The sample complexity of up-to-ϵ multi-dimensional revenue maximization

We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multidimensional settings. Specifically, we study the standard model of n additive bidders whose values for m heterogeneous items are drawn independently. For any such instance and any ϵ > 0, we show that it is possible to learn an ϵ-Bayesian Incentive Compatible auction whose expected revenue is within ϵ of the optimal ϵ-BIC auction from only polynomially many samples. Our approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that aren't necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.