Second-Order Mirror Descent: Convergence in Games Beyond Averaging and Discounting
This work addresses convergence issues in game theory for researchers and practitioners, offering a novel method that is incremental in improving upon existing mirror descent approaches.
The authors tackled the problem of convergence in game-theoretic dynamics by proposing a second-order extension of mirror descent (MD2) that converges to variationally stable states without relying on averaging or discounting techniques, achieving exponential convergence rates for strong states and no-regret guarantees.
In this paper, we propose a second-order extension of the continuous-time game-theoretic mirror descent (MD) dynamics, referred to as MD2, which provably converges to mere (but not necessarily strict) variationally stable states (VSS) without using common auxiliary techniques such as time-averaging or discounting. We show that MD2 enjoys no-regret as well as an exponential rate of convergence towards strong VSS upon a slight modification. MD2 can also be used to derive many novel continuous-time primal-space dynamics. We then use stochastic approximation techniques to provide a convergence guarantee of discrete-time MD2 with noisy observations towards interior mere VSS. Selected simulations are provided to illustrate our results.