Equilibrium Bandits: Learning Optimal Equilibria of Unknown Dynamics
This work addresses decision-making in systems with unknown dynamics for applications like epidemic and game control, presenting a novel algorithmic approach with proven theoretical guarantees.
The paper tackles the problem of learning optimal equilibria in unknown dynamic systems where actions lead to asymptotic equilibria, but waiting for convergence incurs time costs. The authors propose the UECB algorithm, which achieves a regret bound of O(log(T) + τ_c log(τ_c) + τ_c log log(T)), with τ_c as the worst-case convergence time, and demonstrate its application in epidemic and game control.
Consider a decision-maker that can pick one out of $K$ actions to control an unknown system, for $T$ turns. The actions are interpreted as different configurations or policies. Holding the same action fixed, the system asymptotically converges to a unique equilibrium, as a function of this action. The dynamics of the system are unknown to the decision-maker, which can only observe a noisy reward at the end of every turn. The decision-maker wants to maximize its accumulated reward over the $T$ turns. Learning what equilibria are better results in higher rewards, but waiting for the system to converge to equilibrium costs valuable time. Existing bandit algorithms, either stochastic or adversarial, achieve linear (trivial) regret for this problem. We present a novel algorithm, termed Upper Equilibrium Concentration Bound (UECB), that knows to switch an action quickly if it is not worth it to wait until the equilibrium is reached. This is enabled by employing convergence bounds to determine how far the system is from equilibrium. We prove that UECB achieves a regret of $\mathcal{O}(\log(T)+τ_c\log(τ_c)+τ_c\log\log(T))$ for this equilibrium bandit problem where $τ_c$ is the worst case approximate convergence time to equilibrium. We then show that both epidemic control and game control are special cases of equilibrium bandits, where $τ_c\log τ_c$ typically dominates the regret. We then test UECB numerically for both of these applications.