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For each of $T$ time steps, $m$ experts report probability distributions over$n$ outcomes; we wish to learn to aggregate these forecasts in a way thatattains a no-regret guarantee. We focus on the fundamental and practicalaggregation method known as logarithmic pooling -- a weighted average of logodds -- which is in a certain sense the optimal choice of pooling method if oneis interested in minimizing log loss (as we take to be our loss function). Weconsider the problem of learning the best set of parameters (i.e. expertweights) in an online adversarial setting. We assume (by necessity) that theadversarial choices of outcomes and forecasts are consistent, in the sense thatexperts report calibrated forecasts. Imposing this constraint creates a (to ourknowledge) novel semi-adversarial setting in which the adversary retains alarge amount of flexibility. In this setting, we present an algorithm based ononline mirror descent that learns expert weights in a way that attains$O(\sqrt{T} \log T)$ expected regret as compared with the best weights inhindsight.

No-Regret Learning with Unbounded Losses: The Case of Logarithmic Pooling