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Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions

Author Hadley Black

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ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Property testing
  • learning
  • Boolean functions
  • monotonicity
  • k-monotonicity

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Abstract

We study monotonicity testing of functions f : {0,1}^d → {0,1} using sample-based algorithms, which are only allowed to observe the value of f on points drawn independently from the uniform distribution. A classic result by Bshouty-Tamon (J. ACM 1996) proved that monotone functions can be learned with exp(Õ(min{(1/ε)√d,d})) samples and it is not hard to show that this bound extends to testing. Prior to our work the only lower bound for this problem was Ω(√{exp(d)/ε}) in the small ε parameter regime, when ε = O(d^{-3/2}), due to Goldreich-Goldwasser-Lehman-Ron-Samorodnitsky (Combinatorica 2000). Thus, the sample complexity of monotonicity testing was wide open for ε ≫ d^{-3/2}. We resolve this question, obtaining a nearly tight lower bound of exp(Ω(min{(1/ε)√d,d})) for all ε at most a sufficiently small constant. In fact, we prove a much more general result, showing that the sample complexity of k-monotonicity testing and learning for functions f : {0,1}^d → [r] is exp(Ω(min{(rk/ε)√d,d})). For testing with one-sided error we show that the sample complexity is exp(Ω(d)).
Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of d,k,r,1/ε in the exponent) of exp(Θ̃(min{(rk/ε)√d,d})) on the sample complexity of testing and learning measurable k-monotone functions f : ℝ^d → [r] under product distributions. Our upper bound improves upon the previous bound of exp(Õ(min{(k/ε²)√d,d})) by Harms-Yoshida (ICALP 2022) for Boolean functions (r = 2).

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Hadley Black. Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 37:1-37:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.37

Author Details

Hadley Black
  • University of California, San Diego, USA

Funding

  • Black, Hadley: This work was done while the author was at UCLA supported by NSF award AF:Small 2007682, NSF Award: Collaborative Research Encore 2217033.

Acknowledgements

We would like to thank Eric Blais and Nathaniel Harms for helpful discussions during the early stages of this work and for their thoughtful feedback. We would also like to thank the anonymous reviewers whose comments helped significantly to improve this write up.

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