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# Adaptivity and Optimism: An Improved Exponentiated Gradient Algorithm

*Proceedings of the 31st International Conference on Machine Learning*, PMLR 32(2):1593-1601, 2014.

#### Abstract

We present an adaptive variant of the exponentiated gradient algorithm. Leveraging the optimistic learning framework of Rakhlin & Sridharan (2012), we obtain regret bounds that in the learning from experts setting depend on the variance and path length of the best expert, improving on results by Hazan & Kale (2008) and Chiang et al. (2012), and resolving an open problem posed by Kale (2012). Our techniques naturally extend to matrix-valued loss functions, where we present an adaptive matrix exponentiated gradient algorithm. To obtain the optimal regret bound in the matrix case, we generalize the Follow-the-Regularized-Leader algorithm to vector-valued payoffs, which may be of independent interest.