# American Institute of Mathematical Sciences

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

## Approachability, regret and calibration: Implications and equivalences

 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

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

show all references

##### References:
 [1] Barak Shani, Eilon Solan. Strong approachability. Journal of Dynamics & Games, 2014, 1 (3) : 507-535. doi: 10.3934/jdg.2014.1.507 [2] 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 [3] Shie Mannor, Vianney Perchet, Gilles Stoltz. A primal condition for approachability with partial monitoring. Journal of Dynamics & Games, 2014, 1 (3) : 447-469. doi: 10.3934/jdg.2014.1.447 [4] 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 & Management Optimization, 2017, 13 (5) : 1-31. doi: 10.3934/jimo.2019025 [5] Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005 [6] Ying Ji, Shaojian Qu, Yeming Dai. A new approach for worst-case regret portfolio optimization problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 761-770. doi: 10.3934/dcdss.2019050 [7] Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249 [8] 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 [9] Saeid Abbasi-Parizi, Majid Aminnayeri, Mahdi Bashiri. Robust solution for a minimax regret hub location problem in a fuzzy-stochastic environment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1271-1295. doi: 10.3934/jimo.2018083 [10] Quanyi Liang, Kairong Liu, Gang Meng, Zhikun She. Minimization of the lowest eigenvalue for a vibrating beam. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2079-2092. doi: 10.3934/dcds.2018085 [11] Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic & Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010 [12] M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems & Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219 [13] María Andrea Arias Serna, María Eugenia Puerta Yepes, César Edinson Escalante Coterio, Gerardo Arango Ospina. $(Q,r)$ Model with $CVaR_α$ of costs minimization. Journal of Industrial & Management Optimization, 2017, 13 (1) : 135-146. doi: 10.3934/jimo.2016008 [14] Naoufel Ben Abdallah, Irene M. Gamba, Giuseppe Toscani. On the minimization problem of sub-linear convex functionals. Kinetic & Related Models, 2011, 4 (4) : 857-871. doi: 10.3934/krm.2011.4.857 [15] Xavier Dubois de La Sablonière, Benjamin Mauroy, Yannick Privat. Shape minimization of the dissipated energy in dyadic trees. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 767-799. doi: 10.3934/dcdsb.2011.16.767 [16] Tai Chiu Edwin Cheng, Bertrand Miao-Tsong Lin, Hsiao-Lan Huang. Talent hold cost minimization in film production. Journal of Industrial & Management Optimization, 2017, 13 (1) : 223-235. doi: 10.3934/jimo.2016013 [17] Zhi-Feng Pang, Yu-Fei Yang. Semismooth Newton method for minimization of the LLT model. Inverse Problems & Imaging, 2009, 3 (4) : 677-691. doi: 10.3934/ipi.2009.3.677 [18] Xiaoli Yang, Jin Liang, Bei Hu. Minimization of carbon abatement cost: Modeling, analysis and simulation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2939-2969. doi: 10.3934/dcdsb.2017158 [19] Hongwei Lou, Xueyuan Yin. Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors. Mathematical Control & Related Fields, 2018, 8 (3&4) : 855-877. doi: 10.3934/mcrf.2018038 [20] Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070

Impact Factor: