Partition bound is quadratically tight for product distributions

Abstract: Let $$f : {0,1}ⁿ \times {0,1}ⁿ \rightarrow {0,1}$$ be a 2-party function. For every product distribution $\mu$ on ${0,1}ⁿ \times {0,1}<sup>n$, we show that $$\mathsf{CC}^{\mu_{0.49}(f)} = O\left(\left(\log \mathsf{prt}<em>{1/8}(f) \cdot \log \log \mathsf{prt}</em>{1/8}(f)\right)<sup>2\right),$$ where $\mathsf{CC}<sup>\mu_\varepsilon(f)$ is the distributional communication complexity of $f$ with error at most $\varepsilon$ under the distribution $\mu$ and $\mathsf{prt}<em>{1/8}(f)$ is the {\em partition bound} of $f$, as defined by Jain and Klauck [{\em Proc. 25th CCC}, 2010]. We also prove a similar bound in terms of $\mathsf{IC}</em>{1/8}(f)$, the {\em information complexity} of $f$, namely, $$\mathsf{CC}^{\mu_{0.49}(f)} = O\left(\left(\mathsf{IC}<em>{1/8}(f) \cdot \log \mathsf{IC}</em>{1/8}(f)\right)<sup>2\right).$$ The latter bound was recently and independently established by Kol [{\em Proc. 48th STOC}, 2016] using a different technique. We show a similar result for query complexity under product distributions. Let $g : {0,1}ⁿ \rightarrow {0,1}$ be a function. For every bit-wise product distribution $\mu$ on ${0,1}<sup>n$, we show that $$\mathsf{QC}^{\mu_{0.49}(g)} = O\left(\left( \log \mathsf{qprt}<em>{1/8}(g) \cdot \log \log\mathsf{qprt}</em>{1/8}(g) \right)² \right),$$ where $\mathsf{QC}<sup>\mu_{\varepsilon}(g)$ is the distributional query complexity of $f$ with error at most $\varepsilon$ under the distribution $\mu$ and $\mathsf{qprt}_{1/8}(g))$ is the {\em query partition bound} of the function $g$. Partition bounds were introduced (in both communication complexity and query complexity models) to provide LP-based lower bounds for randomized communication complexity and randomized query complexity. Our results demonstrate that these lower bounds are polynomially tight for {\em product} distributions.