A Lower Bound for Distribution-Free Monotonicity Testing

We consider monotonicity testing of functions f:[n]d→ {0,1}, in the property testing framework of Rubinfeld and Sudan [23] and Goldreich, Goldwasser and Ron [14]. Specifically, we consider the framework of distribution-free property testing, where the distance between functions is measured with respect to a fixed but unknown distribution D on the domain and the testing algorithms have an oracle access to random sampling from the domain according to this distribution D. We show that, though in the uniform distribution case, testing of boolean functions defined over the boolean hypercube can be done using query complexity that is polynomial in $\frac{1}{\epsilon}$ and in the dimension d, in the distribution-free setting such testing requires a number of queries that is exponential in d. Therefore, in the high-dimensional case (in oppose to the low-dimensional case), the gap between the query complexity for the uniform and the distribution-free settings is exponential.

[1]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[2]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[3]  Umesh V. Vazirani,et al.  An Introduction to Computational Learning Theory , 1994 .

[4]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[5]  Dana Ron,et al.  Property Testing in Bounded Degree Graphs , 1997, STOC.

[6]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[7]  Noga Alon,et al.  Efficient Testing of Large Graphs , 2000, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[8]  Ronitt Rubinfeld,et al.  Fast Approximate PCPs for Multidimensional Bin-Packing Problems , 1999, RANDOM-APPROX.

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

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

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

[12]  Eldar Fischer,et al.  Testing of matrix properties , 2001, STOC '01.

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

[14]  Luca Trevisan,et al.  Three theorems regarding testing graph properties , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[15]  Artur Czumaj,et al.  Testing Hypergraph Coloring , 2001, ICALP.

[16]  Sanguthevar Rajasekaran Handbook of randomized computing , 2001 .

[17]  Kenji Obata,et al.  A lower bound for testing 3-colorability in bounded-degree graphs , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

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

[19]  Dana Ron,et al.  Tight Bounds for Testing Bipartiteness in General Graphs , 2004, RANDOM-APPROX.

[20]  E. Fischer THE ART OF UNINFORMED DECISIONS: A PRIMER TO PROPERTY TESTING , 2004 .

[21]  Eyal Kushilevitz,et al.  Distribution-Free Connectivity Testing , 2004, APPROX-RANDOM.

[22]  Eyal Kushilevitz,et al.  Distribution-Free Property-Testing , 2007, SIAM J. Comput..