  In online learning an algorithm plays against an environment with losses
possibly picked by an adversary at each round. The generality of this framework
includes problems that are not adversarial, for example offline optimization,
or saddle point problems (i.e. min max optimization). However, online
algorithms are typically not designed to leverage additional structure present
in non-adversarial problems. Recently, slight modifications to well-known
online algorithms such as optimism and adaptive step sizes have been used in
several domains to accelerate online learning -- recovering optimal rates in
offline smooth optimization, and accelerating convergence to saddle points or
social welfare in smooth games. In this work we introduce optimism and adaptive
stepsizes to Lagrangian hedging, a class of online algorithms that includes
regret-matching, and hedge (i.e. multiplicative weights). Our results include:
a general general regret bound; a path length regret bound for a fixed smooth
loss, applicable to an optimistic variant of regret-matching and
regret-matching+; optimistic regret bounds for $\Phi$ regret, a framework that
includes external, internal, and swap regret; and optimistic bounds for a
family of algorithms that includes regret-matching+ as a special case.
