Near-Optimal No-Regret Learning Dynamics for General Convex Games

A recent line of work has established uncoupled learning dynamics such that, when employed by all players in a game, each player's regret after T repetitions grows polylogarithmically in T, an exponential improvement over the traditional guarantees within the no-regret framework. However, so far these results have only been limited to certain classes of games with structured strategy spaces---such as normal-form and extensive-form games. The question as to whether O(\mathrm{polylog} T) regret bounds can be obtained for general convex and compact strategy sets---as is the case in many fundamental models in economics and multiagent systems---while retaining efficient strategy updates is an important question. In this paper, we answer this in the positive by establishing the first uncoupled learning algorithm with O(\log T) per-player regret in general convex games, that is, games with concave utility functions supported on arbitrary convex and compact strategy sets. Our learning dynamics are based on an instantiation of optimistic follow-the-regularized-leader over an appropriately lifted space using a self-concordant regularizer that is peculiarly not a barrier for the feasible region.