Adding a single state memory optimally accelerates symmetric linear maps

TL;DR

This paper proves that adding a single memory slot optimally accelerates symmetric linear maps and that additional memory does not improve convergence when only eigenvalue bounds are known.

Contribution

It establishes that one memory slot is optimal for accelerating symmetric linear maps under limited spectral information.

Findings

Adding one memory slot accelerates convergence.

More memory slots do not improve acceleration under eigenvalue bounds.

Optimal acceleration is achieved with a single memory slot.

Abstract

Previous papers have proposed to add memory registers to the dynamics of discrete-time linear systems in order to accelerate their convergence. In particular, it has been proved that adding one memory slot per agent allows faster convergence towards average consensus. We here prove that this situation \emph{cannot} be improved by adding more memory slots, when the knowledge about the self-adjoint linear map to be accelerated reduces to bounds on its extreme eigenvalues.