  We examine the problem of regret minimization when the learner is involved in
a continuous game with other optimizing agents: in this case, if all players
follow a no-regret algorithm, it is possible to achieve significantly lower
regret relative to fully adversarial environments. We study this problem in the
context of variationally stable games (a class of continuous games which
includes all convex-concave and monotone games), and when the players only have
access to noisy estimates of their individual payoff gradients. If the noise is
additive, the game-theoretic and purely adversarial settings enjoy similar
regret guarantees; however, if the noise is multiplicative, we show that the
learners can, in fact, achieve constant regret. We achieve this faster rate via
an optimistic gradient scheme with learning rate separation -- that is, the
method's extrapolation and update steps are tuned to different schedules,
depending on the noise profile. Subsequently, to eliminate the need for
delicate hyperparameter tuning, we propose a fully adaptive method that attains
nearly the same guarantees as its non-adapted counterpart, while operating
without knowledge of either the game or of the noise profile.
