# Online Learning: Beyond Regret

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Alexander Rakhlin,
Karthik Sridharan,
Ambuj Tewari
;

Proceedings of the 24th Annual Conference on Learning Theory, PMLR 19:559-594, 2011.

#### Abstract

We study online learnability of a wide class of problems, extending the results of \citeRakSriTew10 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 \citeRakSriTew10. 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.

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