Mathematical Programming

, Volume 173, Issue 1–2, pp 465–507 | Cite as

Learning in games with continuous action sets and unknown payoff functions

  • Panayotis MertikopoulosEmail author
  • Zhengyuan Zhou
Full Length Paper Series A


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.


Continuous games Dual averaging Variational stability Fenchel coupling Nash equilibrium 

Mathematics Subject Classification

Primary 91A26 90C15 Secondary 90C33 68Q32 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.CNRS, Inria, LIGUniv. Grenoble AlpesGrenobleFrance
  2. 2.Department of Electrical EngineeringStanford UniversityStanfordUSA

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