的结果，我们表明，由Borgs等人[2007]介绍的tatonnement风格的拍卖管理动态不太可能有效地收敛重复的第二价格拍卖。这推翻了Borgs等人的猜想。[2007]，假设复杂性类PPAD不等于P。我们的硬度结果也意味着存在一个细化的供应感知市场均衡，这是很难用简单的线性效用计算。
