  We revisit the problem of solving two-player zero-sum games in the
decentralized setting. We propose a simple algorithmic framework that
simultaneously achieves the best rates for honest regret as well as adversarial
regret, and in addition resolves the open problem of removing the logarithmic
terms in convergence to the value of the game. We achieve this goal in three
steps. First, we provide a novel analysis of the optimistic mirror descent
(OMD), showing that it can be modified to guarantee fast convergence for both
honest regret and value of the game, when the players are playing
collaboratively. Second, we propose a new algorithm, dubbed as robust
optimistic mirror descent (ROMD), which attains optimal adversarial regret
without knowing the time horizon beforehand. Finally, we propose a simple
signaling scheme, which enables us to bridge OMD and ROMD to achieve the best
of both worlds. Numerical examples are presented to support our theoretical
claims and show that our non-adaptive ROMD algorithm can be competitive to OMD
with adaptive step-size selection.
