26769 articles – 20450 references  [version française]
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COLT - 24th Conference on Learning Theory - 2011, Budapest : Hongrie (2011)
Minimax Policies for Combinatorial Prediction Games
Jean-Yves Audibert1, 2, Sébastien Bubeck3, Gábor Lugosi4, 5

We address the online linear optimization problem when the actions of the forecaster are represented by binary vectors. Our goal is to understand the magnitude of the minimax regret for the worst possible set of actions. We study the problem under three different assumptions for the feedback: full information, and the partial information models of the so-called "semi-bandit", and "bandit" problems. We consider both L -, and L2-type of restrictions for the losses assigned by the adversary. We formulate a general strategy using Bregman projections on top of a potential-based gradient descent, which generalizes the ones studied in the series of papers Gyorgy et al. (2007), Dani et al. (2008), Abernethy et al. (2008), Cesa-Bianchi and Lugosi (2009), Helmbold and Warmuth (2009), Koolen et al. (2010), Uchiya et al. (2010), Kale et al. (2010) and Audibert and Bubeck (2010). We provide simple proofs that recover most of the previous results. We propose new upper bounds for the semi-bandit game. Moreover we derive lower bounds for all three feedback assumptions. With the only exception of the bandit game, the upper and lower bounds are tight, up to a constant factor. Finally, we answer a question asked by Koolen et al. (2010) by showing that the exponentially weighted average forecaster is suboptimal against L adversaries.
2:  INRIA Paris - Rocquencourt - SIERRA
3:  CRM - Centre de Recerca Matemàtica
4:  ICREA - Institució Catalana de Recerca i Estudis Avançats [Barcelona]
5:  Universitat Pompeu Fabra