  Recently, accelerated algorithms using the anchoring mechanism for minimax
optimization and fixed-point problems have been proposed, and matching
complexity lower bounds establish their optimality. In this work, we present
the surprising observation that the optimal acceleration mechanism in minimax
optimization and fixed-point problems is not unique. Our new algorithms achieve
exactly the same worst-case convergence rates as existing anchor-based methods
while using materially different acceleration mechanisms. Specifically, these
new algorithms are dual to the prior anchor-based accelerated methods in the
sense of H-duality. This finding opens a new avenue of research on accelerated
algorithms since we now have a family of methods that empirically exhibit
varied characteristics while having the same optimal worst-case guarantee.
