First-order (coarse) correlated equilibria in non-concave games

We investigate first-order notions of correlated equilibria; distributions of actions for smooth, potentially non-concave games such that players do not incur any regret against small modifications to their strategies along a set of continuous vector fields. We define two such notions, based on local deviations and on stationarity of the distribution, and identify the notion of coarseness as the setting where the associated vector fields are in fact gradient fields. For coarse equilibria, we prove that online (projected) gradient decent has a universal approximation property for both variants of equilibrium. In the non-coarse setting, we instead reduce the problem of finding an equilibrium to fixed-point computation via the usual framework of $\Phi$-regret minimisation, and identify tractable instances. Finally, we study the primal-dual framework to our notion of first-order equilibria. For coarse equilibria defined by a family of functions, we find that a dual bound on the worst-case expectation of a performance metric takes the form of a generalised Lyapunov function for the dynamics of the game. Specifically, usual primal-dual price of anarchy analysis for coarse correlated equilibria as well as the smoothness framework of Roughgarden are both equivalent to a problem of general Lyapunov function estimation. For non-coarse equilibria, we instead observe a vector field fit problem for the gradient dynamics of the game. These follow from containment results in normal form games; the usual notion of a (coarse) correlated equilibria is equivalent to our first-order local notions of (coarse) correlated equilibria with respect to an appropriately chosen set of vector fields.