Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, who wish to trade a good. We study this problem from the perspective of a profit-maximizing broker within an online learning framework, where the agents' valuations are generated by a smooth adversary. We devise a learning algorithm that guarantees a $\tilde{O}(\sqrt{T})$ regret bound, which is tight in the time horizon $T$ up to poly-logarithmic factors. This matches the minimax rate for the stochastic i.i.d. case, and is also well separated from the adversarial setting, where sublinear-regret is unattainable. By extending the strong regret guarantees from the i.i.d. case to the smooth adversary, we significantly broaden the scope of settings where such fast rate is achievable, while closing an important gap in the regret landscape of this fundamental economic problem. To overcome the challenges posed by this adversary, we leverage a continuity property of smooth instances and combines this with a hierarchical net-construction of the broker's action space, which is analyzed via algorithmic chaining. We showcase the applicability of these techniques by deriving a similarly tight $\tilde{O}(\sqrt{T})$ regret bound for a related mechanism design model: the joint ads problem.

翻译：双边贸易模型描述了中介在两个策略性主体——卖方与买方——之间进行贸易撮合的任务。我们从在线学习框架下追求利润最大化的经纪人视角研究该问题，其中主体估值由平滑对手生成。我们设计了一种学习算法，保证达到 $\tilde{O}(\sqrt{T})$ 的遗憾界，这在时间范围 $T$ 上达到紧致（仅相差多对数因子）。该结果匹配随机独立同分布情形的极小化最优率，并与对抗性设置（其中次线性遗憾不可实现）显著区分。通过将强遗憾保证从独立同分布情形扩展至平滑对手，我们显著拓宽了可实现该快速率的场景范围，同时填补了这一基本经济问题遗憾格局中的重要空白。为应对该对手带来的挑战，我们利用平滑实例的连续性性质，并将其与经纪人行动空间的分层网络构造相结合，通过算法链式分析进行论证。我们通过推导相关机制设计模型（联合广告问题）中同样紧致的 $\tilde{O}(\sqrt{T})$ 遗憾界，展示了这些技术的适用性。