Random-Walk Perturbations for Online Combinatorial Optimization

We study online combinatorial optimization problems that a learner is interested in minimizing its cumulative regret in the presence of switching costs. To solve such problems, we propose a version of the follow-the-perturbed-leader algorithm in which the cumulative losses are perturbed by independent symmetric random walks. In the general setting, our forecaster is shown to enjoy near-optimal guarantees on both quantities of interest, making it the best known efficient algorithm for the studied problem. In the special case of prediction with expert advice, we show that the forecaster achieves an expected regret of the optimal order O(n log N)
^1/2
), where n is the time horizon and N is the number of experts, while guaranteeing that the predictions are switched at most O(n log N)
^1/2
) times, in expectation.