  We study the question of obtaining last-iterate convergence rates for
no-regret learning algorithms in multi-player games. We show that the
optimistic gradient (OG) algorithm with a constant step-size, which is
no-regret, achieves a last-iterate rate of $O(1/\sqrt{T})$ with respect to the
gap function in smooth monotone games. This result addresses a question of
Mertikopoulos & Zhou (2018), who asked whether extra-gradient approaches (such
as OG) can be applied to achieve improved guarantees in the multi-agent
learning setting. The proof of our upper bound uses a new technique centered
around an adaptive choice of potential function at each iteration. We also show
that the $O(1/\sqrt{T})$ rate is tight for all $p$-SCLI algorithms, which
includes OG as a special case. As a byproduct of our lower bound analysis we
additionally present a proof of a conjecture of Arjevani et al. (2015) which is
more direct than previous approaches.
