NDLI: Learning Sums of Independent Integer Random Variables

Content Provider	IEEE Xplore Digital Library
Author	Daskalakis, C. Diakonikolas, I. ODonnell, R. Servedio, R.A. Li-Yang Tan
Copyright Year	2013
Abstract	Let S = $X_{1}+···+X_{n}$ be a sum of n independent integer random variables $X_{i},$ where each $X_{i}$ is supported on {0, 1, ..., k - 1} but otherwise may have an arbitrary distribution (in particular the Xi's need not be identically distributed). How many samples are required to learn the distribution S to high accuracy? In this paper we show that the answer is completely independent of n, and moreover we give a computationally efficient algorithm which achieves this low sample complexity. More precisely, our algorithm learns any such S to ε-accuracy (with respect to the total variation distance between distributions) using poly(k, 1/ε) samples, independent of n. Its running time is poly(k, 1/ε) in the standard word RAM model. Thus we give a broad generalization of the main result of [DDS12b] which gave a similar learning result for the special case k = 2 (when the distribution S is a Poisson Binomial Distribution). Prior to this work, no nontrivial results were known for learning these distributions even in the case k = 3. A key difficulty is that, in contrast to the case of k = 2, sums of independent {0, 1, 2}-valued random variables may behave very differently from (discretized) normal distributions, and in fact may be rather complicated - they are not log-concave, they can be Θ(n)-modal, there is no relationship between Kolmogorov distance and total variation distance for the class, etc. Nevertheless, the heart of our learning result is a new limit theorem which characterizes what the sum of an arbitrary number of arbitrary independent {0, 1, ... , k-1}-valued random variables may look like. Previous limit theorems in this setting made strong assumptions on the “shift invariance” of the random variables Xi in order to force a discretized normal limit. We believe that our new limit theorem, as the first result for truly arbitrary sums of independent {0, 1, ... , k-1}-valued random variables, is of independent interest.
Starting Page	217
Ending Page	226
File Size	302258
Page Count	10
File Format	PDF
ISBN	9780769551357
ISSN	02725428
DOI	10.1109/FOCS.2013.31
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2013-10-26
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Random variables Digital TV Gaussian distribution Accuracy Complexity theory Approximation methods Standards sums of independent integer random variables limit theorem discrete distribution learning
Content Type	Text
Resource Type	Article

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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Learning Sums of Independent Integer Random Variables

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