Streaming Algorithms for Support-Aware Histograms

Justin Y. Chen*, Piotr Indyk, Tal Wagner

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

Abstract

Histograms, i.e., piece-wise constant approximations, are a popular tool used to represent data distributions. Traditionally, the difference between the histogram and the underlying distribution (i.e., the approximation error) is measured using the Lp norm, which sums the differences between the two functions over all items in the domain. Although useful in many applications, the drawback of this error measure is that it treats approximation errors of all items in the same way, irrespective of whether the mass of an item is important for the downstream application that uses the approximation. As a result, even relatively simple distributions cannot be approximated by succinct histograms without incurring large error. In this paper, we address this issue by adapting the definition of approximation so that only the errors of the items that belong to the support of the distribution are considered. Under this definition, we develop efficient 1-pass and 2-pass streaming algorithms that compute near-optimal histograms in sub-linear space. We also present lower bounds on the space complexity of this problem. Surprisingly, under this notion of error, there is an exponential gap in the space complexity of 1-pass and 2-pass streaming algorithms. Finally, we demonstrate the utility of our algorithms on a collection of real and synthetic data sets.

Original languageEnglish
Pages (from-to)3184-3203
Number of pages20
JournalProceedings of Machine Learning Research
Volume162
StatePublished - 2022
Externally publishedYes
Event39th International Conference on Machine Learning, ICML 2022 - Baltimore, United States
Duration: 17 Jul 202223 Jul 2022

Funding

FundersFunder number
GIST-MIT1745302
NSF
National Science FoundationDMS-2022448

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