Online Learning: Beyond Regret

We study online learnability of a wide class of problems, extending the results of Rakhlin et al. (2010a) to general notions of performance measure well beyond external regret. Our framework simultaneously captures such well-known notions as internal and general Φ-regret, learning with non-additive global cost functions, Blackwell's approachability, calibration of forecasters, and more. We show that learnability in all these situations is due to control of the same three quantities: a martingale convergence term, a term describing the ability to perform well if future is known, and a generalization of sequential Rademacher complexity, studied in Rakhlin et al. (2010a). Since we directly study complexity of the problem instead of focusing on efficient algorithms, we are able to improve and extend many known results which have been previously derived via an algorithmic construction. Disciplines Computer Sciences | Statistics and Probability This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/statistics_papers/133  Workshop and Conference Proceedings 19 (2011) 559–594 24th Annual Conference on Learning Theory Online Learning: Beyond Regret Alexander Rakhlin Department of Statistics University of Pennsylvania Karthik Sridharan TTI-Chicago Ambuj Tewari Department of Computer Science University of Texas at  Sham Kakade, Ulrike von Luxburg  We study online learnability of a wide class of problems, extending the results of Rakhlin et al. (2010a) to general notions of performance measure well beyond external regret. Our framework simultaneously captures such well-known notions as internal and general Φregret, learning with non-additive global cost functions, Blackwell’s approachability, calibration of forecasters, and more. We show that learnability in all these situations is due to control of the same three quantities: a martingale convergence term, a term describing the ability to perform well if future is known, and a generalization of sequential Rademacher complexity, studied in Rakhlin et al. (2010a). Since we directly study complexity of the problem instead of focusing on efficient algorithms, we are able to improve and extend many known results which have been previously derived via an algorithmic construction.We study online learnability of a wide class of problems, extending the results of Rakhlin et al. (2010a) to general notions of performance measure well beyond external regret. Our framework simultaneously captures such well-known notions as internal and general Φregret, learning with non-additive global cost functions, Blackwell’s approachability, calibration of forecasters, and more. We show that learnability in all these situations is due to control of the same three quantities: a martingale convergence term, a term describing the ability to perform well if future is known, and a generalization of sequential Rademacher complexity, studied in Rakhlin et al. (2010a). Since we directly study complexity of the problem instead of focusing on efficient algorithms, we are able to improve and extend many known results which have been previously derived via an algorithmic construction.