Local (coarse) correlated equilibria in non-concave games

We investigate local notions of correlated equilibria, distributions of actions for smooth games such that players do not incur any regret against modifications of their strategies along a set of continuous vector fields. Our analysis shows that such equilibria are intrinsically linked to the projected gradient dynamics of the game. We identify the equivalent of coarse equilibria in this setting when no regret is incurred against any gradient field of a differentiable function. As a result, such equilibria are approximable when all players employ online (projected) gradient ascent with equal step-sizes as learning algorithms, and when their compact and convex action sets either (1) possess a smooth boundary, or (2) are polyhedra over which linear optimisation is ``trivial''. As a consequence, primal-dual proofs of performance guarantees for local coarse equilibria take the form of a generalised Lyapunov function for the gradient dynamics of the game. Adapting the regret matching framework to our setting, we also show that general local correlated equilibria are approximable when the set of vector fields is finite, given access to a fixed-point oracle for linear or conical combinations. For the class of affine-linear vector fields, which subsumes correlated equilibria of normal form games as a special case, such a fixed-point turns out to be the solution of a convex quadratic minimisation problem. Our results are independent of concavity assumptions on players' utilities.