This paper investigates the convergence of learning dynamics in Stackelberg games. In the class of games we consider, there is a hierarchical game being played between a leader and a follower with continuous action spaces. We show that in zero-sum games, the only stable attractors of the Stackelberg gradient dynamics are Stackelberg equilibria. This insight allows us to develop a gradient-based update for the leader that converges to Stackelberg equilibria in zero-sum games and the set of stable attractors in general-sum games. We then consider a follower employing a gradient-play update rule instead of a best response strategy and propose a two-timescale algorithm with similar asymptotic convergence results. For this algorithm, we also provide finite-time high probability bounds for local convergence to a neighborhood of a stable Stackelberg equilibrium in general-sum games.
06/04/2019 ∙ by Tanner Fiez, et al. ∙ 0 ∙ share
Convergence Analysis of Gradient-Based Learning with Non-Uniform Learning Rates in Non-Cooperative Multi-Agent Settings
Considering a class of gradient-based multi-agent learning algorithms in non-cooperative settings, we provide local convergence guarantees to a neighborhood of a stable local Nash equilibrium. In particular, we consider continuous games where agents learn in (i) deterministic settings with oracle access to their gradient and (ii) stochastic settings with an unbiased estimator of their gradient. Utilizing the minimum and maximum singular values of the game Jacobian, we provide finite-time convergence guarantees in the deterministic case. On the other hand, in the stochastic case, we provide concentration bounds guaranteeing that with high probability agents will converge to a neighborhood of a stable local Nash equilibrium in finite time. Different than other works in this vein, we also study the effects of non-uniform learning rates on the learning dynamics and convergence rates. We find that much like preconditioning in optimization, non-uniform learning rates cause a distortion in the vector field which can, in turn, change the rate of convergence and the shape of the region of attraction. The analysis is supported by numerical examples that illustrate different aspects of the theory. We conclude with discussion of the results and open questions.
05/30/2019 ∙ by Benjamin Chasnov, et al. ∙ 0 ∙ share
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