Non-exponentially weighted aggregation: regret bounds for unbounded loss functions

We tackle the problem of online optimization with a general, possibly unbounded, loss function. It is well known that when the loss is bounded, the exponentially weighted aggregation strategy (EWA) leads to a regret in $\sqrt{T}$ after $T$ steps. In this paper, we study a generalized aggregation strategy, where the weights do no longer depend exponentially on the losses. Our strategy is defined as the minimizer of the expected losses plus a regularizer. When this regularizer is the Kullback-Leibler divergence, we obtain EWA as a special case, but using alternative divergences lead to regret bounds for unbounded losses, at the cost of a worst regret bound in some cases.