First-price auctions have very recently swept the online advertising industry, replacing second-price auctions as the predominant auction mechanism on many platforms. This shift has brought forth important challenges for a bidder: how should one bid in a first-price auction, where unlike in second-price auctions, it is no longer optimal to bid one's private value truthfully and hard to know the others' bidding behaviors? In this paper, we take an online learning angle and address the fundamental problem of learning to bid in repeated first-price auctions, where both the bidder's private valuations and other bidders' bids can be arbitrary. We develop the first minimax optimal online bidding algorithm that achieves an $\widetilde{O}(\sqrt{T})$ regret when competing with the set of all Lipschitz bidding policies, a strong oracle that contains a rich set of bidding strategies. This novel algorithm is built on the insight that the presence of a good expert can be leveraged to improve performance, as well as an original hierarchical expert-chaining structure, both of which could be of independent interest in online learning. Further, by exploiting the product structure that exists in the problem, we modify this algorithm--in its vanilla form statistically optimal but computationally infeasible--to a computationally efficient and space efficient algorithm that also retains the same $\widetilde{O}(\sqrt{T})$ minimax optimal regret guarantee. Additionally, through an impossibility result, we highlight that one is unlikely to compete this favorably with a stronger oracle (than the considered Lipschitz bidding policies). Finally, we test our algorithm on three real-world first-price auction datasets obtained from Verizon Media and demonstrate our algorithm's superior performance compared to several existing bidding algorithms.

中文翻译：

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学习在对抗性首价拍卖中以最佳方式和有效地出价

第一价格拍卖最近席卷了网络广告行业，取代第二价格拍卖成为许多平台上的主要拍卖机制。这种转变给竞拍者带来了重大挑战：一等价拍卖应该如何出价，而与二价拍卖不同的是，真实地竞拍自己的私人价值不再是最优的，而且很难知道其他人的竞拍行为? 在本文中，我们从在线学习的角度解决了学习在重复的第一价格拍卖中出价的基本问题，其中出价者的私人估价和其他出价者的出价都可以是任意的。我们开发了第一个 minimax 最优在线竞价算法，该算法在与所有 Lipschitz 竞价策略集竞争时实现了 $\widetilde{O}(\sqrt{T})$ 后悔，一个强大的预言机，包含一组丰富的投标策略。这种新颖的算法建立在这样一个见解之上，即可以利用优秀专家的存在来提高性能，以及原始的分层专家链结构，这两者都可能对在线学习产生独立的兴趣。此外，通过利用问题中存在的乘积结构，我们将这个算法——以其统计上最优但在计算上不可行的普通形式——修改为一个计算高效且空间高效的算法，该算法也保留了相同的 $\widetilde{O}( \sqrt{T})$ minimax 最优后悔保证。此外，通过一个不可能的结果，我们强调一个人不太可能与更强大的预言机（比考虑的 Lipschitz 投标政策）进行有利的竞争。最后，