Supermartingales in prediction with expert advice

This paper compares two methods of prediction with expert advice, the Aggregating Algorithm and the Defensive Forecasting, in two different settings. The first setting is traditional, with a countable number of experts and a finite number of outcomes. Surprisingly, these two methods of fundamentally different origin lead to identical procedures. In the second setting the experts can give advice conditional on the learner's future decision. Both methods can be used in the new setting and give the same performance guarantees as in the traditional setting. However, whereas defensive forecasting can be applied directly, the AA requires substantial modifications.

[1]  L. J. Savage Elicitation of Personal Probabilities and Expectations , 1971 .

[2]  Dean P. Foster,et al.  Regret in the On-Line Decision Problem , 1999 .

[3]  A. Dawid The geometry of proper scoring rules , 2007 .

[4]  W. Hoeffding Probability Inequalities for sums of Bounded Random Variables , 1963 .

[5]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1997, Texts in Computer Science.

[6]  G. Shafer,et al.  Probability and Finance: It's Only a Game! , 2001 .

[7]  M. A. Girshick,et al.  Theory of games and statistical decisions , 1955 .

[8]  Gábor Lugosi,et al.  Learning correlated equilibria in games with compact sets of strategies , 2007, Games Econ. Behav..

[9]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

[10]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[11]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[12]  Vladimir Vovk,et al.  Prediction with expert advice for the Brier game , 2007, ICML '08.

[13]  M. Shubik,et al.  Theory of Games and Statistical Decisions. , 1955 .

[14]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[15]  D. Fudenberg,et al.  Conditional Universal Consistency , 1999 .

[16]  Vladimir Vovk,et al.  Loss functions, complexities, and the Legendre transformation , 2001, Theor. Comput. Sci..

[17]  Vladimir Vovk,et al.  Prediction with Expert Evaluators' Advice , 2009, ALT.

[18]  Ravi P. Agarwal,et al.  Fixed Point Theory and Applications: Contractions , 2001 .

[19]  Yishay Mansour,et al.  From External to Internal Regret , 2005, J. Mach. Learn. Res..

[20]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[21]  Vladimir Vovk,et al.  Metric entropy in competitive on-line prediction , 2006, ArXiv.

[22]  Vladimir Vovk,et al.  Continuous and randomized defensive forecasting: unified view , 2007, ArXiv.

[23]  A. Dawid,et al.  Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory , 2004, math/0410076.

[24]  Gábor Lugosi,et al.  Prediction, learning, and games , 2006 .

[25]  Vladimir Vovk Defensive Prediction with Expert Advice , 2005, ALT.

[26]  Vladimir Vovk,et al.  Aggregating strategies , 1990, COLT '90.

[27]  Vladimir Vovk,et al.  A game of prediction with expert advice , 1995, COLT '95.

[28]  Vladimir Vovk,et al.  Defensive forecasting for optimal prediction with expert advice , 2007, ArXiv.

[29]  F. Delbaen Probability and Finance: It's Only a Game! , 2002 .

[30]  Gábor Lugosi,et al.  Internal Regret in On-Line Portfolio Selection , 2005, Machine Learning.

[31]  Vladimir Vovk,et al.  On-Line Regression Competitive with Reproducing Kernel Hilbert Spaces , 2005, TAMC.

[32]  Péter Gács,et al.  Uniform test of algorithmic randomness over a general space , 2003, Theor. Comput. Sci..

[33]  Vladimir Vovk Competitive on-line learning with a convex loss function , 2005, ArXiv.

[34]  Vladimir Vovk,et al.  Supermartingales in prediction with expert advice , 2010 .

[35]  David Haussler,et al.  Sequential Prediction of Individual Sequences Under General Loss Functions , 1998, IEEE Trans. Inf. Theory.