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BIU AI and ML Learning Club, July 7 – Local Glivenko-Cantelli (or: estimating the mean in infinite dimensions)

July 7 @ 12:00 pm - 1:00 pm IDT

On July 7,  Prof. Aryeh Kontorovich from the Tel Aviv University will give a talk titled: Local Glivenko-Cantelli (or: estimating the mean in infinite dimensions)

Abstract:

If μ is a distribution over the d-dimensional Boolean cube {0,1}ᵈ, our goal is to estimate its mean p∈[0,1]ᵈ based on n iid draws from μ. Specifically, we consider the empirical mean estimator p̂ and study the maximal deviation M=max_{j∈[d]}|p̂(j)-p(j)|. In the classical Universal Glivenko-Cantelli setting, we seek distribution-free (i.e., independent of μ) bounds on M. This regime is well-understood: for all μ, we have 𝔼[M]≲√log(d)/n up to universal constants, and the bound is tight.
Our present work seeks to establish dimension-free (i.e., without an explicit dependence on d) estimates on M, including those that hold for d=∞. As such bounds must necessarily depend on μ, we refer to this regime as Local Glivenko-Cantelli, and are aware of very few previous bounds of this type — which are quite sub-optimal. Already the special case of product measures μ is quite non-trivial. We give necessary and sufficient conditions on μ for 𝔼[M]→0, and discover a novel sub-Gamma-type maximal inequality for shifted Bernoullis.

 

BIO:

Aryeh Kontorovich received his undergraduate degree in mathematics with a certificate in applied mathematics from Princeton University in 2001. His M.Sc. and Ph.D. are from Carnegie Mellon University, where he graduated in 2007. After a postdoctoral fellowship at the Weizmann Institute of Science, he joined the Computer Science department at Ben-Gurion University of the Negev in 2009, where he is currently a full professor. His research interests are mainly in machine learning, with a focus on probability, statistics, Markov chains, and metric spaces.

He served as the director of the Ben-Gurion University Data Science Research Center during 2021-2022.

Details

Date:
July 7
Time:
12:00 pm - 1:00 pm IDT
Event Categories:
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Venue

חדר ישיבות 329, הנדסה

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