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DOI: 10.4230/LIPIcs.ITCS.2020.37

Learning and Testing Variable Partitions

Authors: Andrej Bogdanov and Baoxiang Wang

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract

Let F be a multivariate function from a product set Σ^n to an Abelian group G. A k-partition of F with cost δ is a partition of the set of variables V into k non-empty subsets (X_1, ̇s, X_k) such that F(V) is δ-close to F_1(X_1)+ ̇s+F_k(X_k) for some F_1, ̇s, F_k with respect to a given error metric. We study algorithms for agnostically learning k partitions and testing k-partitionability over various groups and error metrics given query access to F. In particular we show that 1) Given a function that has a k-partition of cost δ, a partition of cost O(k n^2)(δ + ε) can be learned in time Õ(n^2 poly 1/ε) for any ε > 0. In contrast, for k = 2 and n = 3 learning a partition of cost δ + ε is NP-hard. 2) When F is real-valued and the error metric is the 2-norm, a 2-partition of cost √(δ^2 + ε) can be learned in time Õ(n^5/ε^2). 3) When F is Z_q-valued and the error metric is Hamming weight, k-partitionability is testable with one-sided error and O(kn^3/ε) non-adaptive queries. We also show that even two-sided testers require Ω(n) queries when k = 2. This work was motivated by reinforcement learning control tasks in which the set of control variables can be partitioned. The partitioning reduces the task into multiple lower-dimensional ones that are relatively easier to learn. Our second algorithm empirically increases the scores attained over previous heuristic partitioning methods applied in this context.

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Andrej Bogdanov and Baoxiang Wang. Learning and Testing Variable Partitions. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 37:1-37:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bogdanov_et_al:LIPIcs.ITCS.2020.37,
  author =	{Bogdanov, Andrej and Wang, Baoxiang},
  title =	{{Learning and Testing Variable Partitions}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{37:1--37:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.37},
  URN =		{urn:nbn:de:0030-drops-117221},
  doi =		{10.4230/LIPIcs.ITCS.2020.37},
  annote =	{Keywords: partitioning, agnostic learning, property testing, sublinear-time algorithms, hypergraph cut, reinforcement learning}
}

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Learning and Testing Variable Partitions

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