Approachability, regret and calibration: Implications and equivalences

Previous Article
Dynamics of large cooperative pulsed-coupled networks
JDG Home
This Issue
Next Article

April 2014, 1(2): 181-254. doi: 10.3934/jdg.2014.1.181

Approachability, regret and calibration: Implications and equivalences

Vianney Perchet ^1,

1.	Université Paris-Diderot, Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, 8 place FM/13, Paris, France

Received January 2013 Revised January 2014 Published March 2014

Abstract
Related Papers

Blackwell approachability, regret minimization and calibration are three criteria used to evaluate a strategy (or an algorithm) in sequential decision problems, described as repeated games between a player and Nature. Although they have at first sight not much in common, links between them have been discovered: for instance, both consistent and calibrated strategies can be constructed by following, in some auxiliary game, an approachability strategy.
We gather seminal and recent results, develop and generalize Blackwell's elegant theory in several directions. The final objectives is to show how approachability can be used as a basic powerful tool to exhibit a new class of intuitive algorithms, based on simple geometric properties. In order to be complete, we also prove that approachability can be seen as a byproduct of the very existence of consistent or calibrated strategies.

Keywords: Blackwell's approachability, calibration., regret minimization.

Mathematics Subject Classification: Primary: 91A25, 68W27; Secondary: 91A2.

Citation: Vianney Perchet. Approachability, regret and calibration: Implications and equivalences. Journal of Dynamics and Games, 2014, 1 (2) : 181-254. doi: 10.3934/jdg.2014.1.181

References:

