Testing +/- 1-Weight Halfspaces

We consider the problem of testing whether a Boolean function f:{ − 1,1} [superscript n] →{ − 1,1} is a ±1-weight halfspace, i.e. a function of the form f(x) = sgn(w [subscript 1] x [subscript 1] + w [subscript 2] x [subscript 2 ]+ ⋯ + w [subscript n] x [subscript n] ) where the weights w i take values in { − 1,1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±-weight halfspace versus ε-far from all such halfspaces we prove that nonadaptive algorithms must make Ω(logn) queries. We complement this lower bound with a sublinear upper bound showing that $O(\sqrt{n}\cdot$poly$(\frac{1}{\epsilon}))$ queries suffice

Testing (subclasses of) halfspaces

We address the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w . x − θ). We consider halfspaces over the continuous domain 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 ε-far from any halfspace using only poly(1) queries, independent of the dimension n. In contrast to the case of general halfspaces, we show that testing natural subclasses of halfspaces can be markedly harder; for the class of { − 1,1}-weight halfspaces, we show that a tester must make at least Ω(logn) queries. We complement this lower bound with an upper bound showing that O(√n) queries suffice.National Basic Research Program of China (grant 2007CB807900)National Basic Research Program of China (grant 2007CB807901)National Natural Science Foundation (China) (grant 60553001

Learning Boolean Halfspaces with Small Weights from Membership Queries

We consider the problem of proper learning a Boolean Halfspace with integer weights $\{0,1,\ldots,t\}$ from membership queries only. The best known algorithm for this problem is an adaptive algorithm that asks $n^{O(t^5)}$ membership queries where the best lower bound for the number of membership queries is $n^t$ [Learning Threshold Functions with Small Weights Using Membership Queries. COLT 1999] In this paper we close this gap and give an adaptive proper learning algorithm with two rounds that asks $n^{O(t)}$ membership queries. We also give a non-adaptive proper learning algorithm that asks $n^{O(t^3)}$ membership queries

Public projects, Boolean functions and the borders of Border's theorem

Border's theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian single-item environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border's theorem either restrict attention to relatively simple settings, or resort to approximation. This paper identifies a complexity-theoretic barrier that indicates, assuming standard complexity class separations, that Border's theorem cannot be extended significantly beyond the state-of-the-art. We also identify a surprisingly tight connection between Myerson's optimal auction theory, when applied to public project settings, and some fundamental results in the analysis of Boolean functions.Comment: Accepted to ACM EC 201