Exponentially fast convergence to (strict) equilibrium via hedging

TL;DR

This paper demonstrates that the hedge variant of exponential weights learning converges exponentially fast to strict equilibria in N-player games, even under imperfect information, with convergence speed depending on step-size policies.

Contribution

It establishes exponential convergence rates for the hedge algorithm in game learning, including under uncertainty, and characterizes conditions for local convergence.

Findings

Exponential convergence occurs when players have perfect payoff information.

Under uncertainty, convergence remains with high probability using conservative step-sizes.

The convergence rate is exponential, proportional to the sum of step-sizes over time.

Abstract

Motivated by applications to data networks where fast convergence is essential, we analyze the problem of learning in generic N-person games that admit a Nash equilibrium in pure strategies. Specifically, we consider a scenario where players interact repeatedly and try to learn from past experience by small adjustments based on local - and possibly imperfect - payoff information. For concreteness, we focus on the so-called "hedge" variant of the exponential weights algorithm where players select an action with probability proportional to the exponential of the action's cumulative payoff over time. When players have perfect information on their mixed payoffs, the algorithm converges locally to a strict equilibrium and the rate of convergence is exponentially fast - of the order of $\mathcal{O}(\exp(-a\sum_{j=1}^{t}\gamma_{j}))$ where $a>0$ is a constant and $\gamma_{j}$ is the algorithm's step-size. In the presence of uncertainty, convergence requires a more conservative step-size policy, but with high probability, the algorithm remains locally convergent and achieves an exponential convergence rate.