[1]	J. Abernethy, P. Bartlett and E. Hazan, Blackwell approachability and low-regret learning are equivalent, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 27-46.
[2]	S. As Soulaimani, M. Quincampoix and S. Sorin, Repeated games and qualitative differential games: Approachability and comparison of strategies, SIAM J. Control Optim., 48 (2009), 2461-2479. doi: 10.1137/090749098.
[3]	P. Auer, N. Cesa-Bianchi and C. Gentile, Adaptive and self-confident on-line learning algorithms, J. Comput. System Sci., 64 (2002), 48-75, Special issue on COLT 2000 (Palo Alto, CA). doi: 10.1006/jcss.2001.1795.
[4]	R. J. Aumann, Subjectivity and correlation in randomized strategies, J. Math. Econom., 1 (1974), 67-96. doi: 10.1016/0304-4068(74)90037-8.
[5]	R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, MA, 1995, With the collaboration of Richard E. Stearns.
[6]	M. Benaïm and M. Faure, Consistency of vanishingly smooth fictitious play, Math. Oper. Res., 38 (2013), 437-450. doi: 10.1287/moor.1120.0568.
[7]	M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions. II. Applications, Math. Oper. Res., 31 (2006), 673-695. doi: 10.1287/moor.1060.0213.
[8]	A. Bernstein, S. Mannor and N. Shimkin, Opportunistic strategies for generalized no-regret problems, J. Mach. Learn. Res.: Workshop Conf. Proc., 30 (2013), 158-171.
[9]	A. Bernstein and N. Shimkin, Response-based approachability and its application to generalized no-regret algorithms, Manuscript.
[10]	L. J. Billera and B. Sturmfels, Fiber polytopes, The Annals of Mathematics, 135 (1992), 527-549. doi: 10.2307/2946575.
[11]	D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific J. Math., 6 (1956), 1-8. doi: 10.2140/pjm.1956.6.1.
[12]	D. Blackwell, Controlled random walks, in Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, 1956, 336-338.
[13]	D. Blackwell and M. A. Girshick, Theory of Games and Statistical Decisions, John Wiley and Sons, Inc., New York, 1954.
[14]	A. Blum and Y. Mansour, From external to internal regret, in Learning theory, vol. 3559 of Lecture Notes in Comput. Sci., Springer, Berlin, (2005), 621-636. doi: 10.1007/11503415_42.
[15]	S. Bubeck, Introduction to online optimization, Manuscript.
[16]	N. Cesa-Bianchi and G. Lugosi, Potential-based algorithms in on-line prediction and game theory, Machine Learning, 51 (2003), 239-261.
[17]	N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511546921.
[18]	X. Chen and H. White, Laws of large numbers for Hilbert space-valued mixingales with applications, Econometric Theory, 12 (1996), 284-304. doi: 10.1017/S0266466600006599.
[19]	A. P. Dawid, The well-calibrated Bayesian, J. Amer. Statist. Assoc., 77 (1982), 605-613. doi: 10.1080/01621459.1982.10477856.
[20]	A. P. Dawid, Self-calibrating priors do not exist: Comment, J. Amer. Statist. Assoc., 80 (1985), 339-342. doi: 10.1080/01621459.1985.10478117.
[21]	L. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi equations, Indiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040.
[22]	K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42.
[23]	K. Fan, A minimax inequality and applications, in Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, (1972), 103-113.
[24]	W. Feller, An Introduction to Probability Theory and its Applications. Vol. I, Third edition, John Wiley & Sons Inc., New York, 1968.
[25]	D. P. Foster, A proof of calibration via blackwell's approachability theorem, Games and Economic Behavior, 29 (1999), 73-78. doi: 10.1006/game.1999.0719.
[26]	D. P. Foster, A. Rakhlin, K. Sridharan and A. Tewari, Complexity-based approach to calibration with checking rules, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 293-314.
[27]	D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Games Econom. Behav., 21 (1997), 40-55. doi: 10.1006/game.1997.0595.
[28]	D. P. Foster and R. V. Vohra, Asymptotic calibration, Biometrika, 85 (1998), 379-390. doi: 10.1093/biomet/85.2.379.
[29]	D. P. Foster and R. V. Vohra, Regret in the on-line decision problem, Games Econom. Behav., 29 (1999), 7-35. doi: 10.1006/game.1999.0740.
[30]	D. P. Foster and R. V. Vohra, Calibration: Respice, adspice, prospice, Advances in Economics and Econometrics, 1 (2013), 423-442. doi: 10.1017/CBO9781139060011.014.
[31]	D. Fudenberg and D. M. Kreps, Learning mixed equilibria, Games Econom. Behav., 5 (1993), 320-367. doi: 10.1006/game.1993.1021.
