Skip to main content
Erschienen in: Dynamic Games and Applications 1/2023

31.01.2022

Continuous Time Learning Algorithms in Optimization and Game Theory

verfasst von: Sylvain Sorin

Erschienen in: Dynamic Games and Applications | Ausgabe 1/2023

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The purpose of this work is the comparison of learning algorithms in continuous time used in optimization and game theory. The first three are issued from no-regret dynamics and cover in particular “Replicator dynamics” and “Local projection dynamics”. Then we study “Conditional gradient” versus “Global projection” dynamics and finally “Frank-Wolfe” versus “Best reply” dynamics. Important similarities occur when considering potential or dissipative games.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Literatur
1.
Zurück zum Zitat Akin E (1979) The geometry of population genetics. Lecture notes in biomathematics, vol 31. Springer, BerlinCrossRef Akin E (1979) The geometry of population genetics. Lecture notes in biomathematics, vol 31. Springer, BerlinCrossRef
2.
Zurück zum Zitat Alvarez F, Bolte J, Brahic O (2004) Hessian Riemannian gradient flows in convex programming. SIAM J Control Optim 43:477–501MathSciNetMATHCrossRef Alvarez F, Bolte J, Brahic O (2004) Hessian Riemannian gradient flows in convex programming. SIAM J Control Optim 43:477–501MathSciNetMATHCrossRef
3.
Zurück zum Zitat Antipin AS (1994) Minimization of convex functions on convex sets by means of differential equations. Differ Equ 30:1365–1375MathSciNetMATH Antipin AS (1994) Minimization of convex functions on convex sets by means of differential equations. Differ Equ 30:1365–1375MathSciNetMATH
4.
Zurück zum Zitat Attouch H, Teboulle M (2004) Regularized Lotka-Volterra dynamical system as continuous proximal-like method in optimization. J Optim Theory Appl 121:541–570MathSciNetMATHCrossRef Attouch H, Teboulle M (2004) Regularized Lotka-Volterra dynamical system as continuous proximal-like method in optimization. J Optim Theory Appl 121:541–570MathSciNetMATHCrossRef
7.
Zurück zum Zitat Beck A, Teboulle M (2003) Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper Res Lett 31:167–175MathSciNetMATHCrossRef Beck A, Teboulle M (2003) Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper Res Lett 31:167–175MathSciNetMATHCrossRef
8.
Zurück zum Zitat Benaim M, Hofbauer J, Sorin S (2005) Stochastic approximations and differential inclusions. SIAM J Control Optim 44:328–348MathSciNetMATHCrossRef Benaim M, Hofbauer J, Sorin S (2005) Stochastic approximations and differential inclusions. SIAM J Control Optim 44:328–348MathSciNetMATHCrossRef
9.
Zurück zum Zitat Benaim M, Hofbauer J, Sorin S (2006) Stochastic approximations and differential inclusions. Part II: applications. Math Oper Res 31:673–695MathSciNetMATHCrossRef Benaim M, Hofbauer J, Sorin S (2006) Stochastic approximations and differential inclusions. Part II: applications. Math Oper Res 31:673–695MathSciNetMATHCrossRef
10.
Zurück zum Zitat Benaim M, Hofbauer J, Sorin S (2012) Perturbations of set-valued dynamical systems, with applications to game theory. Dyn Games Appl 2:195–205MathSciNetMATHCrossRef Benaim M, Hofbauer J, Sorin S (2012) Perturbations of set-valued dynamical systems, with applications to game theory. Dyn Games Appl 2:195–205MathSciNetMATHCrossRef
12.
Zurück zum Zitat Bolte J, Teboulle M (2003) Barrier operators and associated gradient-like dynamical systems for constrained minimization problems. SIAM J Control Optim 42:1266–1292MathSciNetMATHCrossRef Bolte J, Teboulle M (2003) Barrier operators and associated gradient-like dynamical systems for constrained minimization problems. SIAM J Control Optim 42:1266–1292MathSciNetMATHCrossRef
13.
Zurück zum Zitat Brézis H (1973) Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland, AmsterdamMATH Brézis H (1973) Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland, AmsterdamMATH
14.
Zurück zum Zitat Brown GW (1951) Iterative solutions of games by fictitious play. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, Hoboken, pp 374–376 Brown GW (1951) Iterative solutions of games by fictitious play. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, Hoboken, pp 374–376
15.
Zurück zum Zitat Brown GW, von Neumann J (1950) Solutions of games by differential equations. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, I. Annals of mathematical studies, vol 24. Princeton University Press, Princeton, pp 73–79 Brown GW, von Neumann J (1950) Solutions of games by differential equations. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, I. Annals of mathematical studies, vol 24. Princeton University Press, Princeton, pp 73–79
16.
19.
Zurück zum Zitat Facchinei F, Pang J (2007) Finite-dimensional variational inequalities and complementarity problems. Springer, BerlinMATH Facchinei F, Pang J (2007) Finite-dimensional variational inequalities and complementarity problems. Springer, BerlinMATH
22.
