Beyond Talagrand functions: new lower bounds for testing monotonicity and unateness

We prove a lower bound of Ω(n1/3) for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function f:{0,1}n→ {0,1} is monotone versus far from monotone. This improves the recent lower bound of Ω(n1/4) for the same problem by Belovs and Blais (STOC'16). Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity we prove a lower bound of Ω(√n) for two-sided, adaptive algorithms and a lower bound of Ω(n) for one-sided, non-adaptive algorithms for testing unateness, a natural generalization of monotonicity. The latter matches the linear upper bounds by Khot and Shinkar (RANDOM'16) and by Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri (2017).

[1]  Eldar Fischer.  On the strength of comparisons in property testing , 2004, Inf. Comput..

[2]  Ronitt Rubinfeld,et al.  Sublinear algorithms for testing monotone and unimodal distributions , 2004, STOC '04.

[3]  Seshadhri Comandur,et al.  Optimal bounds for monotonicity and lipschitz testing over hypercubes and hypergrids , 2013, STOC '13.

[4]  Seshadhri Comandur,et al.  An Optimal Lower Bound for Monotonicity Testing over Hypergrids , 2014, Theory Comput..

[5]  Bernard Chazelle,et al.  Estimating the distance to a monotone function , 2007, Random Struct. Algorithms.

[6]  Hazel Shirley.  Learning to save a life , 2008 .

[7]  Sourav Chakraborty,et al.  Monotonicity testing and shortest-path routing on the cube , 2010, Comb..

[8]  Dana Ron,et al.  Testing Monotonicity , 2000, Comb..

[9]  Subhash Khot,et al.  An Õ(n) Queries Adaptive Tester for Unateness , 2016, Electron. Colloquium Comput. Complex..

[10]  Eyal Kushilevitz,et al.  Testing Monotonicity over Graph Products , 2004, ICALP.

[11]  Sofya Raskhodnikova,et al.  Testing Unateness of Real-Valued Functions , 2016, ArXiv.

[12]  Alan Frieze,et al.  Random Structures and Algorithms , 2014 .

[13]  Eric Blais,et al.  Quantum Algorithm for Monotonicity Testing on the Hypercube , 2015, Theory Comput..

[14]  Dana Ron,et al.  Property Testing: A Learning Theory Perspective , 2007, COLT.

[15]  Michel Talagrand,et al.  How much are increasing sets positively correlated? , 1996, Comb..

[16]  Oded Goldreich,et al.  Property Testing - Current Research and Surveys , 2010, Property Testing.

[17]  Seshadhri Comandur,et al.  A o(n) monotonicity tester for boolean functions over the hypercube , 2013, STOC '13.

[18]  Sofya Raskhodnikova,et al.  Lower Bounds for Testing Properties of Functions over Hypergrid Domains , 2014, 2014 IEEE 29th Conference on Computational Complexity (CCC).

[19]  J. Davenport.  Editor , 1960 .

[20]  Ronitt Rubinfeld,et al.  Spot-Checkers , 2000, J. Comput. Syst. Sci..

[21]  Seshadhri Comandur,et al.  An o(n) Monotonicity Tester for Boolean Functions over the Hypercube , 2016, SIAM J. Comput..

[22]  Sofya Raskhodnikova,et al.  Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps , 2017, ICALP.

[23]  Rocco A. Servedio,et al.  Boolean Function Monotonicity Testing Requires (Almost) n 1/2 Non-adaptive Queries , 2014, STOC.

[24]  Dana Ron,et al.  Algorithmic and Analysis Techniques in Property Testing , 2010, Found. Trends Theor. Comput. Sci..

[25]  Eric Blais,et al.  A polynomial lower bound for testing monotonicity , 2016, STOC.

[26]  Seshadhri Comandur,et al.  A Õ(n) Non-Adaptive Tester for Unateness , 2016, Electron. Colloquium Comput. Complex..

[27]  Dana Ron,et al.  Improved Testing Algorithms for Monotonicity , 1999, Electron. Colloquium Comput. Complex..

[28]  Ronitt Rubinfeld,et al.  Spot-checkers , 1998, STOC '98.

[29]  Seshadhri Comandur,et al.  A $\widetilde{O}(n)$ Non-Adaptive Tester for Unateness , 2016, ArXiv.

[30]  Joshua Brody,et al.  Property Testing Lower Bounds via Communication Complexity , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[31]  Elchanan Mossel,et al.  On the noise sensitivity of monotone functions , 2003, Random Struct. Algorithms.

[32]  Subhash Khot,et al.  On Monotonicity Testing and Boolean Isoperimetric Type Theorems , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[33]  Ronitt Rubinfeld,et al.  Approximating the Influence of Monotone Boolean Functions in $O(\sqrt{n})$ Query Complexity , 2011, APPROX-RANDOM.

[34]  Rocco A. Servedio,et al.  New Algorithms and Lower Bounds for Monotonicity Testing , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[35]  Ronitt Rubinfeld,et al.  Monotonicity testing over general poset domains , 2002, STOC '02.

[36]  Rocco A. Servedio,et al.  Testing monotone high-dimensional distributions , 2005, STOC '05.