[32]	D. Fudenberg and D. K. Levine, Conditional universal consistency, Games Econom. Behav., 29 (1999), 104-130. doi: 10.1006/game.1998.0705.
[33]	D. Fudenberg and D. K. Levine, An easier way to calibrate, Games Econom. Behav., 29 (1999), 131-137. doi: 10.1006/game.1999.0726.
[34]	F. Gul, D. Pearce and E. Stachetti, A bound on the proportion of pure strategy equilibria in generic games, Math. Oper. Res., 18 (1993), 548-552. doi: 10.1287/moor.18.3.548.
[35]	P. Hall and C. C. Heyde, Martingale Limit Theory and its Application, Academic Press Inc., New York, 1980, Probability and Mathematical Statistics.
[36]	J. Hannan, Approximation to bayes risk in repeated play, in Contributions to the Theory of Games, vol. III of Annals of Mathematics Studies 39, Princeton University Press, Princeton, N. J., 1957, 97-139.
[37]	S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica, 68 (2000), 1127-1150. doi: 10.1111/1468-0262.00153.
[38]	S. Hart and A. Mas-Colell, A general class of adaptive strategies, J. Econom. Theory, 98 (2001), 26-54. doi: 10.1006/jeth.2000.2746.
[39]	S. Hart and A. Mas-Colell, Regret-based continuous-time dynamics, Games Econom. Behav., 45 (2003), 375-394. doi: 10.1016/S0899-8256(03)00178-7.
[40]	E. Hazan and S. M. Kakade, (weak) calibration is computationaly hard, J. Mach. Learn. Res.: Workshop Conf. Proc., 23 (2012), 3.1-3.10.
[41]	J. Hofbauer and W. H. Sandholm, On the global convergence of stochastic fictitious play, Econometrica, 70 (2002), 2265-2294. doi: 10.1111/1468-0262.00376.
[42]	J. Hofbauer, S. Sorin and Y. Viossat, Time average replicator and best-reply dynamics, Math. Oper. Res., 34 (2009), 263-269. doi: 10.1287/moor.1080.0359.
[43]	S. M. Kakade and D. P. Foster, Deterministic calibration and Nash equilibrium, in Learning theory, vol. 3120 of Lecture Notes in Comput. Sci., Springer, Berlin, (2004), 33-48. doi: 10.1007/978-3-540-27819-1_3.
[44]	O. Kallenberg and R. Sztencel, Some dimension-free features of vector-valued martingales, Probability Theory and Related Fields, 88 (1991), 215-247. doi: 10.1007/BF01212560.
[45]	E. Kohlberg, Optimal strategies in repeated games with incomplete information, Internat. J. Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399.
[46]	J. Kwon, Hilbert Distance, Bounded Convex Functions, and Application to the Exponential Weight Algorithm, Master's thesis, ENS Lyon, 2012.
[47]	E. Lehrer, Any inspection is manipulable, Econometrica, 69 (2001), 1333-1347. doi: 10.1111/1468-0262.00244.
[48]	E. Lehrer, Approachability in infinite dimensional spaces, Internat. J. Game Theory, 31 (2002), 253-268. doi: 10.1007/s001820200115.
[49]	E. Lehrer, A wide range no-regret theorem, Games Econom. Behav., 42 (2003), 101-115. doi: 10.1016/S0899-8256(03)00032-0.
[50]	E. Lehrer and E. Solan, Excludability and bounded computational capacity, Math. Oper. Res., 31 (2006), 637-648. doi: 10.1287/moor.1060.0211.
[51]	E. Lehrer and E. Solan, Learning to play partially-specified equilibrium, Manuscript.
[52]	E. Lehrer and E. Solan, Approachability with bounded memory, Games Econom. Behav., 66 (2009), 995-1004. doi: 10.1016/j.geb.2007.07.011.
[53]	E. Lehrer and S. Sorin, Minmax via differential inclusion, J. Conv. Analysis, 14 (2007), 271-273.
[54]	N. Littlestone and M. Warmuth, The weighted majority algorithm, Information and Computation, 108 (1994), 212-261. doi: 10.1006/inco.1994.1009.
[55]	R. D. Luce and H. Raiffa, Games and Decisions: Introduction and Critical Survey, John Wiley & Sons Inc., New York, N. Y., 1957.
[56]	S. Mannor and N. Shimkin, Regret minimization in repeated matrix games with variable stage duration, Games Econom. Behav., 63 (2008), 227-258. doi: 10.1016/j.geb.2007.07.006.
[57]	S. Mannor and G. Stoltz, A geometric proof of calibration, Math. Oper. Res., 35 (2010), 721-727. doi: 10.1287/moor.1100.0465.
[58]	S. Mannor, G. Stoltz and V. Perchet, Robust approachability and regret minimization in games with partial monitoring, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 515-536.
[59]	S. Mannor, G. Stoltz and V. Perchet, Set-valued approachability, with applications to regret minimization in games with partial monitoring, Manuscript.
[60]	S. Mannor and J. N. Tsitsiklis, Approachability in repeated games: Computational aspects and a Stackelberg variant, Games Econom. Behav., 66 (2009), 315-325. doi: 10.1016/j.geb.2008.03.008.
