Strong XOR Lemma for Communication with Bounded Rounds : (extended abstract)

In this paper, we prove a strong XOR lemma for bounded-round two-player randomized communication. For a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow\{0,1\}$, the n-fold XOR function $f^{\oplus n}:\mathcal{X}^{n}\times \mathcal{Y}^{n}\rightarrow\{0,1\}$ maps n input pairs $(X_{1},\ldots,\ X_{n},\ Y_{1},\ldots\,\ Y_{n})$ to the XOR of the n output bits $f(X_{1},\ Y_{1})\oplus\cdots\oplus f(X_{n},\ Y_{n})$. We prove that if every r-round communication protocols that computes f with probability 2/3 uses at least C bits of communication, then any r-round protocol that computes $f^{\oplus n}$ with probability $1/2+\exp(-O(n))$ must use $n\cdot(r^{-O(r)}\cdot C-1)$ bits. When r is a constant and C is sufficiently large, this is $\Omega(n\cdot C)$ bits. It matches the communication cost and the success probability of the trivial protocol that computes the n bits $f(X_{i},\ Y_{i})$ independently and outputs their XOR, up to a constant factor in n. A similar XOR lemma has been proved for f whose communication lower bound can be obtained via bounding the discrepancy [17]. By the equivalence between the discrepancy and the correlation with 2-bit communication protocols [19], our new XOR lemma implies the previous result.