Decentralized Optimal Equilibrium Learning Over Dynamic Networks

We study decentralized bandit learning in discounted infinite-horizon, general-sum Markov games with continuous actions. Agents observe the public state, but only their own realized rewards, and use simultaneous perturbation stochastic approximation to update their actions. Under strong monotonicity of the negative running-reward gradients and sufficiently small time steps, we prove that the fully decentralized procedure converges almost surely to the unique stationary Markov perfect equilibrium. We obtain an $O(n^{-1/2})$ convergence rate and present a natural extension to episodic finite-horizon learning.

For decades, static solution concepts (Nash, Correlated, and Coarse Correlated Equilibria) and the Price of Anarchy (PoA) have formed the bedrock of algorithmic game theory, with no-regret learning proving fast convergence to such game-theoretic equilibria. We show that reducing multi-agent learning to static equilibrium and black-box regret analysis obscures underlying dynamic disequilibrium and game theoretic bounds. First, interior Nash equilibria lack $C^1$ vector field information, meaning agents cannot distinguish aligned from strictly opposing incentives. Inheriting this geometry, the worst-case pure Nash equilibria dictating robust PoA bounds manifest as topologically unstable strict saddles, and in canonical congestion games, as global repellers supported on almost everywhere strictly dominated strategies. Anchoring efficiency guarantees to these unstable states causes algebraic sensitivity; we prove that accommodating all strictly positive affine costs renders the PoA unbounded. Furthermore, projecting learning trajectories onto the discrete simplex of correlated play systematically accommodates non-rationalizable behavior. Evaluating dynamics via Coarse Correlated Equilibria or proximal refinements fails to preclude strictly dominated strategies. Moreover, optimal $O(1/T)$ swap-regret minimization does not preclude macroscopic turbulence, manifesting as chaotic limit sets even in minimal games. Finally, we examine the non-atomic limit of congestion games. Though considered highly stable with tight sub-linear $Θ(p/\ln p)$ PoA bounds (where $p$ is the polynomial degree), we prove that under discrete-time learning, the unique equilibrium destabilizes into Li-Yorke chaos and global attractors whose time-averaged inefficiency degrades exponentially as $2^p$. These results necessitate re-evaluating worst-case equilibrium frameworks for dynamically grounded metrics.

We study decentralized online convex optimization when agents communicate over a graph and messages may be compressed. Classical decentralized online methods typically require learning-rate choices that depend on the horizon, comparator scale, or other problem parameters, while compressed communication introduces additional disagreement that must be controlled. We propose DECO-EF (DEcentralized COin-betting with Error Feedback), a decentralized parameter-free online learning algorithm that combines coin-betting predictions with compressed difference-based gossip. Each agent maintains a clean accumulated state and a compressed tracker, and communicates only compressed state differences during gossip steps. The method is parameter-free in the online-learning sense: it does not tune to the horizon, the comparator norm, or the learning rate. We prove expected comparator-adaptive network-regret bounds for DECO-EF under compressed communication. To the best of our knowledge, this gives the first expected sublinear network-regret guarantees for parameter-free decentralized online learning under compressed communication.