We develop a novel technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f : {0, 1}n â†’ {0, 1} and let Af be the matrix whose columns are each anâ€¦ (More)

We introduce the notion of a halfspace matrix, which is a sign matrix A with rows indexed by linear threshold functions f, columns indexed by inputs x âˆˆ {âˆ’Â 1, 1} n , and the entries given by A f,x =â€¦ (More)

2006 47th Annual IEEE Symposium on Foundations ofâ€¦

2006

We give the first representation-independent hardness results for PAC learning intersections of halfspaces, a central concept class in computational learning theory. Our hardness results are derivedâ€¦ (More)

We prove that AC<sup>0</sup> cannot be efficiently simulated by MAJÂºMAJ circuits. Namely, we construct an AC<sup>0</sup> circuit of depth 3 that requires MAJÂºMAJ circuits of sizeâ€¦ (More)

Reinforcement learning (RL) is a powerful abstraction of sequential decision making that has an established theoretical foundation and has proven effective in a variety of small, simulated domains.â€¦ (More)

2008 49th Annual IEEE Symposium on Foundations ofâ€¦

2008

We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e.,f(x,y)=D(|x Lambda y|), where D:{0,1,...,n}-rarr{0,1} is a given predicateâ€¦ (More)

In the gap Hamming distance problem, two parties must determine whether their respective strings x,y âˆˆ {0,1}n are at Hamming distance less than n/2âˆ’ âˆš n or greater than n/2+ âˆš n. In a recent tour deâ€¦ (More)

The sign-rank of a matrix A = [Aij ] with Â±1 entries is the least rank of a real matrix B = [Bij ] with AijBij > 0 for all i, j. We obtain the first exponential lower bound on the sign-rank of aâ€¦ (More)