Ironing in the Dark

Abstract

This paper presents the first polynomial-time algorithm for position and matroid auction environments that learns, from samples from an unknown bounded valuation distribution, an auction with expected revenue arbitrarily close to the maximum possible. In contrast to most previous work, our results apply to arbitrary (not necessarily regular) distributions and the strongest possible benchmark, the Myerson-optimal auction. Learning a near-optimal auction for an irregular distribution is technically challenging because it requires learning the appropriate "ironed intervals," a delicate global property of the distribution.

Settling the sample complexity of single-parameter revenue maximization

“…Indeed, for the special case of independently and identically distributed (i.i.d.) bidders with [0, 1]-bounded distributions and additive approximation, Roughgarden and Schrijvers [23] showed a sample complexity upper bound ofÕ(n 2 −2 ), which is the only previous example, to our knowledge, with a sub-cubic dependence in −1 .…”

“…See Section 3 for the formal definitions of the dominated empirical distribution and the algorithm. Next we explain the main difference between our algorithm and those in previous works, with the exception of Roughgarden and Schrijvers [23]. Previous works generally pick the optimal auction w.r.t.…”

“…Learning the Virtual Values. An alternative approach (e.g., [10,23]) is to learn the individual value distributions well enough to obtain enough approximately accurate information about the virtual values, which induces a mechanism. Then, we analyze the revenue approximation using the connections between expected revenue and virtual values.…”

Online Auctions and Multi-scale Online Learning

“…We conjecture that our bound for the online posted pricing problem is tight up to log factors, and leave resolving this as an open problem. The third bound is not comparable to the best sample complexity for the multi buyer auction problem by Roughgarden and Schrijvers (2016); it is better than theirs for large ǫ (when 1/ǫ ≤ o(nh)), and is worse for smaller ǫ (when 1/ǫ ≥ ω(nh)). Also, compare these to the corresponding upper bounds for the first two problems by Blum et al (2004); Blum and Hartline (2005), which are respectively h log(1/ǫ) ǫ , and min h log h log log h ǫ 2 , h log log h ǫ 3 .…”

“…Morgenstern and Roughgarden (2015) consider arbitrary distributions with values bounded by h, and gave bounds that are polynomial in n, h, and ǫ −1 . Roughgarden and Schrijvers (2016); Huang et al (2015) give further improvements on the single-and multi-buyer versions respectively; tables 1 and 2 give a comparison of these results with our bounds, for the problems we consider. The dynamic pricing problem has also been studied when there are a given number of copies of the item to sell (limited supply) (Agrawal and Devanur, 2014;Babaioff et al, 2015;Badanidiyuru et al, 2013;Besbes and Zeevi, 2009).…”