  We consider the sample complexity of revenue maximization for multiple
bidders in unrestricted multi-dimensional 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 $\varepsilon>0$, we show
that it is possible to learn an $\varepsilon$-Bayesian Incentive Compatible
auction whose expected revenue is within $\varepsilon$ of the optimal
$\varepsilon$-BIC auction from only polynomially many samples.
  Our fully nonparametric 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 are not
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.

Computer Science and Game Theory;Machine Learning