[61]	D. McFadden, Conditional logit analysis of qualitative choice behavior, Frontiers in econometrics, Academic Press: New York, 1974, 105-142.
[62]	J.-F. Mertens, S. Sorin and S. Zamir, Repeated games, CORE discussion paper, (1994), 9420-9422. doi: 10.1057/9780230226203.3424.
[63]	J. Neveu, Bases Mathématiques du Calcul des Probabilités, Masson et Cie, éditeurs, Paris, 1970.
[64]	J. Neveu, Martingales à Temps Discret, Masson et Cie, éditeurs, Paris, 1972.
[65]	D. Oakes, Self-calibrating priors do not exist, J. Amer. Statist. Assoc., 80 (1985), 339-342, With comments by A. P. Dawid and Mark J. Schervish. doi: 10.1080/01621459.1985.10478117.
[66]	W. Olszewski, Calibration and expert testing, in Handbook of Game Theory (eds. P. Young and S. Zamir), vol. IV, to appear.
[67]	V. Perchet, Calibration and internal no-regret with random signals, Algorithmic learning theory, 68-82, Lecture Notes in Comput. Sci., 5809, Springer, Berlin, 2009. doi: 10.1007/978-3-642-04414-4_10.
[68]	V. Perchet, Approchabilité, Calibration et Regret dans les Jeux à Observations Partielles (in French), PhD thesis, Université Pierre et Marie Curie, 2010.
[69]	V. Perchet, Approachability of convex sets in games with partial monitoring, J. Optim. Theory Appl., 149 (2011), 665-677. doi: 10.1007/s10957-011-9797-3.
[70]	V. Perchet, No-regret with partial monitoring: Calibration-based optimal algorithms, J. Mach. Learn. Res., 12 (2011), 1893-1921.
[71]	V. Perchet, Exponential weights approachability, applications to regret minimization and calibration, Manuscript.
[72]	V. Perchet and M. Quincampoix, Purely informative game: Application to approachability with partial monitoring, Manuscript.
[73]	A. Rakhlin, Lecture notes on online learning, Manuscript.
[74]	A. Rakhlin, K. Sridharan and A. Tewari, Online Learning: Random Averages, Combinatorial Parameters, and Learnability, in Neural Information Processing Systems, 2010.
[75]	A. Rakhlin, K. Sridharan and A. Tewari, Online learning: Beyond regret, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 559-594.
[76]	W. Rudin, Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics, New York, 1974.
[77]	A. Sandroni, R. Smorodinsky and R. V. Vohra, Calibration with many checking rules, Math. Oper. Res., 28 (2003), 141-153. doi: 10.1287/moor.28.1.141.14264.
[78]	E. Seneta, Nonnegative Matrices and Markov Chains, 2nd edition, Springer Series in Statistics, Springer-Verlag, New York, 1981.
[79]	S. Sorin, A First Course on Zero-Sum Repeated Games, Springer-Verlag, 2002.
[80]	S. Sorin, Lectures on Dynamics in Games, Unpublished Lecture Notes, 2008.
[81]	S. Sorin, Exponential weight algorithm in continuous time, Math. Program., 116 (2009), 513-528. doi: 10.1007/s10107-007-0111-y.
[82]	X. Spinat, A necessary and sufficient condition for approachability, Math. Oper. Res., 27 (2002), 31-44. doi: 10.1287/moor.27.1.31.333.
[83]	G. Stoltz, Incomplete Information and Internal Regret in Prediction of Individual Sequences, PhD thesis, Université Paris-Sud, 2005.
[84]	G. Stoltz and G. Lugosi, Internal regret in on-line portfolio selection, Mach. Learn., 59 (2005), 125-159. doi: 10.1007/s10994-005-0465-4.
[85]	G. Stoltz and G. Lugosi, Learning correlated equilibria in games with compact sets of strategies, Games Econom. Behav., 59 (2007), 187-208. doi: 10.1016/j.geb.2006.04.007.
[86]	N. Vieille, Weak approachability, Math. Oper. Res., 17 (1992), 781-791. doi: 10.1287/moor.17.4.781.
[87]	Y. Viossat and A. Zapechelnyuk, No-regret dynamics and fictitious play, J. Econom. Theory, 148 (2013), 825-842. doi: 10.1016/j.jet.2012.07.003.
[88]	V. Vovk, Aggregating strategies, in Proceedings of the 3rd Annual Workshop on Computational Learning Theory, 1990, 372-383.
[89]	V. Vovk, I. Nouretdinov, A. Takemura and G. Shafer, Defensive forecasting for linear protocols, in Algorithmic Learning Theory, vol. 3734 of Lecture Notes in Comput. Sci., Springer, Berlin, (2005), 459-473. doi: 10.1007/11564089_35.
[90]	D. Walkup and R. J.-B. Wets, A lipschitzian characterization of convex polyhedra, Proceedings of the American Mathematical Society, 23 (1969), 167-173. doi: 10.1090/S0002-9939-1969-0246200-8.
[91]	A. Zapechelnyuk, Better-reply dynamics with bounded recall, Math. Oper. Res., 33 (2008), 869-879. doi: 10.1287/moor.1080.0323.
[92]	M. Zinkevich, Online convex programming and generalized infinitesimal gradient ascent, in Proceedings of the Twentieth International Conference on Machine Learning (ICML), 2003.

