We propose a version of the follow-the-perturbed-leader online prediction algorithm in which the cumulative losses are perturbed by independent symmetric random walks. The forecaster is shown to achieve an expected regret of the optimal order O( √ n log N ) where n is the time horizon and N is the number of experts. More importantly, it is shown that the forecaster changes its prediction at most O( √ n log N ) times, in expectation. We also extend the analysis to online combinatorial optimization and show that even in this more general setting, the forecaster rarely switches between experts while having a regret of near-optimal order. Online learning, Online combinatorial optimization, Follow the Perturbed Leader, Random walk I. PRELIMINARIES In this paper we study the problem of online prediction with expert advice, see [1] . The problem may be described as a repeated game between a forecaster and an adversary-the environment. At each time instant t = 1, . . . , n, the forecaster chooses one of the N available actions (often called experts) and suffers a loss ℓ i,t ∈ [0, 1] corresponding to the chosen action i. We consider the so-called oblivious adversary model in which the environment selects all losses before the prediction game starts and reveals the losses ℓ i,t at time t after the forecaster has made its prediction. The losses are deterministic but the forecaster may randomize: at time t, the forecaster chooses a probability distribution p t over the set of N actions and draws a random action I t according to the distribution p t . The prediction protocol is described in Figure 1 . The usual goal for the standard prediction problem is to devise an algorithm such that the cumulative loss L n = n t=1 ℓ It,t is as small as possible, in expectation and/or with high probability (where probability is with respect to the forecaster's randomization). Since we do not make any assumption on how the environment generates the losses ℓ i,t , we cannot
Devroye et al. REF proposed a PFE algorithm whose expected regret and expected number of switches are both O( √ T log n), where n is the size of the action set.
Prediction by Random-Walk Perturbation
ArXiv
ArXiv
Mathematics Computer Science