4.7.5 Random Restart

It may seem like a randomized algorithm that only succeeds, say, 20% of the time is not very useful if you need an algorithm to succeed, say, 99% of the time. However, a randomized algorithm that succeeds some of the time can be extended to an algorithm that succeeds more often by running it multiple times, using a random restart, and reporting any solution found.

Whereas a random walk, and its variants, are evaluated empirically by running experiments, the performance of random restart can also be predicted ahead of time, because the runs following a random restart are independent of each other.

An algorithm that succeed with probability $p$, that is run $n$ times or until a solution is found, will find a solution with probability $1-(1-p)^{n}$. It fails to find a solution if it fails for each retry, and each retry is independent of the others.

For example, an algorithm with $p=0.5$ tried 5 times, will find a solution around 96.9% of the time; tried 10 times it will find a solution 99.9% of the time. If each run succeeded with a probability $p=0.1$, running it 10 times will succeed 65% of the time, and running it 44 times will give a 99% success rate.

A run-time distribution allows us to predict how the algorithm will work with random restart after a certain number of steps. Intuitively, a random restart will repeat the lower left corner of the run-time distribution, suitably scaled down, at the stage where the restart occurs. A random restart after a certain number of greedy descent steps will make any algorithm that sometimes finds a solution into an algorithm that always finds a solution, given that one exists, if it is run for long enough.

A random restart can be expensive if there are many variables. A partial restart randomly assigns just some not all of the variables, say 100 variables, or 10% of the variables, to move to another part of the search space. While this is often effective, the above theoretical analysis does not work because the partial restarts are not independent of each other.