Submultiplicative Glivenko-Cantelli and uniform convergence of revenues

Noga Alon, Moshe Babaioff, Yannai A. Gonczarowski, Yishay Mansour, Shay Moran, Amir Yehudayoff

Research output: Contribution to journalConference articlepeer-review

Abstract

In this work we derive a variant of the classic Glivenko-Cantelli Theorem, which asserts uniform convergence of the empirical Cumulative Distribution Function (CDF) to the CDF of the underlying distribution. Our variant allows for tighter convergence bounds for extreme values of the CDF. We apply our bound in the context of revenue learning, which is a well-studied problem in economics and algorithmic game theory. We derive sample-complexity bounds on the uniform convergence rate of the empirical revenues to the true revenues, assuming a bound on the kth moment of the valuations, for any (possibly fractional) k > 1. For uniform convergence in the limit, we give a complete characterization and a zero-one law: if the first moment of the valuations is finite, then uniform convergence almost surely occurs; conversely, if the first moment is infinite, then uniform convergence almost never occurs.

Original languageEnglish
Pages (from-to)1657-1666
Number of pages10
JournalAdvances in Neural Information Processing Systems
Volume2017-December
StatePublished - 2017
Event31st Annual Conference on Neural Information Processing Systems, NIPS 2017 - Long Beach, United States
Duration: 4 Dec 20179 Dec 2017

Funding

FundersFunder number
Israel-USA bi-national Science Foundation
Israeli Academy of Sciences
National Science Foundations and the Simons Foundations1162/15
Microsoft Research
European Metrology Programme for Innovation and Research740282
European Research Council
German-Israeli Foundation for Scientific Research and Development
United States-Israel Binational Science Foundation2014389
Israel Academy of Sciences and Humanities1435/14
Israel Science Foundation
Israeli Centers for Research Excellence4/11

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