@report {1616777,
	title = {Towards minimax policies for online linear optimization with bandit feedback},
	year = {2012},
	institution = {ArXiv Report},
	address = {Conference on Learning Theory (COLT)},
	abstract = {We address the online linear optimization problem with bandit feedback. Our contribution is twofold. First, we provide an algorithm (based on exponential weights) with a regret of order dnlogN--------√ for any finite action set with N actions, under the assumption that the instantaneous loss is bounded by 1. This shaves off an extraneous d--√ factor compared to previous works, and gives a regret bound of order dnlogn------√ for any compact set of actions. Without further assumptions on the action set, this last bound is minimax optimal up to a logarithmic factor. Interestingly, our result also shows that the minimax regret for bandit linear optimization with expert advice in d dimension is the same as for the basic d-armed bandit with expert advice. Our second contribution is to show how to use the Mirror Descent algorithm to obtain computationally efficient strategies with minimax optimal regret bounds in specific examples. More precisely we study two canonical action sets: the hypercube and the Euclidean ball. In the former case, we obtain the first computationally efficient algorithm with a dn-√ regret, thus improving by a factor dlogn------√ over the best known result for a computationally efficient algorithm. In the latter case, our approach gives the first algorithm with a dnlogn-------√ regret, again shaving off an extraneous d--√ compared to previous works.},
	url = {http://arxiv.org/abs/1202.3079},
	author = {Sebastien Bubeck and Nicolo Cesa-Bianchi and Sham M. Kakade}
}
