Weak Composition Theorems for Randomized Communication

We present two results towards the randomized communication complexity of composed functions. Our contributions include: - \textbf{Lifting-by-Influence.} For any binary function $f$ and any outer gadget $g:\{0,1\}^n \rightarrow \mathbb{R}$, we show that up to additive terms, computing $g \circ f^n$ information-theoretically requires solving at least $\operatorname{Inf}(g)$ instances of $f$, where $\operatorname{Inf}(g)$ denotes the total influence of $g$. To our knowledge, this is the first composition lower bound for randomized communication that applies to \emph{every} choice of $f$ and $g$. - \textbf{Symmetric Composition.} When $g$ is a symmetric function with a value transition at Hamming weight $d$, we strengthen the number of required instances to $\min\{d,n+1-d\}$. As an implication, this recovers the previously known direct-sum theorem, XOR lemma, and majority lemma. Our proofs develop novel and distinctive usages of the classical \emph{input embedding} argument of Barak, Braverman, Chen, and Rao (SICOMP 2013). As an additional ingredient, we introduce a new public sampling framework that avoids auxiliary communication during the recovery phase of the protocol, ultimately reducing the extra cost from $O(n)$ bits of information to just $O(1)$.