Constant regret for sequence prediction with limited advice

We investigate the problem of cumulative regret minimization for individual sequence prediction with respect to the best expert in a finite family of size K under limited access to information. We assume that in each round, the learner can predict using a convex combination of at most p experts for prediction, then they can observe a posteriori the losses of at most m experts. We assume that the loss function is range-bounded and exp-concave. In the standard multi-armed bandits setting, when the learner is allowed to play only one expert per round and observe only its feedback, known optimal regret bounds are of the order O($\sqrt$ KT). We show that allowing the learner to play one additional expert per round and observe one additional feedback improves substantially the guarantees on regret. We provide a strategy combining only p = 2 experts per round for prediction and observing m $\ge$ 2 experts' losses. Its randomized regret (wrt. internal randomization of the learners' strategy) is of order O (K/m) log(K$\delta$ --1) with probability 1 -- $\delta$, i.e., is independent of the horizon T ("constant" or "fast rate" regret) if (p $\ge$ 2 and m $\ge$ 3). We prove that this rate is optimal up to a logarithmic factor in K. In the case p = m = 2, we provide an upper bound of order O(K 2 log(K$\delta$ --1)), with probability 1 -- $\delta$. Our strategies do not require any prior knowledge of the horizon T nor of the confidence parameter $\delta$. Finally, we show that if the learner is constrained to observe only one expert feedback per round, the worst-case regret is the "slow rate" $\Omega$($\sqrt$ KT), suggesting that synchronous observation of at least two experts per round is necessary to have a constant regret.