Abstract
In the distribution-free property testing model, the distance between functions is measured with respect to an arbitrary and unknown probability distribution \(\mathcal{D}\) over the input domain. We consider distribution-free testing of several basic Boolean function classes over {0,1}n, namely monotone conjunctions, general conjunctions, decision lists, and linear threshold functions. We prove that for each of these function classes, Ω((n/logn)1/5) oracle calls are required for any distribution-free testing algorithm. Since each of these function classes is known to be distribution-free properly learnable (and hence testable) using Θ(n) oracle calls, our lower bounds are within a polynomial factor of the best possible.
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Glasner, D., Servedio, R.A. (2007). Distribution-Free Testing Lower Bounds for Basic Boolean Functions. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_36
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DOI: https://doi.org/10.1007/978-3-540-74208-1_36
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