  We consider the problem of minimizing a smooth convex function by reducing
the optimization to computing the Nash equilibrium of a particular zero-sum
convex-concave game. Zero-sum games can be solved using online learning
dynamics, where a classical technique involves simulating two no-regret
algorithms that play against each other and, after $T$ rounds, the average
iterate is guaranteed to solve the original optimization problem with error
decaying as $O(\log T/T)$. In this paper we show that the technique can be
enhanced to a rate of $O(1/T^2)$ by extending recent work \cite{RS13,SALS15}
that leverages \textit{optimistic learning} to speed up equilibrium
computation. The resulting optimization algorithm derived from this analysis
coincides \textit{exactly} with the well-known \NA \cite{N83a} method, and
indeed the same story allows us to recover several variants of the Nesterov's
algorithm via small tweaks. We are also able to establish the accelerated
linear rate for a function which is both strongly-convex and smooth. This
methodology unifies a number of different iterative optimization methods: we
show that the \HB algorithm is precisely the non-optimistic variant of \NA, and
recent prior work already established a similar perspective on \FW
\cite{AW17,ALLW18}.
