  In this paper we study the smooth convex-concave saddle point problem.
Specifically, we analyze the last iterate convergence properties of the
Extragradient (EG) algorithm. It is well known that the ergodic (averaged)
iterates of EG converge at a rate of $O(1/T)$ (Nemirovski, 2004). In this
paper, we show that the last iterate of EG converges at a rate of
$O(1/\sqrt{T})$. To the best of our knowledge, this is the first paper to
provide a convergence rate guarantee for the last iterate of EG for the smooth
convex-concave saddle point problem. Moreover, we show that this rate is tight
by proving a lower bound of $\Omega(1/\sqrt{T})$ for the last iterate. This
lower bound therefore shows a quadratic separation of the convergence rates of
ergodic and last iterates in smooth convex-concave saddle point problems.
