On Fortification of Projection Games

TL;DR

This paper improves the fortification method for projection games by introducing stronger graph structures called fortifiers, which ensure the necessary distribution guarantees for proving the Parallel Repetition Theorem.

Contribution

It introduces the concept of fortifiers, stronger graphs with $ ext{ell}_1$ and $ ext{ell}_2$ guarantees, to fix previous limitations in fortification techniques for projection games.

Findings

Fortifiers are graphs with $ ext{ell}_1$ and $ ext{ell}_2$ guarantees.

Expander graphs with sufficient spectral gap serve as good fortifiers.

Using fortifiers with $ ext{ell}_2$ guarantees is necessary for robustness.

Abstract

A recent result of Moshkovitz \cite{Moshkovitz14} presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in \cite{Moshkovitz14} to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both $\ell_1$ and $\ell_2$ guarantees on induced distributions from large subsets. We then show that an expander with sufficient spectral gap, or a bi-regular extractor with stronger parameters (the latter is also the construction used in an independent update \cite{Moshkovitz15} of \cite{Moshkovitz14} with an alternate argument), is a good fortifier. We also show that using a fortifier (in particular $\ell_2$ guarantees) is necessary for obtaining the robustness required for fortification.