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A Stochastic View of Optimal Regret through Minimax Duality

Abernethy, Jacob, Agarwal, Alekh, & Bartlett, Peter L. (2009) A Stochastic View of Optimal Regret through Minimax Duality. In Dasgupta , S & Klivans, A (Eds.) Proceedings of the 22nd Annual Conference on Learning Theory, Montreal, Quebec.

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Abstract

We study the regret of optimal strategies for online convex optimization games. Using von Neumann's minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary's action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen's inequality for a concave functional--the minimizer over the player's actions of expected loss--defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary.

Peter L. Bartlett, Alexander Rakhlin

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ID Code: 42511
Item Type: Conference Paper
Additional URLs:
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > STATISTICS (010400)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Past > Schools > Mathematical Sciences
Copyright Owner: Contact author
Deposited On: 12 Aug 2011 13:49
Last Modified: 01 Mar 2012 13:42

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