Learning in games with continuous action sets and unknown payoff functions (1608.07310v2)

Abstract: This paper examines the convergence of no-regret learning in games with continuous action sets. For concreteness, we focus on learning via "dual averaging", a widely used class of no-regret learning schemes where players take small steps along their individual payoff gradients and then "mirror" the output back to their action sets. In terms of feedback, we assume that players can only estimate their payoff gradients up to a zero-mean error with bounded variance. To study the convergence of the induced sequence of play, we introduce the notion of variational stability, and we show that stable equilibria are locally attracting with high probability whereas globally stable equilibria are globally attracting with probability 1. We also discuss some applications to mixed-strategy learning in finite games, and we provide explicit estimates of the method's convergence speed.

• The paper introduces variational stability to extend equilibrium analysis in continuous games, ensuring that incremental gradient-based updates lead toward Nash equilibria.
• It proves that dual averaging methods converge to Nash equilibria under established regularity conditions with almost sure and high-probability guarantees.
• Explicit numerical bounds and convergence rates are provided, highlighting efficiency improvements for adaptive learning in multi-agent systems.

Essay on "Learning in Games with Continuous Action Sets and Unknown Payoff Functions"

The paper "Learning in Games with Continuous Action Sets and Unknown Payoff Functions" by Panayotis Mertikopoulos and Zhengyuan Zhou addresses a prominent challenge in game theory and online learning: the convergence of no-regret learning dynamics in games where players' actions are continuous and their payoff functions are initially unknown. The authors focus on a class of algorithms commonly utilized in online optimization called dual averaging (DA). This approach involves players taking incremental steps along estimated gradients of their payoff functions and then mirroring these steps back onto their feasible action sets.

Contributions and Core Findings:

1. Variational Stability: The paper introduces the concept of variational stability (VS) as an extension of the notion of evolutionarily stable strategies in population games. Variational stability provides a useful framework for analyzing the convergence properties of learning dynamics in games. It captures the notion that players' strategies tend towards an equilibrium point, where unilateral deviations do not increase a player's payoff. The authors show that if a game is variationally stable, it leads to convergence of learning dynamics to equilibria.
2. Convergence Results: The primary contribution is proving that under certain regularity conditions (such as bounded gradient estimation errors and certain continuity assumptions), the DA algorithm will converge to a Nash equilibrium (NE) or a stable set of equilibria in both deterministic and stochastic settings. For globally stable set scenarios, the convergence is almost sure, while for locally stable configurations, the convergence happens with high probability.
3. Numerical Bounds and Convergence Rates: The authors provide explicit bounds and convergence rates for the dual averaging method. They demonstrate that the ergodic average of the players' strategies converges to the equilibrium with a rate of convergence described by the decay of the equilibrium gap function. Particularly, for cases of strongly stable equilibria, they show sharper convergence rates, highlighting the efficiency of the DA scheme compared to traditional gradient-based methods.
4. Applications to Finite and Zero-Sum Games: The work extends the analysis to mixed-strategy extensions of finite games and zero-sum games. It identifies conditions under which dominated strategies are eliminated and strict equilibria are approached, further illustrating the applicability of the theoretical results.
5. Implications for Multi-Agent Systems: This paper is significant for its implications in the context of multi-agent learning systems, providing a robust framework for analyzing equilibrium convergence when payoff functions are unknown, subject to estimation error, and decisions occur in a continuous space.

Implications and Future Directions:

The research addresses the practical necessity of adaptive algorithms in multi-agent systems where payoff landscapes are complex and noisy. The identification of variational stability as a key indicator of convergence is particularly valuable for theorists and practitioners alike seeking scalable solutions in fields like economic modeling, network optimization, and distributed control systems.

Future research could extend these ideas by relaxing the assumptions on feedback noise structure or exploring scenarios with bounded rationality and adaptive strategies over time. The method's flexibility also invites exploration into different forms of regularization and their impact on convergence speeds, providing new directions for enhancing algorithmic efficiency.

Overall, the paper constitutes a substantial contribution to the literature on no-regret learning in games with continuous action spaces, offering novel insights and formalism that pave the way for developing more robust learning dynamics in real-world scenarios.