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A Strong Direct Sum Theorem for Distributional Query Complexity

Authors Guy Blanc, Caleb Koch, Carmen Strassle, Li-Yang Tan

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Author Details

Guy Blanc
  • Department of Computer Science, Stanford University, CA, USA
Caleb Koch
  • Department of Computer Science, Stanford University, CA, USA
Carmen Strassle
  • Department of Computer Science, Stanford University, CA, USA
Li-Yang Tan
  • Department of Computer Science, Stanford University, CA, USA

Funding

The authors are supported by NSF awards 1942123, 2211237, 2224246 and a Google Research Scholar award. Guy is also supported by a Jane Street Graduate Research Fellowship, Caleb by an NDSEG fellowship, and Carmen by a Stanford Computer Science Distinguished Fellowship.

Cite As Get BibTex

Guy Blanc, Caleb Koch, Carmen Strassle, and Li-Yang Tan. A Strong Direct Sum Theorem for Distributional Query Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 16:1-16:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.16

Abstract

Consider the expected query complexity of computing the k-fold direct product f^{⊗ k} of a function f to error ε with respect to a distribution μ^k. One strategy is to sequentially compute each of the k copies to error ε/k with respect to μ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. 
Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem that holds for all functions in the standard query model had been elusive.
A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new resilience lemma that accompanies it, showing that the hardcore of f^{⊗k} is likely to remain dense under arbitrary partitions of the input space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Query complexity
  • direct product theorem
  • hardcore theorem

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