# CS Theory Seminar

**Learning Sums of Independent Commonly Supported Integer Random Variables**

**Speaker:**Anindya De, Northwestern University

**Location:**Annenberg 314

**Abstract:** This talk is about learning Sums of Independent Commonly Supported Integer Random Variables, or SICSIRVs. For S a finite set of natural numbers, an "S-supported SICSIRV" is a random variable which is the sum of N independent random variables each of which is supported on S.

So, for example, if S = {0,1}, then an S-supported SICSIRV is a sum of N independent but not necessarily identically distributed Bernoulli random variables (a.k.a. a Poisson Binomial random variable).

Daskalakis et. al. (STOC 12, FOCS 13) have shown that for S={0,1,...,k-1}, there is an algorithm for learning an unknown S-SICSIRV using poly(k,1/\epsilon) draws from the distribution and running in poly(k,1/\epsilon) time, independent of N.

In this work we investigate the problem of learning an unknown S-SICSIRV where S={a_1,...,a_k} may be any finite set. We show that when |S|=3, it is possible to learn any S-SICSIRV with poly(1/\epsilon) samples, independent of the values a_1,a_2,a_3; when |S| >= 4, it is possible to learn any S-SICSIRV with poly(1/\epsilon, \log \log a_k) samples; and already when |S|=4, at least \log \log a_k samples may be required for learning.

We thus establish a sharp transition in the complexity of learning S-SICSIRVs when the size of S goes from three to four.

Joint work with Phil Long and Rocco Servedio.

**Series**Theory of Computing Seminar Series

**Contact:** Bonnie Leung bjleung@caltech.edu