The Complexity of Gradient Descent (Invited Talk)

Author Rahul Savani

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Rahul Savani
  • Department of Computer Science, University of Liverpool, UK

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Rahul Savani. The Complexity of Gradient Descent (Invited Talk). In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 5:1-5:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


PPAD and PLS are successful classes that capture the complexity of important game-theoretic problems. For example, finding a mixed Nash equilibrium in a bimatrix game is PPAD-complete, and finding a pure Nash equilibrium in a congestion game is PLS-complete. Many important problems, such as solving a Simple Stochastic Game or finding a mixed Nash equilibrium of a congestion game, lie in both classes. It was strongly believed that their intersection, PPAD ∩ PLS, does not have natural complete problems. We show that it does: any problem that lies in both classes can be reduced in polynomial time to the problem of finding a stationary point of a continuously differentiable function on the domain [0,1]². Thus, as PPAD captures problems that can be solved by Lemke-Howson type complementary pivoting algorithms, and PLS captures problems that can be solved by local search, we show that PPAD ∩ PLS exactly captures problems that can be solved by Gradient Descent. This is joint work with John Fearnley, Paul Goldberg, and Alexandros Hollender. It appeared at STOC'21, where it was given a Best Paper Award [Fearnley et al., 2021].

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Mathematical optimization
  • Mathematics of computing → Continuous functions
  • Computational Complexity
  • Continuous Optimization
  • TFNP
  • PPAD
  • PLS
  • CLS


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  1. Yakov Babichenko and Aviad Rubinstein. Settling the Complexity of Nash Equilibrium in Congestion Games. In Proc. of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 1426-1437, 2021. Google Scholar
  2. Constantinos Daskalakis and Christos H. Papadimitriou. Continuous Local Search. In Proc. of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 790-804, 2011. Google Scholar
  3. Kousha Etessami, Christos H. Papadimitriou, Aviad Rubinstein, and Mihalis Yannakakis. Tarski’s Theorem, Supermodular Games, and the Complexity of Equilibria. In Proc. of the 11th Innovations in Theoretical Computer Science Conference (ITCS), pages 18:1-18:19, 2020. Google Scholar
  4. John Fearnley, Paul W. Goldberg, Alexandros Hollender, and Rahul Savani. The Complexity of Gradient Descent: CLS = PPAD ∩ PLS. In Proc. of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 46-59, 2021. Google Scholar
  5. John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique End of Potential Line. Journal of Computer and System Sciences, 114:1-35, 2020. Google Scholar
  6. John Fearnley and Rahul Savani. A Faster Algorithm for Finding Tarski Fixed Points. In Proc. of the 38th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 29:1-29:16, 2021. Google Scholar
  7. Frédéric Meunier, Wolfgang Mulzer, Pauline Sarrabezolles, and Yannik Stein. The Rainbow at the End of the Line - A PPAD Formulation of the Colorful Carathéodory Theorem with applications. In Proc. of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1342-1351, 2017. Google Scholar
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