show all references

References:

[1]	J. Abernethy, P. Bartlett and E. Hazan, Blackwell approachability and low-regret learning are equivalent, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 27-46.
[2]	S. As Soulaimani, M. Quincampoix and S. Sorin, Repeated games and qualitative differential games: Approachability and comparison of strategies, SIAM J. Control Optim., 48 (2009), 2461-2479. doi: 10.1137/090749098.
[3]	P. Auer, N. Cesa-Bianchi and C. Gentile, Adaptive and self-confident on-line learning algorithms, J. Comput. System Sci., 64 (2002), 48-75, Special issue on COLT 2000 (Palo Alto, CA). doi: 10.1006/jcss.2001.1795.
[4]	R. J. Aumann, Subjectivity and correlation in randomized strategies, J. Math. Econom., 1 (1974), 67-96. doi: 10.1016/0304-4068(74)90037-8.
[5]	R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, MA, 1995, With the collaboration of Richard E. Stearns.
[6]	M. Benaïm and M. Faure, Consistency of vanishingly smooth fictitious play, Math. Oper. Res., 38 (2013), 437-450. doi: 10.1287/moor.1120.0568.
[7]	M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions. II. Applications, Math. Oper. Res., 31 (2006), 673-695. doi: 10.1287/moor.1060.0213.
[8]	A. Bernstein, S. Mannor and N. Shimkin, Opportunistic strategies for generalized no-regret problems, J. Mach. Learn. Res.: Workshop Conf. Proc., 30 (2013), 158-171.
[9]	A. Bernstein and N. Shimkin, Response-based approachability and its application to generalized no-regret algorithms, Manuscript.
[10]	L. J. Billera and B. Sturmfels, Fiber polytopes, The Annals of Mathematics, 135 (1992), 527-549. doi: 10.2307/2946575.
[11]	D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific J. Math., 6 (1956), 1-8. doi: 10.2140/pjm.1956.6.1.
[12]	D. Blackwell, Controlled random walks, in Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, 1956, 336-338.
[13]	D. Blackwell and M. A. Girshick, Theory of Games and Statistical Decisions, John Wiley and Sons, Inc., New York, 1954.
[14]	A. Blum and Y. Mansour, From external to internal regret, in Learning theory, vol. 3559 of Lecture Notes in Comput. Sci., Springer, Berlin, (2005), 621-636. doi: 10.1007/11503415_42.
[15]	S. Bubeck, Introduction to online optimization, Manuscript.
[16]	N. Cesa-Bianchi and G. Lugosi, Potential-based algorithms in on-line prediction and game theory, Machine Learning, 51 (2003), 239-261.
[17]	N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511546921.
[18]	X. Chen and H. White, Laws of large numbers for Hilbert space-valued mixingales with applications, Econometric Theory, 12 (1996), 284-304. doi: 10.1017/S0266466600006599.
[19]	A. P. Dawid, The well-calibrated Bayesian, J. Amer. Statist. Assoc., 77 (1982), 605-613. doi: 10.1080/01621459.1982.10477856.
[20]	A. P. Dawid, Self-calibrating priors do not exist: Comment, J. Amer. Statist. Assoc., 80 (1985), 339-342. doi: 10.1080/01621459.1985.10478117.
[21]	L. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi equations, Indiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040.
[22]	K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42.
[23]	K. Fan, A minimax inequality and applications, in Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, (1972), 103-113.
[24]	W. Feller, An Introduction to Probability Theory and its Applications. Vol. I, Third edition, John Wiley & Sons Inc., New York, 1968.
[25]	D. P. Foster, A proof of calibration via blackwell's approachability theorem, Games and Economic Behavior, 29 (1999), 73-78. doi: 10.1006/game.1999.0719.
[26]	D. P. Foster, A. Rakhlin, K. Sridharan and A. Tewari, Complexity-based approach to calibration with checking rules, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 293-314.
[27]	D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Games Econom. Behav., 21 (1997), 40-55. doi: 10.1006/game.1997.0595.
[28]	D. P. Foster and R. V. Vohra, Asymptotic calibration, Biometrika, 85 (1998), 379-390. doi: 10.1093/biomet/85.2.379.
[29]	D. P. Foster and R. V. Vohra, Regret in the on-line decision problem, Games Econom. Behav., 29 (1999), 7-35. doi: 10.1006/game.1999.0740.
[30]	D. P. Foster and R. V. Vohra, Calibration: Respice, adspice, prospice, Advances in Economics and Econometrics, 1 (2013), 423-442. doi: 10.1017/CBO9781139060011.014.
[31]	D. Fudenberg and D. M. Kreps, Learning mixed equilibria, Games Econom. Behav., 5 (1993), 320-367. doi: 10.1006/game.1993.1021.
