  In this work, we establish a frequency-domain framework for analyzing
gradient-based algorithms in linear minimax optimization problems;
specifically, our approach is based on the Z-transform, a powerful tool applied
in Control Theory and Signal Processing in order to characterize linear
discrete-time systems. We employ our framework to obtain the first tight
analysis of stability of Optimistic Gradient Descent/Ascent (OGDA), a natural
variant of Gradient Descent/Ascent that was shown to exhibit last-iterate
convergence in bilinear games by Daskalakis et al.
\cite{DBLP:journals/corr/abs-1711-00141}. Importantly, our analysis is
considerably simpler and more concise than the existing ones.
  Moreover, building on the intuition of OGDA, we consider a general family of
gradient-based algorithms that augment the memory of the optimization through
multiple historical steps. We reduce the convergence -- to a saddle-point -- of
the dynamics in bilinear games to the stability of a polynomial, for which
efficient algorithmic schemes are well-established. As an immediate corollary,
we obtain a broad class of algorithms -- that contains OGDA as a special case
-- with a last-iterate convergence guarantee to the space of Nash equilibria of
the game.
