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On the Complexity of Computing Sparse Equilibria and Lower Bounds for No-Regret Learning in Games

Authors Ioannis Anagnostides, Alkis Kalavasis, Tuomas Sandholm, Manolis Zampetakis



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Ioannis Anagnostides
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Alkis Kalavasis
  • Department of Computer Science, Yale University, New Haven, CT, USA
Tuomas Sandholm
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Manolis Zampetakis
  • Department of Computer Science, Yale University, New Haven, CT, USA

Acknowledgements

We are grateful to the anonymous ITCS reviewers for their helpful feedback.

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Ioannis Anagnostides, Alkis Kalavasis, Tuomas Sandholm, and Manolis Zampetakis. On the Complexity of Computing Sparse Equilibria and Lower Bounds for No-Regret Learning in Games. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.5

Abstract

Characterizing the performance of no-regret dynamics in multi-player games is a foundational problem at the interface of online learning and game theory. Recent results have revealed that when all players adopt specific learning algorithms, it is possible to improve exponentially over what is predicted by the overly pessimistic no-regret framework in the traditional adversarial regime, thereby leading to faster convergence to the set of coarse correlated equilibria (CCE) - a standard game-theoretic equilibrium concept. Yet, despite considerable recent progress, the fundamental complexity barriers for learning in normal- and extensive-form games are poorly understood. In this paper, we make a step towards closing this gap by first showing that - barring major complexity breakthroughs - any polynomial-time learning algorithms in extensive-form games need at least 2^{log^{1/2 - o(1)} |𝒯|} iterations for the average regret to reach below even an absolute constant, where |𝒯| is the number of nodes in the game. This establishes a superpolynomial separation between no-regret learning in normal- and extensive-form games, as in the former class a logarithmic number of iterations suffices to achieve constant average regret. Furthermore, our results imply that algorithms such as multiplicative weights update, as well as its optimistic counterpart, require at least 2^{(log log m)^{1/2 - o(1)}} iterations to attain an O(1)-CCE in m-action normal-form games under any parameterization. These are the first non-trivial - and dimension-dependent - lower bounds in that setting for the most well-studied algorithms in the literature. From a technical standpoint, we follow a beautiful connection recently made by Foster, Golowich, and Kakade (ICML '23) between sparse CCE and Nash equilibria in the context of Markov games. Consequently, our lower bounds rule out polynomial-time algorithms well beyond the traditional online learning framework, capturing techniques commonly used for accelerating centralized equilibrium computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Convergence and learning in games
Keywords
  • No-regret learning
  • extensive-form games
  • multiplicative weights update
  • optimism
  • lower bounds

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