[32]	D. Fudenberg and D. K. Levine, Conditional universal consistency, Games Econom. Behav., 29 (1999), 104-130. doi: 10.1006/game.1998.0705.
[33]	D. Fudenberg and D. K. Levine, An easier way to calibrate, Games Econom. Behav., 29 (1999), 131-137. doi: 10.1006/game.1999.0726.
[34]	F. Gul, D. Pearce and E. Stachetti, A bound on the proportion of pure strategy equilibria in generic games, Math. Oper. Res., 18 (1993), 548-552. doi: 10.1287/moor.18.3.548.
[35]	P. Hall and C. C. Heyde, Martingale Limit Theory and its Application, Academic Press Inc., New York, 1980, Probability and Mathematical Statistics.
[36]	J. Hannan, Approximation to bayes risk in repeated play, in Contributions to the Theory of Games, vol. III of Annals of Mathematics Studies 39, Princeton University Press, Princeton, N. J., 1957, 97-139.
[37]	S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica, 68 (2000), 1127-1150. doi: 10.1111/1468-0262.00153.
[38]	S. Hart and A. Mas-Colell, A general class of adaptive strategies, J. Econom. Theory, 98 (2001), 26-54. doi: 10.1006/jeth.2000.2746.
[39]	S. Hart and A. Mas-Colell, Regret-based continuous-time dynamics, Games Econom. Behav., 45 (2003), 375-394. doi: 10.1016/S0899-8256(03)00178-7.
[40]	E. Hazan and S. M. Kakade, (weak) calibration is computationaly hard, J. Mach. Learn. Res.: Workshop Conf. Proc., 23 (2012), 3.1-3.10.
[41]	J. Hofbauer and W. H. Sandholm, On the global convergence of stochastic fictitious play, Econometrica, 70 (2002), 2265-2294. doi: 10.1111/1468-0262.00376.
[42]	J. Hofbauer, S. Sorin and Y. Viossat, Time average replicator and best-reply dynamics, Math. Oper. Res., 34 (2009), 263-269. doi: 10.1287/moor.1080.0359.
[43]	S. M. Kakade and D. P. Foster, Deterministic calibration and Nash equilibrium, in Learning theory, vol. 3120 of Lecture Notes in Comput. Sci., Springer, Berlin, (2004), 33-48. doi: 10.1007/978-3-540-27819-1_3.
[44]	O. Kallenberg and R. Sztencel, Some dimension-free features of vector-valued martingales, Probability Theory and Related Fields, 88 (1991), 215-247. doi: 10.1007/BF01212560.
[45]	E. Kohlberg, Optimal strategies in repeated games with incomplete information, Internat. J. Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399.
[46]	J. Kwon, Hilbert Distance, Bounded Convex Functions, and Application to the Exponential Weight Algorithm, Master's thesis, ENS Lyon, 2012.
[47]	E. Lehrer, Any inspection is manipulable, Econometrica, 69 (2001), 1333-1347. doi: 10.1111/1468-0262.00244.
[48]	E. Lehrer, Approachability in infinite dimensional spaces, Internat. J. Game Theory, 31 (2002), 253-268. doi: 10.1007/s001820200115.
[49]	E. Lehrer, A wide range no-regret theorem, Games Econom. Behav., 42 (2003), 101-115. doi: 10.1016/S0899-8256(03)00032-0.
[50]	E. Lehrer and E. Solan, Excludability and bounded computational capacity, Math. Oper. Res., 31 (2006), 637-648. doi: 10.1287/moor.1060.0211.
[51]	E. Lehrer and E. Solan, Learning to play partially-specified equilibrium, Manuscript.
[52]	E. Lehrer and E. Solan, Approachability with bounded memory, Games Econom. Behav., 66 (2009), 995-1004. doi: 10.1016/j.geb.2007.07.011.
[53]	E. Lehrer and S. Sorin, Minmax via differential inclusion, J. Conv. Analysis, 14 (2007), 271-273.
[54]	N. Littlestone and M. Warmuth, The weighted majority algorithm, Information and Computation, 108 (1994), 212-261. doi: 10.1006/inco.1994.1009.
[55]	R. D. Luce and H. Raiffa, Games and Decisions: Introduction and Critical Survey, John Wiley & Sons Inc., New York, N. Y., 1957.
[56]	S. Mannor and N. Shimkin, Regret minimization in repeated matrix games with variable stage duration, Games Econom. Behav., 63 (2008), 227-258. doi: 10.1016/j.geb.2007.07.006.
[57]	S. Mannor and G. Stoltz, A geometric proof of calibration, Math. Oper. Res., 35 (2010), 721-727. doi: 10.1287/moor.1100.0465.
[58]	S. Mannor, G. Stoltz and V. Perchet, Robust approachability and regret minimization in games with partial monitoring, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 515-536.
[59]	S. Mannor, G. Stoltz and V. Perchet, Set-valued approachability, with applications to regret minimization in games with partial monitoring, Manuscript.
[60]	S. Mannor and J. N. Tsitsiklis, Approachability in repeated games: Computational aspects and a Stackelberg variant, Games Econom. Behav., 66 (2009), 315-325. doi: 10.1016/j.geb.2008.03.008.
