Efficient tracking of a growing number of experts

We consider a variation on the problem of prediction with expert advice, where new forecasters that were unknown until then may appear at each round. As often in prediction with expert advice, designing an algorithm that achieves near-optimal regret guarantees is straightforward, using aggregation of experts. However, when the comparison class is sufficiently rich, for instance when the best expert and the set of experts itself changes over time, such strategies naively require to maintain a prohibitive number of weights (typically exponential with the time horizon). By contrast, designing strategies that both achieve a near-optimal regret and maintain a reasonable number of weights is highly non-trivial. We consider three increasingly challenging objectives (simple regret, shifting regret and sparse shifting regret) that extend existing notions defined for a fixed expert ensemble; in each case, we design strategies that achieve tight regret bounds, adaptive to the parameters of the comparison class, while being computationally inexpensive. Moreover, our algorithms are anytime, agnostic to the number of incoming experts and completely parameter-free. Such remarkable results are made possible thanks to two simple but highly effective recipes: first the "abstention trick" that comes from the specialist framework and enables to handle the least challenging notions of regret, but is limited when addressing more sophisticated objectives. Second, the "muting trick" that we introduce to give more flexibility. We show how to combine these two tricks in order to handle the most challenging class of comparison strategies.