Restart Strategies in a Continuous Setting

Abstract

Restarting is a technique frequently employed in randomized algorithms. After some number of computation steps, the state of the algorithm is reinitialized with a new, independent random seed. Luby et al. (Inf. Process. Lett. 47(4), 173–180, 1993) introduced a universal restart strategy. They showed that their strategy is an optimal universal strategy in the worst case. However, the optimality result has only been shown for discrete processes. In this work, it is shown that their result does not translate into a continuous setting. Furthermore, we show that there are no (asymptotically) optimal strategies in a continuous setting. Nevertheless, we obtain an optimal universal strategy on a restricted class of continuous probability distributions. Furthermore, as a side result, we show that the expected value under restarts for the lognormal distribution tends towards 0. Finally, the results are illustrated using simulations.