[61]	D. McFadden, Conditional logit analysis of qualitative choice behavior, Frontiers in econometrics, Academic Press: New York, 1974, 105-142.
[62]	J.-F. Mertens, S. Sorin and S. Zamir, Repeated games, CORE discussion paper, (1994), 9420-9422. doi: 10.1057/9780230226203.3424.
[63]	J. Neveu, Bases Mathématiques du Calcul des Probabilités, Masson et Cie, éditeurs, Paris, 1970.
[64]	J. Neveu, Martingales à Temps Discret, Masson et Cie, éditeurs, Paris, 1972.
[65]	D. Oakes, Self-calibrating priors do not exist, J. Amer. Statist. Assoc., 80 (1985), 339-342, With comments by A. P. Dawid and Mark J. Schervish. doi: 10.1080/01621459.1985.10478117.
[66]	W. Olszewski, Calibration and expert testing, in Handbook of Game Theory (eds. P. Young and S. Zamir), vol. IV, to appear.
[67]	V. Perchet, Calibration and internal no-regret with random signals, Algorithmic learning theory, 68-82, Lecture Notes in Comput. Sci., 5809, Springer, Berlin, 2009. doi: 10.1007/978-3-642-04414-4_10.
[68]	V. Perchet, Approchabilité, Calibration et Regret dans les Jeux à Observations Partielles (in French), PhD thesis, Université Pierre et Marie Curie, 2010.
[69]	V. Perchet, Approachability of convex sets in games with partial monitoring, J. Optim. Theory Appl., 149 (2011), 665-677. doi: 10.1007/s10957-011-9797-3.
[70]	V. Perchet, No-regret with partial monitoring: Calibration-based optimal algorithms, J. Mach. Learn. Res., 12 (2011), 1893-1921.
[71]	V. Perchet, Exponential weights approachability, applications to regret minimization and calibration, Manuscript.
[72]	V. Perchet and M. Quincampoix, Purely informative game: Application to approachability with partial monitoring, Manuscript.
[73]	A. Rakhlin, Lecture notes on online learning, Manuscript.
[74]	A. Rakhlin, K. Sridharan and A. Tewari, Online Learning: Random Averages, Combinatorial Parameters, and Learnability, in Neural Information Processing Systems, 2010.
[75]	A. Rakhlin, K. Sridharan and A. Tewari, Online learning: Beyond regret, J. Mach. Learn. Res.: Workshop Conf. Proc., 19 (2011), 559-594.
[76]	W. Rudin, Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics, New York, 1974.
[77]	A. Sandroni, R. Smorodinsky and R. V. Vohra, Calibration with many checking rules, Math. Oper. Res., 28 (2003), 141-153. doi: 10.1287/moor.28.1.141.14264.
[78]	E. Seneta, Nonnegative Matrices and Markov Chains, 2nd edition, Springer Series in Statistics, Springer-Verlag, New York, 1981.
[79]	S. Sorin, A First Course on Zero-Sum Repeated Games, Springer-Verlag, 2002.
[80]	S. Sorin, Lectures on Dynamics in Games, Unpublished Lecture Notes, 2008.
[81]	S. Sorin, Exponential weight algorithm in continuous time, Math. Program., 116 (2009), 513-528. doi: 10.1007/s10107-007-0111-y.
[82]	X. Spinat, A necessary and sufficient condition for approachability, Math. Oper. Res., 27 (2002), 31-44. doi: 10.1287/moor.27.1.31.333.
[83]	G. Stoltz, Incomplete Information and Internal Regret in Prediction of Individual Sequences, PhD thesis, Université Paris-Sud, 2005.
[84]	G. Stoltz and G. Lugosi, Internal regret in on-line portfolio selection, Mach. Learn., 59 (2005), 125-159. doi: 10.1007/s10994-005-0465-4.
[85]	G. Stoltz and G. Lugosi, Learning correlated equilibria in games with compact sets of strategies, Games Econom. Behav., 59 (2007), 187-208. doi: 10.1016/j.geb.2006.04.007.
[86]	N. Vieille, Weak approachability, Math. Oper. Res., 17 (1992), 781-791. doi: 10.1287/moor.17.4.781.
[87]	Y. Viossat and A. Zapechelnyuk, No-regret dynamics and fictitious play, J. Econom. Theory, 148 (2013), 825-842. doi: 10.1016/j.jet.2012.07.003.
[88]	V. Vovk, Aggregating strategies, in Proceedings of the 3rd Annual Workshop on Computational Learning Theory, 1990, 372-383.
[89]	V. Vovk, I. Nouretdinov, A. Takemura and G. Shafer, Defensive forecasting for linear protocols, in Algorithmic Learning Theory, vol. 3734 of Lecture Notes in Comput. Sci., Springer, Berlin, (2005), 459-473. doi: 10.1007/11564089_35.
[90]	D. Walkup and R. J.-B. Wets, A lipschitzian characterization of convex polyhedra, Proceedings of the American Mathematical Society, 23 (1969), 167-173. doi: 10.1090/S0002-9939-1969-0246200-8.
[91]	A. Zapechelnyuk, Better-reply dynamics with bounded recall, Math. Oper. Res., 33 (2008), 869-879. doi: 10.1287/moor.1080.0323.
[92]	M. Zinkevich, Online convex programming and generalized infinitesimal gradient ascent, in Proceedings of the Twentieth International Conference on Machine Learning (ICML), 2003.