Zurück zum Zitat Friesz TL, Bernstein D, Mehta NJ, Tobin RL, Ganjalizadeh S (1994) Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper Res 42:1120–1136MathSciNetMATHCrossRef Friesz TL, Bernstein D, Mehta NJ, Tobin RL, Ganjalizadeh S (1994) Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper Res 42:1120–1136MathSciNetMATHCrossRef
24.
Zurück zum Zitat Hart S, Mas-Colell A (2003) Uncoupled dynamics do not lead to Nash equilibrium. Am Econ Rev 93:1830–1836CrossRef Hart S, Mas-Colell A (2003) Uncoupled dynamics do not lead to Nash equilibrium. Am Econ Rev 93:1830–1836CrossRef
26.
Zurück zum Zitat Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeMATHCrossRef Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeMATHCrossRef
27.
Zurück zum Zitat Hofbauer J, Sorin S (2006) Best response dynamics for continuous zero-sum games. Discrete Contin Dyn Syst Ser B 6:215–224MathSciNetMATH Hofbauer J, Sorin S (2006) Best response dynamics for continuous zero-sum games. Discrete Contin Dyn Syst Ser B 6:215–224MathSciNetMATH
28.
29.
Zurück zum Zitat Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic Press, LondonMATH Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic Press, LondonMATH
31.
32.
33.
34.
36.
Zurück zum Zitat Mertikopoulos P, Zhou Z (2019) Learning in games with continuous action sets and unknown payoff functions. Math Program 173:465–507MathSciNetMATHCrossRef Mertikopoulos P, Zhou Z (2019) Learning in games with continuous action sets and unknown payoff functions. Math Program 173:465–507MathSciNetMATHCrossRef
39.
40.
Zurück zum Zitat Moreau JJ (1965) Proximité et dualité dans un espace hilbertien. Bull Soc Math Fr 93:273–299MATHCrossRef Moreau JJ (1965) Proximité et dualité dans un espace hilbertien. Bull Soc Math Fr 93:273–299MATHCrossRef
43.
Zurück zum Zitat Nemirovski A, Yudin D (1983) Problem complexity and method efficiency in optimization. Wiley, Hoboken Nemirovski A, Yudin D (1983) Problem complexity and method efficiency in optimization. Wiley, Hoboken
46.
Zurück zum Zitat Opial Z (1967) Weak Convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc 73:591–597MathSciNetMATHCrossRef Opial Z (1967) Weak Convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc 73:591–597MathSciNetMATHCrossRef
48.
Zurück zum Zitat Polyak B (1987) Introduction to optimization. In: Optimization software Polyak B (1987) Introduction to optimization. In: Optimization software
49.
50.
Zurück zum Zitat Rockafellar RT (1970) Monotone operators associated with saddle-functions and minmax problems. In: Browder F (ed) Nonlinear functional analysis. Proceedings of symposia in pure math, vol 18. AMS, pp 241–250 Rockafellar RT (1970) Monotone operators associated with saddle-functions and minmax problems. In: Browder F (ed) Nonlinear functional analysis. Proceedings of symposia in pure math, vol 18. AMS, pp 241–250
51.
54.
Zurück zum Zitat Sandholm WH (2011) Population games and evolutionary dynamics. MIT Press, CambridgeMATH Sandholm WH (2011) Population games and evolutionary dynamics. MIT Press, CambridgeMATH
55.
Zurück zum Zitat Sandholm WH (2015) Population games and deterministic evolutionary dynamics. In: Young HP, Zamir S (eds) Handbook of game theory IV. Elsevier, Amsterdam, pp 703–778 Sandholm WH (2015) Population games and deterministic evolutionary dynamics. In: Young HP, Zamir S (eds) Handbook of game theory IV. Elsevier, Amsterdam, pp 703–778
56.
57.
Zurück zum Zitat Shahshahani S (1979) A new mathematical framework for the study of linkage and selection. In: Memoirs of the American Mathematical Society, vol 211 Shahshahani S (1979) A new mathematical framework for the study of linkage and selection. In: Memoirs of the American Mathematical Society, vol 211
58.
Zurück zum Zitat Smith MJ (1979) The existence, uniqueness and stability of traffic equilibria. Transp Res Part B 13:295–304MathSciNetCrossRef Smith MJ (1979) The existence, uniqueness and stability of traffic equilibria. Transp Res Part B 13:295–304MathSciNetCrossRef
60.
Zurück zum Zitat Sorin S (2011) On some global and unilateral adaptive dynamics. In: Sigmund K (ed) Evolutionary game dynamics. Proceedings of symposia in applied mathematics, vol 69. A.M.S., pp 81–109 Sorin S (2011) On some global and unilateral adaptive dynamics. In: Sigmund K (ed) Evolutionary game dynamics. Proceedings of symposia in applied mathematics, vol 69. A.M.S., pp 81–109
62.
Zurück zum Zitat Sorin S (2021) No-regret algorithms in on-line learning, games and convex optimization. In: Mathematical programming (to appear) Sorin S (2021) No-regret algorithms in on-line learning, games and convex optimization. In: Mathematical programming (to appear)
68.
Zurück zum Zitat Wardrop G (1952) Some theoretical aspects of road traffic research communication networks. Proc Inst Civ Eng Part 2 1:325–378 Wardrop G (1952) Some theoretical aspects of road traffic research communication networks. Proc Inst Civ Eng Part 2 1:325–378
Metadaten
Titel
Continuous Time Learning Algorithms in Optimization and Game Theory
verfasst von
Sylvain Sorin
Publikationsdatum
31.01.2022
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 1/2023
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00423-x

Weitere Artikel der Ausgabe 1/2023

Dynamic Games and Applications 1/2023 Zur Ausgabe

Premium Partner