Direct product via round-preserving compression

We obtain a strong direct product theorem for two-party bounded round communication complexity. Let suc_r (μ, f, C) denote the maximum success probability of an r-round communication protocol that uses at most C bits of communication in computing f(x,y) when (x,y) ∼ μ. Jain et al. [12] have recently showed that if suc_r(μ, f, C) ≤ 2/3 and T ≪, (C - Ω(r²))·n/r, then suc^r(μ ⁿ, fⁿ, T) ≤ exp(-Ω(n/r²)). Here we prove that if suc_7r(μ, f, C) ≤ 2/3 and T ≪ (C - Ω(r log r))·n then suc_r(μⁿ, fⁿ, T) ≤ exp(-Ω(n)). Up to a log r factor, our result asymptotically matches the upper bound on suc_7r (μⁿ ,fⁿ, T) given by the trivial solution which applies the per-copy optimal protocol independently to each coordinate. The proof relies on a compression scheme that improves the tradeoff between the number of rounds and the communication complexity over known compression schemes.