Testing Juntas Research Articles

Distribution-free Junta Testing

We study the problem of testing whether an unknown n -variable Boolean function is a k -junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1} n . Our first main result is that distribution-free k -junta testing can be performed, with one-sided error, by an adaptive algorithm that uses Õ( k 2 )/ϵ queries (independent of n ). Complementing this, our second main result is a lower bound showing that any non-adaptive distribution-free k -junta testing algorithm must make Ω(2 k /3 ) queries even to test to accuracy ϵ = 1/3. These bounds establish that while the optimal query complexity of non-adaptive k -junta testing is 2 Θ( k ) , for adaptive testing it is poly( k ), and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.

Testing Halfspaces

This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e., a function of the form $f(x)=\mathrm{sgn}(w\cdot x-\theta)$. We consider halfspaces over the continuous domain $\mathbf{R}^n$ (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube $\{-1,1\}^n$ (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are $\epsilon$-far from any halfspace using only $\mathrm{poly}(\frac{1}{\epsilon})$ queries, independent of the dimension n. Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: The first gives an exact relationship between the expected value of a halfspace f and the sum of the squares of f's degree-1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube $\{-1,1\}^n$ (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree-1 Fourier coefficients are small. Dealing with general halfspaces over $\{-1,1\}^n$ poses significant additional complications and requires other ingredients. These include “cross-consistency” versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree-1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [E. Fischer, G. Kindler, D. Ron, S. Safra, and A. Samorodnitsky, Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, 2002, pp. 103–112].

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: (1) whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; (2) with no access to any classical or quantum membership (“black-box”) queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; (3) which require only a few quantum examples but possibly many classical random examples (which are considered quite “cheap” relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: (1) We give an algorithm for testing k-juntas to accuracy ε that uses O(k/ϵ) quantum examples. This improves on the number of examples used by the best known classical algorithm. (2) We establish the following lower bound: any FS-based k-junta testing algorithm requires $$\Omega(\sqrt{k})$$ queries. (3) We give an algorithm for learning k-juntas to accuracy ϵ that uses O(ϵ−1 k log k) quantum examples and O(2 k log(1/ϵ)) random examples. We show that this learning algorithm is close to optimal by giving a related lower bound.