Fast rates for prediction with limited expert advice

We investigate the problem of minimizing the excess generalization error with respect to the best expert prediction in a finite family in the stochastic setting, under limited access to information. We consider that the learner has only access to a limited number of expert advices per training round, as well as for prediction. Assuming that the loss function is Lipschitz and strongly convex, we show that if we are allowed to see the advice of only one expert per round in the training phase, or to use the advice of only one expert for prediction in the test phase, the worst-case excess risk is ${\Omega}(1/\sqrt{T})$ with probability lower bounded by a constant. However, if we are allowed to see at least two actively chosen expert advices per training round and use at least two experts for prediction, the fast rate $\mathcal{O}(1/T)$ can be achieved. We design novel algorithms achieving this rate in this setting, and in the setting where the learner have a budget constraint on the total number of observed experts advices, and give precise instance-dependent bounds on the number of training rounds needed to achieve a given generalization error precision.