Distribution-Free Testing Lower Bounds for Basic Boolean Functions

In the distribution-free property testing model, the distance between functions is measured with respect to an arbitrary and unknown probability distribution D over the input domain. We consider distribution-free testing of several basic Boolean function classes over {0, 1}, namely monotone conjunctions, general conjunctions, decision lists, and linear threshold functions. We prove that for each of these function classes, Ω((n/ log n)) 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.