[1]	Barak Shani, Eilon Solan. Strong approachability. Journal of Dynamics and Games, 2014, 1 (3) : 507-535. doi: 10.3934/jdg.2014.1.507
[2]	Dario Bauso, Thomas W. L. Norman. Approachability in population games. Journal of Dynamics and Games, 2020, 7 (4) : 269-289. doi: 10.3934/jdg.2020019
[3]	Richard Tapia. My reflections on the Blackwell-Tapia prize. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1669-1672. doi: 10.3934/mbe.2013.10.1669
[4]	Shie Mannor, Vianney Perchet, Gilles Stoltz. A primal condition for approachability with partial monitoring. Journal of Dynamics and Games, 2014, 1 (3) : 447-469. doi: 10.3934/jdg.2014.1.447
[5]	Wenxiu Gong, Zuoliang Xu. An alternative tree method for calibration of the local volatility. Journal of Industrial and Management Optimization, 2022, 18 (1) : 137-156. doi: 10.3934/jimo.2020146
[6]	Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics and Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
[7]	Ying Ji, Shaojian Qu, Yeming Dai. A new approach for worst-case regret portfolio optimization problem. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 761-770. doi: 10.3934/dcdss.2019050
[8]	Laurent Devineau, Pierre-Edouard Arrouy, Paul Bonnefoy, Alexandre Boumezoued. Fast calibration of the Libor market model with stochastic volatility and displaced diffusion. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1699-1729. doi: 10.3934/jimo.2019025
[9]	Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial and Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[10]	Gabriel Jouan, Anne Cuzol, Valérie Monbet, Goulven Monnier. Gaussian mixture models for clustering and calibration of ensemble weather forecasts. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022037
[11]	Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249
[12]	Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 2 (2) : 107-126. doi: 10.3934/mfc.2019009
[13]	Saeid Abbasi-Parizi, Majid Aminnayeri, Mahdi Bashiri. Robust solution for a minimax regret hub location problem in a fuzzy-stochastic environment. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1271-1295. doi: 10.3934/jimo.2018083
[14]	Yan Zhou, Chi Kin Chan, Kar Hung Wong. The impacts of retailers' regret aversion on a random multi-period supply chain network. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2651-2675. doi: 10.3934/jimo.2021086
[15]	Quanyi Liang, Kairong Liu, Gang Meng, Zhikun She. Minimization of the lowest eigenvalue for a vibrating beam. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2079-2092. doi: 10.3934/dcds.2018085
[16]	Hongwei Lou, Xueyuan Yin. Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors. Mathematical Control and Related Fields, 2018, 8 (3&4) : 855-877. doi: 10.3934/mcrf.2018038
[17]	Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic and Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010
[18]	Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[19]	Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1757-1772. doi: 10.3934/dcdss.2020103
[20]	M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems and Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Vianney Perchet
on Google Scholar
Vianney Perchet

[Back to Top]