The Pseudo-Dimension of Near-Optimal Auctions

This paper develops a general approach, rooted in statistical learning theory, to learning an approximately revenue-maximizing auction from data. We introduce $t$-level auctions to interpolate between simple auctions, such as welfare maximization with reserve prices, and optimal auctions, thereby balancing the competing demands of expressivity and simplicity. We prove that such auctions have small representation error, in the sense that for every product distribution $F$ over bidders valuations, there exists a $t$-level auction with small $t$ and expected revenue close to optimal. We show that the set of $t$-level auctions has modest pseudo-dimension (for polynomial $t$) and therefore leads to small learning error. One consequence of our results is that, in arbitrary single-parameter settings, one can learn a mechanism with expected revenue arbitrarily close to optimal from a polynomial number of samples.

Universally Utility-Maximizing Privacy Mechanisms

A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Privacy can be rigorously quantified using the framework of {em differential privacy}, which requires that a mechanisms output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a {em geometric mechanism} $M^*$ -- a discrete variant of the simple and well-studied Laplace mechanism -- that is {em simultaneously expected loss-minimizing} for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: {em every} potential user $u$, no matter what its side information and preferences, derives as much utility from $M^*$ as from interacting with a differentially private mechanism $M_u$ that is optimally tailored to $u$.