Optimal No-regret Learning in Repeated First-price Auctions

We study online learning in repeated first-price auctions where a bidder, only observing the winning bid at the end of each auction, learns to adaptively bid in order to maximize her cumulative payoff. To achieve this goal, the bidder faces a censored feedback: if she wins the bid, then she is not able to observe the highest bid of the other bidders, which we assume is \textit{iid} drawn from an unknown distribution. In this paper, we develop the first learning algorithm that achieves a near-optimal $\widetilde{O}(\sqrt{T})$ regret bound, by exploiting two structural properties of first-price auctions, i.e. the specific feedback structure and payoff function. The feedback in first-price auctions combines the graph feedback across actions (bids), the cross learning across contexts (private values), and a partial order over the contexts; we generalize it as the partially ordered contextual bandits. We establish both strengths and weaknesses of this framework, by showing a curious separation that a regret nearly independent of the action/context sizes is possible under stochastic contexts, but is impossible under adversarial contexts. In particular, this framework leads to an $O(\sqrt{T}\log^{2.5}T)$ regret for first-price auctions when the bidder's private values are \emph{iid}. Despite the limitation of the above framework, we further exploit the special payoff function of first-price auctions to develop a sample-efficient algorithm even in the presence of adversarially generated private values. We establish an $O(\sqrt{T}\log^3 T)$ regret bound for this algorithm, hence providing a complete characterization of optimal learning guarantees for first-price auctions.