Sample Complexity of Multi-Item Profit Maximization

We study the design of pricing mechanisms and auctions when the mechanism designer does not know the distribution of buyers' values. Instead the mechanism designer receives a set of samples from this distribution and his goal is to use the sample to design a pricing mechanism or auction with high expected profit. We provide generalization guarantees which bound the difference between average profit on the sample and expected profit over the distribution. These bounds are directly proportional to the intrinsic complexity of the mechanism class the designer is optimizing over. We present a single, overarching theorem that uses empirical Rademacher complexity to measure the intrinsic complexity of a variety of widely-studied single- and multi-item auction classes, including affine maximizer auctions, mixed-bundling auctions, and second-price item auctions. Despite the extensive applicability of our main theorem, we match and improve over the best-known generalization guarantees for many auction classes. This all-encompassing theorem also applies to multi- and single-item pricing mechanisms in both multi- and single-unit settings, such as linear and non-linear pricing mechanisms. Finally, our central theorem allows us to easily derive generalization guarantees for every class in several finely grained hierarchies of auction and pricing mechanism classes. We demonstrate how to determine the precise level in a hierarchy with the optimal tradeoff between profit and generalization using structural profit maximization. The mechanism classes we study are significantly different from well-understood function classes typically found in machine learning, so bounding their complexity requires a sharp understanding of the interplay between mechanism parameters and buyer valuations.