Blackwell Approachability and No-Regret Learning are Equivalent

Jacob Abernethy, Peter L. Bartlett, Elad Hazan
Proceedings of the 24th Annual Conference on Learning Theory, PMLR 19:27-46, 2011.

Abstract

We consider the celebrated Blackwell Approachability Theorem for two-player games with vector payoffs. Blackwell himself previously showed that the theorem implies the existence of a “no-regret” algorithm for a simple online learning problem. We show that this relationship is in fact much stronger, that Blackwell’s result is equivalent to, in a very strong sense, the problem of regret minimization for Online Linear Optimization. We show that any algorithm for one such problem can be efficiently converted into an algorithm for the other. We provide one novel application of this reduction: the first efficient algorithm for calibrated forecasting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v19-abernethy11b, title = {Blackwell Approachability and No-Regret Learning are Equivalent}, author = {Abernethy, Jacob and Bartlett, Peter L. and Hazan, Elad}, booktitle = {Proceedings of the 24th Annual Conference on Learning Theory}, pages = {27--46}, year = {2011}, editor = {Kakade, Sham M. and von Luxburg, Ulrike}, volume = {19}, series = {Proceedings of Machine Learning Research}, address = {Budapest, Hungary}, month = {09--11 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v19/abernethy11b/abernethy11b.pdf}, url = {https://proceedings.mlr.press/v19/abernethy11b.html}, abstract = {We consider the celebrated Blackwell Approachability Theorem for two-player games with vector payoffs. Blackwell himself previously showed that the theorem implies the existence of a “no-regret” algorithm for a simple online learning problem. We show that this relationship is in fact much stronger, that Blackwell’s result is equivalent to, in a very strong sense, the problem of regret minimization for Online Linear Optimization. We show that any algorithm for one such problem can be efficiently converted into an algorithm for the other. We provide one novel application of this reduction: the first efficient algorithm for calibrated forecasting.} }
Endnote
%0 Conference Paper %T Blackwell Approachability and No-Regret Learning are Equivalent %A Jacob Abernethy %A Peter L. Bartlett %A Elad Hazan %B Proceedings of the 24th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2011 %E Sham M. Kakade %E Ulrike von Luxburg %F pmlr-v19-abernethy11b %I PMLR %P 27--46 %U https://proceedings.mlr.press/v19/abernethy11b.html %V 19 %X We consider the celebrated Blackwell Approachability Theorem for two-player games with vector payoffs. Blackwell himself previously showed that the theorem implies the existence of a “no-regret” algorithm for a simple online learning problem. We show that this relationship is in fact much stronger, that Blackwell’s result is equivalent to, in a very strong sense, the problem of regret minimization for Online Linear Optimization. We show that any algorithm for one such problem can be efficiently converted into an algorithm for the other. We provide one novel application of this reduction: the first efficient algorithm for calibrated forecasting.
RIS
TY - CPAPER TI - Blackwell Approachability and No-Regret Learning are Equivalent AU - Jacob Abernethy AU - Peter L. Bartlett AU - Elad Hazan BT - Proceedings of the 24th Annual Conference on Learning Theory DA - 2011/12/21 ED - Sham M. Kakade ED - Ulrike von Luxburg ID - pmlr-v19-abernethy11b PB - PMLR DP - Proceedings of Machine Learning Research VL - 19 SP - 27 EP - 46 L1 - http://proceedings.mlr.press/v19/abernethy11b/abernethy11b.pdf UR - https://proceedings.mlr.press/v19/abernethy11b.html AB - We consider the celebrated Blackwell Approachability Theorem for two-player games with vector payoffs. Blackwell himself previously showed that the theorem implies the existence of a “no-regret” algorithm for a simple online learning problem. We show that this relationship is in fact much stronger, that Blackwell’s result is equivalent to, in a very strong sense, the problem of regret minimization for Online Linear Optimization. We show that any algorithm for one such problem can be efficiently converted into an algorithm for the other. We provide one novel application of this reduction: the first efficient algorithm for calibrated forecasting. ER -
APA
Abernethy, J., Bartlett, P.L. & Hazan, E.. (2011). Blackwell Approachability and No-Regret Learning are Equivalent. Proceedings of the 24th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 19:27-46 Available from https://proceedings.mlr.press/v19/abernethy11b.html.

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