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Exponentially Improved Algorithms and Lower Bounds for Testing Signed Majorities

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Abstract

A signed majority function is a linear threshold function f:{+1,−1}n→{+1,−1} of the form \(f(x)=\operatorname{sign}(\sum_{i=1}^{n} \sigma _{i} x_{i})\) where each σ i ∈{+1,−1}. Signed majority functions are a highly symmetrical subclass of the class of all linear threshold functions, which are functions of the form \(\operatorname{sign} (\sum_{i=1}^{n} w_{i} x_{i} - \theta)\) for arbitrary real w i ,θ.

We study the query complexity of testing whether an unknown f:{+1,−1}n→{+1,−1} is a signed majority function versus ϵ-far from every signed majority function. While it is known (SIAM J. Comput. 39(5):2004–2047, 2010) that the broader class of all linear threshold functions is testable with poly(1/ϵ) queries (independent of n), prior to our work the best upper bound for signed majority functions was \(O(\sqrt{n}) \cdot \mathrm{poly} (1/\epsilon)\) queries (via a non-adaptive algorithm), and the best lower bound was Ω(logn) queries for non-adaptive algorithms (Proceedings of the 13th International Workshop on Approximation, Randomization and Combinatorial Optimization (RANDOM), pp. 646–657, 2009).

As our main results we exponentially improve both these prior bounds for testing signed majority functions:

  • (Upper bound) We give a poly(logn,1/ϵ)-query adaptive algorithm (which is computationally efficient) for this testing problem;

  • (Lower bound) We show that any non-adaptive algorithm for testing the class of signed majorities to constant accuracy must make n Ω(1) queries. This directly implies a lower bound of Ω(logn) queries for any adaptive algorithm.

Our testing algorithm performs a sequence of restrictions together with consistency checks to ensure that each successive restriction is “compatible” with the function prior to restriction. This approach is used to transform the original n-variable testing problem into a testing problem over poly(logn,1/ϵ) variables where a simple direct method can be applied. Analysis of the degree-1 Fourier coefficients plays an important role in our proofs.

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Notes

  1. In particular, in all that follows we make statements that hold with high probability, without explicitly stating this.

  2. This can be done either by running the \(O(\sqrt{n}\cdot\mathrm{poly}(1/\epsilon))\)-queries algorithm of [27] (which runs in poly(n,1/ϵ) time), or by simply running a proper learning algorithm with time and query complexity poly(n,1/ϵ) for the class SMAJ and applying the well-known result that the query complexity of testing a class of functions is essentially upper bounded by the query complexity of proper learning the class [19, Prop. 3.1.1]. To properly learn a function f∈SMAJ, i.e., f=Maj σ for σ∈{+1,−1}n, it suffices to find σ i for each i∈[n] and this can easily be done by performing poly(n) queries.

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Acknowledgements

We would like to thank Ryan O’Donnell and Gilad Tsur for discussions in the initial stages of this work.

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Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Dana Ron

  2. Columbia University, New York, USA

    Rocco A. Servedio

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  1. Dana Ron
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Correspondence to Rocco A. Servedio.

Additional information

D. Ron was supported by ISF grants number 246/08 and 671/13.

R.A. Servedio was supported by NSF grants CCF-0915929 and CCF-1115703.

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Ron, D., Servedio, R.A. Exponentially Improved Algorithms and Lower Bounds for Testing Signed Majorities. Algorithmica 72, 400–429 (2015). https://doi.org/10.1007/s00453-013-9858-0

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