The Taub Faculty of Computer Science Events and Talks

Lifting theorems are theorems that relate the query complexity of a function $f : \{0, 1\}^{n} \rightarrow \{0, 1\}$  to the communication complexity of the composed function fgn, for some “gadget” $g  : \{0, 1\}^{b} \times \{0, 1\}^{b} \rightarrow \{0, 1\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget g.
 
We prove a new lifting theorem that works for all gadgets g that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold.

Our result has two main motivations: First, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.

Joint work with Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, and Toniann Pitassi.