Runtime Distributions and Criteria for Restarts

TL;DR

This paper analyzes how the properties of runtime distributions influence the effectiveness of restart strategies in randomized algorithms, providing criteria for when restarts are beneficial and conditions for optimal restart times.

Contribution

It introduces distribution properties that determine restart usefulness and analyzes key distributions like log-normal, Weibull, and Pareto for these criteria.

Findings

Scale parameters do not affect restart usefulness.

Location parameters have limited influence on restart strategies.

Conditions for optimal restart times are provided for key distributions.

Abstract

Randomized algorithms sometimes employ a restart strategy. After a certain number of steps, the current computation is aborted and restarted with a new, independent random seed. In some cases, this results in an improved overall expected runtime. This work introduces properties of the underlying runtime distribution which determine whether restarts are advantageous. The most commonly used probability distributions admit the use of a scale and a location parameter. Location parameters shift the density function to the right, while scale parameters affect the spread of the distribution. It is shown that for all distributions scale parameters do not influence the usefulness of restarts and that location parameters only have a limited influence. This result simplifies the analysis of the usefulness of restarts. The most important runtime probability distributions are the log-normal, the Weibull, and the Pareto distribution. In this work, these distributions are analyzed for the usefulness of restarts. Secondly, a condition for the optimal restart time (if it exists) is provided. The log-normal, the Weibull, and the generalized Pareto distribution are analyzed in this respect. Moreover, it is shown that the optimal restart time is also not influenced by scale parameters and that the influence of location parameters is only linear.