In this paper, we study how a budget-constrained bidder should learn to bid adaptively in repeated first-price auctions to maximize cumulative payoff. This problem arises from the recent industry-wide shift from second-price auctions to first-price auctions in display advertising, which renders truthful bidding suboptimal. We propose a simple dual-gradient-descent-based bidding policy that maintains a dual variable for the budget constraint as the bidder consumes the budget. We analyze two settings based on the bidder's knowledge of future private values: (i) an uninformative setting where all distributional knowledge (potentially non-stationary) is entirely unknown, and (ii) an informative setting where a prediction of budget allocation is available in advance. We characterize the performance loss (regret) relative to an optimal policy with complete information. For uninformative setting, we show that the regret is ~O(sqrt(T)) plus a Wasserstein-based variation term capturing non-stationarity, which is order-optimal. In the informative setting, the variation term can be eliminated using predictions, yielding a regret of ~O(sqrt(T)) plus the prediction error. Furthermore, we go beyond the global budget constraint by introducing a refined benchmark based on a per-period budget allocation plan, achieving exactly ~O(sqrt(T)) regret. We also establish robustness guarantees when the baseline policy deviates from the planned allocation, covering both ideal and adversarial deviations.

翻译：本文研究预算受限的投标方如何在重复一价拍卖中学习自适应竞价以最大化累积收益。该问题源于展示广告行业近期从二价拍卖向一价拍卖的集体转型，使得真实报价策略不再最优。我们提出一种基于对偶梯度下降的简洁竞价策略，通过维护预算约束的对偶变量来跟踪预算消耗过程。根据投标方对未来私有价值认知程度的不同，我们分析两种场景：（一）无信息场景——所有分布知识（可能包含非平稳性）完全未知；（二）有信息场景——可预先获得预算分配的预测。我们刻画了相对于具备完全信息的最优策略的性能损失（遗憾值）。对于无信息场景，我们证明遗憾值约为~O(√T)加上表征非平稳性的Wasserstein变分项，该界具有阶数最优性。在有信息场景中，利用预测可消除变分项，得到遗憾值约为~O(√T)加预测误差。此外，我们突破全局预算约束的限制，引入基于周期预算分配计划的精细基准，实现了严格~O(√T)的遗憾值。当基线策略偏离计划分配时，我们同时建立了针对理想偏离和对抗性偏离的鲁棒性保证。