Equilibrium Bandits: Learning Optimal Equilibria of Unknown Dynamics

TL;DR

This paper introduces UECB, a novel bandit algorithm that efficiently learns optimal system equilibria with unknown dynamics, balancing exploration and convergence time to maximize rewards.

Contribution

The paper proposes UECB, the first algorithm to adaptively switch actions based on convergence bounds, achieving sublinear regret in equilibrium bandit problems.

Findings

UECB achieves regret of O(log T + τ_c log τ_c + τ_c log log T).

Both epidemic and game control are special cases with dominant τ_c log τ_c regret.

Numerical tests confirm UECB's effectiveness in applications.

Abstract

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 $\mathcal{O}(\log(T)+\tau_c\log(\tau_c)+\tau_c\log\log(T))$ for this equilibrium bandit problem where $\tau_c$ 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 $\tau_c\log \tau_c$ typically dominates the regret. We then test UECB numerically for both of these applications.